Asymptotic Methods for Solitary Solutions and Compactons

Abstract

This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform, and ancient Chinese mathematics. Hamilton principle and variational principles are also emphasized. The reviewed asymptotic methods are easy to be followed for various applications. Some ideas on this paper are first appeared.

1. Introduction

Soliton was first discovered in 1834 by Russell [1], who observed that a canal boat stopping suddenly gave rise to a solitary wave which traveled down the canal for several miles, without breaking up or losing strength. Russell named this phenomenon the “soliton.”

In a highly informative as well as entertaining article [1], Russell gave an engaging historical account of the important scientific observation:

I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.

His ideas did not earn attention until 1965 when Zabusky and Kruskal [2] began to use a finite difference approach to the study of KdV equation, which was obtained by Korteweg and de Vires [3]. Various analytical methods also led to a complete understanding of solitons, especially the inverse scattering transform proposed by Gardner et al. [4] in 1967. The significance of Russell’s discovery was then fully appreciated. It was discovered that many phenomena in physics, electronics, and biology can be described by a mathematical and physical theory of “soliton.” For a historical account of the scientific development of solitons, the reader is referred to the “Encyclopedia of Complexity and Systems Science,” especially [5, 6]. Some analytical methods leading to our present state of the art are available in several review articles [7–9].

2. Basic Properties of Solitary Solutions and Compactons [5, 6]

A soliton is a special traveling wave that after a collision with another soliton eventually emerges unscathed. Solitons are solutions of partial differential equations that model phenomena like water waves or waves along a weakly anharmonic mass-spring chain. A soliton is a bell-like solution as illustrated in Figure 1.

Figure 1 

Bell-like solitary wave.

The soliton can be written in a standard form, which is 𝑢(𝜉)=𝑝sech²𝑞𝜉=4𝑝𝑒^2𝑞𝜉+𝑒^−2𝑞𝜉,+2 (2.1) where 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑥−𝑐𝑡, and 𝑐 is the wave velocity.

It is obvious that lim_𝜉→∞𝑢(𝜉)=0,lim_{𝜉→−∞}𝑢(𝜉)=0. (2.2) The soliton has exponential tails, which are the basic character of solitary waves. This property allows the exponential function to describe its solution, see Section 5.9 for detailed discussion.

The soliton obeys a superposition-like principle: solitons passing through one another emerge unmodified, see Figure 2.

(a)

(b)

(c)

(d)

(e)

Figure 2 

Collision of two solitary waves.

A compacton is a special solitary traveling wave that, unlike a soliton, does not have exponential tails. A compacton-like solution is a special wave solution which can be expressed by the squares of sinusoidal or cosinoidal functions.

3. Explanation of Compacton Solutions [6]

Compactons are special solitons with finite wavelength. It was Rosenau and Hyman [10] who first found compactons in 1993.

3.1. Compacton: An Oscillatory Wave with No Tails

Now consider a modified version of KdV equation in the form 𝑢_𝑡+𝑢²_𝑥+𝑢²_𝑥𝑥𝑥=0. (3.1) Introducing a complex variable 𝜉 defined as 𝜉=𝑥−𝑐𝑡, where 𝑐 is the velocity of traveling wave, integrating once, and we have −𝑐𝑢+𝑢²+𝑢²_𝜉𝜉=𝐷, (3.2) where 𝐷 is an integral constant, for solitary solution, and we set 𝐷=0.

We rewrite (3.2) in the form 𝜈_𝜉𝜉+𝜈−𝑐𝜈^1/2=0, (3.3) where 𝑢²=𝜈.

In case 𝑐=0, we have periodic solution: 𝜈(𝜉)=𝐴cos𝜉+𝐵sin𝜉. Periodic solution of nonlinear oscillators can be approximated by sinusoidal function. It helps understanding if an equation can be classified as oscillatory by direct inspection of its terms.

We consider two common-order differential equations whose exact solutions are important for physical understanding: 𝑢^′′−𝑘²𝑢𝑢=0,_′′+𝜔²𝑢=0. (3.4)(3.5) Both equations have linear terms with constant coefficients.

The crucial difference between these two very simple equations is the sign of the coefficient of 𝑢 in the second term. This determines whether the solutions are exponential or oscillatory. The general solution of (3.4) is 𝑢=𝐴𝑒^𝑘𝑡+𝐵𝑒^−𝑘𝑡. (3.6) The second equation, (3.5), has a positive coefficient of 𝑢, and in this case, the general solution reads 𝑢=𝐴cos𝜔𝑡+𝐵sin𝜔𝑡. (3.7) This solution describes an oscillation at the angular velocity 𝜔.

Equation (3.3) behaves sometimes like an oscillator when 1−𝑐𝜈^−1/2>0, that is, 𝑢=𝜈^1/2 has a periodic solution, and we assume that 𝜈 can be expressed in the form 𝜈=𝑢²=𝐴²cos⁴𝜔𝜉. (3.8) Substituting (3.8) into (3.3) results in 12𝐴²𝜔²cos²𝜔𝜉−16𝐴²𝜔²cos⁴𝜔𝜉−𝐴²cos⁴𝜔𝜉−𝑐𝐴cos²𝜔𝜉=0. (3.9) We, therefore, have 12𝐴²𝜔²−𝑐𝐴=0,−16𝐴²𝜔²−𝐴²=0. (3.10) Solving the above system, (3.10), yields 1𝜔=44,𝐴=3𝑐. (3.11) We obtain the solution in the form 𝑢=𝜈^1/2=4𝑐3cos²14(𝑥−𝑐𝑡). (3.12) By a careful inspection, 𝜈 can tend to a very small value or even zero, and as a result, 1−𝑐𝜈^−1/2 tends to negative infinite, and (3.3) behaves like (3.4) with 𝑘→∞; the exponential tails vanish completely at the edge of the bell shape (see Figure 3): 𝑢=4𝑐31cos4(𝑥−𝑐𝑡),|𝑥−𝑐𝑡|≤2𝜋,0,otherwise. (3.13) This is a compact wave. Unlike solitons, compacton does not have exponential tails (Figure 3).

Figure 3 

Compacton wave without tails.

3.2. A Criterion for Oscillatory Thermopower Waves

Thermal conduction in fuel/Bi₂Te₃/Al₂O₃ or fuel/Bi₂Te₃/terracotta systems always results in strong oscillation of the output signals. A criterion for oscillatory thermopower waves is much needed.

Recently, Walia et al. proposed a theory of thermopower wave oscillations to describe coupled thermal waves in fuel/Bi₂Te₃/Al₂O₃ or fuel/Bi₂Te₃/terracotta systems [11]. The dimensionless governing equations are as follows [11]: 𝜕𝑢=𝜕𝜕𝑡²𝑢𝜕𝑥²+𝑤𝑒^−1/𝑢−ℓ𝑢−𝑢_𝑎,𝜕𝑤𝜕𝑡=−𝛽𝑤𝑒^−1/𝑢, (3.14)(3.15) where 𝑢 is nondimensional temperature, 𝑤 is the concentration of the fuel, 𝛽 and ℓ are, respectively, parameters related to the properties of the fuel and volumetric heat transfer, and 𝑢_𝑎 is the ambient temperature.

Ignoring the nonlinear term in (3.14), we would have a wave solution. Changing the parameters in the system will result in strong oscillation [11], and an analytical criterion to predict oscillatory thermopower waves is very useful for design of Bi₂Te₃ films.

The system, (3.14) and (3.15), is difficult to solve analytically because of strong nonlinearity. In order to obtain a criterion for oscillatory thermopower waves, some necessary approximations are needed. Equation (3.15) is approximately written in the form 𝜕𝑤1𝜕𝑡=−𝛽𝑤1−𝑢. (3.16)𝑢 in (3.16) is assumed to be a known function; solving 𝑤 in (3.16) results in 1𝑤=exp−𝛽1−𝑢. (3.17) The nonlinear term, 𝑤𝑒^−1/𝑢, in (3.14) is expressed in an approximate form 𝑤𝑒^−1/𝑢1=exp−𝛽1−𝑢𝑒^−1/𝑢=11−𝛽1−𝑢11−𝑢=1−𝛽+2𝛽−1𝑢−𝛽𝑢². (3.18) Equation (3.14) is rewritten in the following equivalent form: 𝜕𝑢=𝜕𝜕𝑡²𝑢𝜕𝑥²+1−𝛽+2𝛽−1𝑢−ℓ𝑢−𝑢_𝑎+𝑓(𝑢,𝑤), (3.19) where 𝑓(𝑢,𝑤) is defined as 𝑓(𝑢,𝑤)=𝑤𝑒^−1/𝑢1−𝛽+2𝛽−1𝑢. (3.20) In order to solve thermopower waves, we make a transform 𝜉=𝑥−𝑐𝑡, (3.21) where 𝑐 is wave speed.

By the transform, (3.21), we convert (3.19) into an ordinary differential equation, which is 𝑢^′′+𝑐𝑢^′+2𝛽−1𝑢−ℓ𝑢+ℓ𝑢_𝑎+1−𝛽+𝑓(𝑢,𝑤)=0. (3.22) We rewrite (3.22) in the form 𝑢^′′+𝑐𝑢^′+2𝛽−1𝑢−ℓ𝑢=𝐹(𝑢,𝑤), (3.23) where 𝐹(𝑢,𝑤)=−ℓ𝑢_𝑎+1−𝛽−𝑓(𝑢,𝑤). (3.24)

Note. Equation (3.23) is exactly equivalent to (3.14). In order to solve (3.23) approximately, we write an iteration formulation 𝑢^′′_𝑛+1+𝑐𝑢^′_𝑛+1+2𝛽−1𝑢_𝑛+1−ℓ𝑢_𝑛+1=𝐹𝑢_𝑛(𝜉),𝑤_𝑛(𝜉). (3.25)
In (3.25), 𝐹 can be considered as a known function of 𝜉. Equation (3.25) is, therefore, similar to a forced nonlinear oscillator.
We search for a periodic solution of (3.25). To this end, we assume that its solution can be expressed in the form 𝑢_𝑛+1(𝜉)=𝐴cos𝜔𝜉+𝜃₀. (3.26) By an analytical method [7], we can obtain the following approximate frequency: 𝜔=22𝛽−1𝐴²−ℓ. (3.27) The assumption, (3.26), follows 𝜔>0, that is, 22𝛽−1𝐴²−ℓ>0. (3.28)

This is a criterion for oscillatory thermopower waves. When 2(2𝛽−1)/𝐴²−ℓ≤0, we can predict thermopower waves without oscillation.

3.3. A Criterion for Gaseous Emission Waves

Lin and Hildemann [12] developed a general mathematical model to predict emissions of volatile organic compounds (VOCs) from hazardous or sanitary landfills. The model includes important mechanisms occurring in unsaturated subsurface landfill environments: biogas flow, leachate flow, diffusion, adsorption, degradation, volatilization, and mass transfer limitations through the top cover. Lin-Hildemann equation for gaseous emission can be expressed as follows [12]: 𝜕𝑢𝜕𝑡+𝑉𝜕𝑢𝜕𝜕𝑧=𝐷²𝑢𝜕𝑧²−𝜇𝑢, (3.29) with the following boundary/initial conditions: 𝑉𝑢−𝐷𝜕𝑢𝑢()𝑢(),𝜕𝑧=−𝐻𝑢at𝑧=0,∞,𝑡=0,𝑧,0=𝐴exp𝛽𝑧 (3.30)(3.31)(3.32) where 𝑢 is the total chemical concentration per unit volume of soil, 𝑉 is the effective emission speed, and 𝐷 is effective diffusion coefficient. The definitions of 𝑉 and 𝐷 are given in [12].

This paper aims at a wave solution of (3.29). By the wave transform, 𝜉=𝑧−𝑐𝑡. (3.33)

Equation (3.29) is converted into an ordinary differential equation, which is 𝐷𝑢^′′+(𝑐+𝑉)𝑢^′−𝜇𝑢=0, (3.34) where 𝑐 is the wave speed.

Considering the boundary condition, (3.31), the solution of (3.34) is 𝑢=𝐴exp(𝑎𝜉)=𝐴exp{𝑎(𝑧−𝑐𝑡)}, (3.35) where 𝑎=𝑉−𝑐−(𝑉−𝑐)²+4𝜇𝐷.2𝐷 (3.36) By the boundary condition, (3.30), we have 𝑉−𝐷𝑎=−𝐻. (3.37) Solving 𝑎 and 𝑐 from (3.36) and (3.37) results in 𝑎=𝑉+𝐻𝐷,𝑐=𝐻+𝜇𝐷.𝑉+𝐻 (3.38)(3.39) Considering the initial condition, we have 𝛽=𝑉+𝐻𝐷<0. (3.40) Equation (3.40) can be written in the form 𝛽=−𝜈^𝐵_𝐺/𝑅_𝐺+𝜈^𝐵_𝐿/𝑅_𝐿+𝐾_𝑇/𝑅_𝐺𝐷^𝐸_𝐺/𝑅_𝐺+𝐷^𝐸_𝐿/𝑅_𝐿<0, (3.41) where 𝜈^𝐵_𝐺, 𝜈^𝐵_𝐿 are the bulk (apparent) gas and water velocities, respectively. 𝐷^𝐸_𝐺, 𝐷^𝐸_𝐿 are the effective gaseous and aqueous diffusion coefficients in soil, respectively; 𝑅_𝐺, 𝑅_𝐿 are phase-partitioning coefficients of gas and liquid, respectively.

Equation (3.41) is the criterion for gaseous emission waves.

For a wave solution, the initial condition should be expressed in the form of (3.32). If the initial condition cannot be expressed in an exponential function, the criterion for gaseous emission waves becomes invalid. The present criterion can easily be extended to various nonlinear cases.

4. Exact Solutions versus Asymptotic Solutions

There is plainly a tendency in the modern nonlinear science community to obtain exact solutions for nonlinear equations. There are many results on the exact solutions of nonlinear equations where the initial or boundary conditions are not considered. These solutions are called mathematical solutions because the physical constraints on the real-world problem that is being modeled are not accounted for. Our main aim, however, is to find solutions of the underlying problem that satisfy all the initial/boundary conditions that exist. These solutions, naturally, are called the physical solutions of the problem. Consider, for example, the well-known KdV equation 𝜕𝑢𝜕𝑡−6𝑢𝜕𝑢+𝜕𝜕𝑥³𝑢𝜕𝑥³=0, (4.1) with a solitary solution 𝑐𝑢=−2sech²𝑐4𝑥−𝑐𝑡−𝜉₀. (4.2) Many mathematical solutions for (4.1) could be found that carry no physical meaning (𝑢=1, for instance, is an exact solution of (4.1) that has no physical meaning at all). Other researchers, on the other hand, begin with some very good initial conditions, say 𝑢()𝑐𝑥,0=−2sech²𝑐4𝑥−𝜉₀, (4.3) and find that the condition is in fact too good to solve the equation. For a travelling solution, for example, we might guess a solution of the form 𝑢()𝑐𝑥,𝑡=−2sech²𝑐4𝑥+𝑎𝑡−𝜉₀, (4.4) where the unknown constant 𝑎 can be determined by substituting (4.4) into (4.1).

An asymptotic approach is, however, to search for an asymptotic solution with physical understanding. If, for example, we feel interest in a solitary solution of (4.1), then we can assume that the solution has the form 𝑢()=1𝑥,𝑡{𝑘({(,𝑎exp𝑥−𝑐𝑡)}+𝑏exp−𝑘𝑥−𝑐𝑡)} (4.5) or 𝑢()=1𝑥,𝑡{𝑘({(,𝑎exp𝑥−𝑐𝑡)}+𝑏exp−𝑘𝑥−𝑐𝑡)}+𝑑 (4.6) where 𝑎, 𝑏, 𝑐, 𝑑, and 𝑘 are unknown constants which can be determined via various methods.

For 𝑁-solitary solutions, we can assume that the solution has the following form: 𝑢(𝑥,𝑡)=^𝑁_𝑖=11𝑎_𝑖exp𝑘_𝑖𝑥−𝑐_𝑖𝑡+𝑏_𝑖exp−𝑘_𝑖𝑥−𝑐_𝑖𝑡+𝑑_𝑖, (4.7) or 𝑢()=∑𝑥,𝑡^𝑛𝑎_𝑛exp𝑞_𝑛𝑥+𝑝_𝑛𝑡∑_𝑚𝑏_𝑚exp𝛼_𝑚𝑥+𝛽_𝑚𝑡, (4.8) where 𝑎_𝑖,𝑏_𝑖,𝑐_𝑖,𝑑_𝑖,and𝑘_𝑖 and 𝛼_𝑖,𝛽_𝑖,𝑞_𝑖,and𝑝_𝑖 are unknown constants to be further determined.

For a two-wave solution, we can assume that 𝑎𝑢=⁻¹exp(−𝜉)+𝑏₁exp(𝜉)+𝑎₀+𝑏₋₁exp−𝜂+𝑏₁exp𝜂𝑐₋₁exp(−𝜉)+𝑐₁exp(𝜉)+𝑏₀+𝑑₋₁exp−𝜂+𝑑₁exp𝜂, (4.9) or 1𝑢=𝑐₋₁exp(−𝜉)+𝑐₁exp(𝜉)+𝑏₀+𝑑₋₁exp−𝜂+𝑑₁𝜂,exp (4.10) where 𝜉=𝑘₁𝑥+𝑤₁𝑡,𝜂=𝑘₂𝑥+𝑤₂𝑡.

Some asymptotic methods are easy and accessible to all nonmathematicians using only pencil and paper. Consider a nonlinear differential equation for corneal shape [13] ℎ^′′𝑏−𝑎ℎ+1+ℎ^′2=0, (4.11) with boundary conditions ℎ(1)=0 and ℎ^′(0)=0.

Hereby we suggest a Taylor series method to find an asymptotic solution [14].

We rewrite (4.11) in the form ℎ^′′=𝑎ℎ−𝑏1+ℎ^′2−1/2. (4.12) Incorporating the boundary condition, ℎ^′(0)=0, we have ℎ^′′(0)=𝑎ℎ(0)−𝑏1+ℎ^′2(0)^−1/2=𝑎ℎ₀−𝑏. (4.13) Differentiating (4.12) with respect to 𝑥 results in ℎ^′′′=𝑎ℎ^′+𝑏1+ℎ^′2−3/2ℎ^′ℎ^′′. (4.14) This yields ℎ^′′′(0)=0. (4.15) Proceeding a similar way as above, we have ℎ⁽⁴⁾(0)=𝑎ℎ^′′(0)+𝑏ℎ^2′′(0)=𝑎𝑎ℎ₀−𝑏+𝑏𝑎ℎ₀−𝑏². (4.16) Applying the Taylor series, we obtain ℎ(𝑥)=ℎ(0)+ℎ^′(0)1𝑥+ℎ2!^′′(0)𝑥²+1ℎ3!^′′′(0)𝑥³+1ℎ4!⁽⁴⁾(0)𝑥⁴, (4.17) or ℎ(𝑥)=ℎ₀+12𝑎ℎ₀−𝑏𝑥²+1𝑎24𝑎ℎ₀−𝑏+𝑏𝑎ℎ₀−𝑏²𝑥⁴. (4.18) Incorporating the boundary condition, ℎ(1)=0, yields ℎ(1)=ℎ₀+12𝑎ℎ₀+1−𝑏𝑎24𝑎ℎ₀−𝑏+𝑏𝑎ℎ₀−𝑏²=0, (4.19) or 𝑎²𝑏ℎ²₀+24+12𝑎+𝑎²−2𝑎𝑏²ℎ₀−12𝑏−𝑎𝑏+𝑏³=0. (4.20) From (4.20), ℎ₀ can be solved, which reads [14] ℎ₀=−24+12𝑎+𝑎²−2𝑎𝑏²+24+12𝑎+𝑎²−2𝑎𝑏²²+4𝑎²𝑏12𝑏+𝑎𝑏−𝑏³2𝑎²𝑏. (4.21) To compare with Okrasiński and Płociniczak’s result, setting 𝑎=𝑏=1, we have ℎ₀=−35+35²+482=0.33956, (4.22) which is very close to Okrasiński and Płociniczak’s result [13].

The accuracy can be further improved if the solution procedure continues.

Comparing the Okrasiński and Płociniczak’s method with our pencil-and-paper method, we conclude that the solution process is accessible to nonmathematicians to solve any nonlinear two-point boundary problems.

5. Asymptotic Methods for Solitary Solutions

The investigation of soliton solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. There are many analytical approaches to the search for solitary solutions, see [5, 6]. Among various methods, the homotopy perturbation method, the variational iteration method, the exp-function method, and the variational approach have been worked out over a number of years by numerous authors, and they have matured into relatively fledged analytical methods for nonlinear equations thanks to the efforts of many researchers.

5.1. Soliton Perturbation

We consider the following perturbed nonlinear evolution equation [5, 15]: 𝑢_𝑇+𝐿(𝑢)+𝑁(𝑢)=𝜀𝑅(𝑢),0<𝜀≪1. (5.1) When 𝜀=0, we have an unperturbed equation 𝑢_𝑇+𝐿(𝑢)+𝑁(𝑢)=0, (5.2) which is assumed to have a solitary solution.

In case 𝜀≠0, but 0<𝜀≪1, we can use perturbation theory and look for approximate solutions of (5.1) which are close to solitary solutions of (5.2).

Using multiple time scales (a slow time 𝜏 and fast time 𝑡 scale such that 𝜕_𝑇=𝜕_𝑡+𝜀𝜕_𝜏), we assume that the soliton solution can be expressed in the form [5, 15] 𝑢(𝑥,𝑇)=𝑢₀(𝜉,𝜏)+𝜀𝑢₁(𝜉,𝜏,𝑡)+𝜀²𝑢₂(𝜉,𝜏,𝑡)+⋯, (5.3) where 𝜉=𝑥−𝑐𝑡, 𝜏 is a slow time, and 𝑡 is a fast time.

Substituting (5.3) into (5.1), then equating like powers of 𝜀, we can obtain a series of linear equations for 𝑢_𝑖(𝑖=0,1,2,3,…), which can be solved sequentially.

In most cases, the nonlinear term 𝑅(𝑢) in (5.1) plays an import role in understanding various solitary phenomena, and the coefficient 𝜀 is not limited to “small parameter.”

5.2. Modified Multitime Expansions [16]

In order to overcome the shortcoming arising in the above solution process, hereby we applied the modified multitime expansions (see Section 2.9 of [16]). To illustrate the method, we consider the following equation: 𝜕𝑢𝜕𝑇+𝑎𝑢𝜕𝑢𝜕𝜕𝑥+𝑏³𝑢𝜕𝑥³+𝜀𝑅(𝑢)=0. (5.4) Introducing the time scales 𝑇_𝑛=𝜀^𝑛𝑡(𝑛=0,1,2,3,…), and using the parameter-expansion method (see Section 5.8), we assume that the solution and the constants 𝑎 and 𝑏 can be expressed as [5, 16] 𝑢=𝑢₀𝑇₀,𝑇₁,𝑇₂,…+𝜀𝑢₁𝑇₀,𝑇₁,𝑇₂,…+𝜀²𝑢₂𝑇₀,𝑇₁,𝑇₂,…+⋯,𝑎=𝑎₀+𝜀𝑎₁+𝜀²𝑎₂+⋯,𝑏=𝑏₀+𝜀𝑏₁+𝜀²𝑏₂+⋯, (5.5)(5.6)(5.7) where the constants 𝑎_𝑖 and 𝑏_𝑖 can be identified by means of no secular terms. Hereby, we define the secular term in a more general form that the term involves time in the form 𝑡^𝑛𝑓, even in case 𝑡^𝑛𝑓 tends to zero when 𝑡→∞.

The equation for 𝑢₀ is 𝜕𝑢₀𝜕𝑡+𝑎₀𝑢₀𝜕𝑢₀𝜕𝑥+𝑏₀𝜕³𝑢₀𝜕𝑥³=0. (5.8) We can choose suitably the values of 𝑎₀ and 𝑏₀, so that the solution of (5.8) can be easily obtained, and involves the basic properties of the original solution.

We use the Duffing equation to illustrate the solution procedure [16] 𝑢^′′+1⋅𝑢+𝜀𝑢³=0,𝑢(0)=𝐴,𝑢^′(0)=0. (5.9) Suppose that the solution can be expressed in (5.5), and the coefficient, 1, can be expanded into 1=𝜔²+𝜀𝜔₁+𝜀²𝜔₂+⋯. (5.10) Substituting (5.5) and (5.10) into (5.9) and collecting terms of the same powers of 𝜀, we have 𝐷²₀𝑢₀+𝜔²𝑢₀𝐷=0,₂₀𝑢₁+𝜔²𝑢₁+𝜔₁𝑢₀+2𝐷₀𝐷₁𝑢₀+𝑢³₀=0, (5.11)(5.12) where 𝐷_𝑖=𝜕²/𝜕𝑇²_𝑖.

Solving (5.11) with the initial conditions 𝑢₀(0)=𝐴 and 𝑢^′₀(0)=0, we have 𝑢₀=𝐴𝑇₁,𝑇₂cos𝜔𝑇₀. (5.13)

Substituting 𝑢₀ into (5.12) results in 𝐷²₀𝑢₁+𝜔²𝑢₁+𝐴𝜔₁+34𝐴²cos𝜔𝑇₀−2𝜔𝐷₁(𝐴)sin𝜔𝑇₀+14𝐴³cos3𝜔𝑇₀=0. (5.14)

Eliminating secular terms needs 𝐷₁(𝐴)𝜔=0,₁3=−4𝐴². (5.15)

If only the first-order approximate solution is searched for, from (5.10), we have 1=𝜔²+𝜀𝜔₁=𝜔²−34𝜀𝐴², (5.16) or 𝜔=31+4𝜀𝐴². (5.17)

The obtained frequency-amplitude relationship, (5.17), is valid for the whole solution domain, and the maximal relative error is less than 7% when 𝜀𝐴²→∞.

5.3. Variational Approach

This section is an elementary introduction to the concepts of the calculus of variations and its applications to solitary solutions. Generally speaking, there exist two basic ways to describe a nonlinear problem: (1) by differential equations (DE) with initial/boundary conditions; (2) by variational principles (VP). The former is widely used, while the later is rarely used in solitary theory. The VP model has many advantages over its DE partner: simple and compact in form while comprehensive in content, encompassing implicitly almost all information characterizing the problem under consideration. Variational methods have been, and continue to be, popular tools for nonlinear problems. When contrasted with other approximate analytical methods, variational methods combine the following two advantages: (1) they provide physical insight into the nature of the solution of the problem; (2) the obtained solutions are the best among all the possible trial functions.

5.3.1. Inverse Problem of Calculus of Variations

The inverse problem of calculus of variations is to establish a variational formulation directly from governing equations and boundary/initial conditions. We will use the semi-inverse method [17, 18] to establish various variational principles directly from the governing equations.

Consider the well-known Korteweg-de Vries (KdV) equation 𝑢_𝑡+𝑎𝑢𝑢_𝑥+𝑏𝑢_𝑥𝑥𝑥=0, (5.18) where 𝑎 and 𝑏 are constants, and the subscripts denote partial differentiations.

We rewrite it in a conserved form 𝜕𝑢+𝜕𝜕𝑡1𝜕𝑥2𝑎𝑢²+𝑏𝑢_𝑥𝑥=0. (5.19) According to the conservation form of (5.19), we can introduce a potential functional Φ defined by Φ_𝑥Φ=𝑢,_𝑡1=−2𝑎𝑢²+𝑏𝑢_𝑥𝑥. (5.20)(5.21) So the KdV equation can be written in the form Φ_𝑥𝑡+𝑎Φ_𝑥Φ_𝑥𝑥+𝑏Φ_{𝑥𝑥𝑥𝑥}=0, (5.22) which can be derived from the following variational principle using the semi-inverse method [17]: 𝐽(Φ)=12Φ_𝑥Φ_𝑡+𝑎6Φ³_𝑥−𝑏2Φ²_𝑥𝑥𝑑𝑡𝑑𝑥. (5.23) In order to obtain a generalized variational principle with two independent fields (Φ,𝑢), we apply the Lagrange multiplier to (5.23) 𝐽()=1Φ,𝑢,𝜆2Φ_𝑥Φ_𝑡+𝑎6Φ³_𝑥−𝑏2Φ²_𝑥𝑥+𝜆Φ_𝑥−𝑢𝑑𝑡𝑑𝑥. (5.24) The stationary condition with respect to 𝑢 results in 𝜆=0. (5.25) The Lagrange multiplier method is not valid for the case. This phenomenon is called Lagrange crisis. Hereby, we suggest three ways to overcome the crisis [18].

(1) The Semi-Inverse Method [17, 18]
Generally, the multiplier can be expressed in the form after identification 𝜆=𝜆𝑢,Φ,𝑢_𝑡,𝑢_𝑥,Φ_𝑡,Φ_𝑥,…. (5.26) We replace the last term including the Lagrange multiplier by a new variable 𝐹, that is, 𝐽()=1Φ,𝑢2Φ_𝑥Φ_𝑡+𝑎6Φ³_𝑥−𝑏2Φ²_𝑥𝑥+𝐹𝑑𝑡𝑑𝑥, (5.27) where 𝐹 is unknown, which can be expressed in the form 𝐹=𝐹𝑢,Φ,𝑢_𝑡,𝑢_𝑥,Φ_𝑡,Φ_𝑥,…. (5.28) Equation (5.27) is such constructed according to the semi-inverse method [17]. To identify 𝐹, making the functional stationary with respect to 𝑢, we have 𝛿𝐹𝛿𝑢=0, (5.29) where 𝛿𝐹/𝛿𝑢 is the variational derivative defined as 𝛿𝐹=𝛿𝑢𝜕𝐹−𝜕𝜕𝑢𝜕𝑡𝜕𝐹𝜕𝑢_𝑡−𝜕𝜕𝑥𝜕𝐹𝜕𝑢_𝑥+𝜕²𝜕𝑥²𝜕𝐹𝜕𝑢_𝑥𝑥+⋯. (5.30) Equation (5.29) should be equivalent to (5.20), and this requires 𝛿𝐹Φ𝛿𝑢=𝑐_𝑥−𝑢, (5.31) where 𝑐 is a nonzero constant. From (5.31), we can determine 𝐹 as follows: 1𝐹=−2𝑐Φ_𝑥−𝑢²=𝑐̃Φ_𝑥−𝑢², (5.32) where 𝑐̃ is a constant, 𝑐̃=−𝑐/2.
We, therefore, obtain the following needed variational principle: 𝐽()=1Φ,𝑢2Φ_𝑥Φ_𝑡+𝑎6Φ³_𝑥−𝑏2Φ²_𝑥𝑥+𝑐̃Φ_𝑥−𝑢²𝑑𝑡𝑑𝑥. (5.33) Its Euler-Lagrange equations are −2𝑐̃Φ_𝑥−𝑢=0,−Φ_𝑥𝑡−𝑎Φ_𝑥Φ_𝑥𝑥−𝑏Φ_{𝑥𝑥𝑥𝑥}−2𝑐̃Φ_𝑥−𝑢_𝑥=0, (5.34) which satisfy the field equations (5.20) and (5.22), respectively.

(2) The Hidden Lagrange Multiplier [18]
Let us come back to (5.25), 𝜆=0, which should be the constraint equation. This means that (5.25) inexplicitly involves a lost constraint equation, so we can identify the multiplier in the form [18] 𝜆=𝑐̃Φ_𝑥−𝑢. (5.35) This results in the same result above.

(3) Replacement of Some Variables in the Original Functional Using the Constraint Equation
Sometimes the Lagrange crisis can be eliminated by replacing some variables in the original variational principle using the constraint equation; a detailed discussion was systematically given in [18].
We replace Φ_𝑥Φ_𝑡 by 𝑢Φ_𝑡 in (5.23) and introduce a Lagrange multiplier in the resultant function 𝐽()=1Φ,𝑢,𝜆2𝑢Φ_𝑡+𝑎6Φ³_𝑥−𝑏2Φ²_𝑥𝑥+𝜆Φ_𝑥−𝑢𝑑𝑡𝑑𝑥. (5.36) Identification of the multiplier yields 1𝜆=2Φ_𝑡. (5.37) Submitting the identified multiplier into (5.36) results in 𝐽()=1Φ,𝑢2𝑢Φ_𝑡+𝑎6Φ³_𝑥−𝑏2Φ²_𝑥𝑥+12Φ_𝑡Φ_𝑥=𝑎−𝑢𝑑𝑡𝑑𝑥6Φ³_𝑥−𝑏2Φ²_𝑥𝑥+12Φ_𝑡Φ_𝑥𝑑𝑡𝑑𝑥. (5.38) We find that the constraint is not eliminated yet, and this is another Lagrange crisis, which can also be eliminated by the semi-inverse method 𝐽()=1Φ,𝑢2𝑢Φ_𝑡+𝑎6Φ³_𝑥−𝑏2Φ²_𝑥𝑥+𝐹𝑑𝑡𝑑𝑥. (5.39) The stationary conditions with respect to 𝑢 and Φ are, respectively, as follows: 12Φ_𝑡+𝛿𝐹−1𝛿𝑢=0,2𝑢_𝑡−𝑎Φ_𝑥Φ_𝑥𝑥−𝑏Φ_{𝑥𝑥𝑥𝑥}+𝛿𝐹𝛿Φ=0. (5.40) According to (5.20) and (5.21), we have 𝛿𝐹1𝛿𝑢=−2Φ_𝑡=14𝑎𝑢²+12𝑏𝑢_𝑥𝑥,𝛿𝐹=1𝛿Φ2𝑢_𝑡+𝑎Φ_𝑥Φ_𝑥𝑥+𝑏Φ_{𝑥𝑥𝑥𝑥}=12𝑢_𝑡−Φ_𝑥𝑡=12Φ_𝑥𝑡−Φ_𝑥𝑡1=−2Φ_𝑥𝑡. (5.41) From (5.41), 𝐹 can be determined as 1𝐹=12𝑎𝑢³−14𝑏𝑢_𝑥²+14Φ_𝑥Φ_𝑡. (5.42) We, therefore, obtain the following variational principle: 𝐽()=1Φ,𝑢2𝑢Φ_𝑡+𝑎6Φ³_𝑥−𝑏2Φ²_𝑥𝑥+112𝑎𝑢³−14𝑏𝑢_𝑥²+14Φ_𝑥Φ_𝑡𝑑𝑡𝑑𝑥. (5.43) By the semi-inverse method, we can obtain various different two-field variational principles, and we write here the following one for reference: 𝐽()=1Φ,𝑢2Φ_𝑥Φ_𝑡1+𝑎3𝑢³−12𝑢²Φ_𝑥−𝑏2Φ²_𝑥𝑥𝑑𝑡𝑑𝑥. (5.44) The potential in the above functionals (5.23), (5.33), and (5.44) requires second order of differentiation (Φ_𝑥𝑥), leading to the complications in the finite element calculation. For the purpose of simplification in finite element computation, we often introduce some additional variables to reduce the order of differentiations. This is of course equivalent to introducing some additional constraints in the variational principle. Generally, we can eliminate the introduced constraints by the Lagrange multiplier method, but as illustrated above, the method might fail.

Now we introduce a new variable 𝐸 defined as 𝑎𝐸=2𝑢²+𝑏𝑢_𝑥𝑥. (5.45) By the semi-inverse method [17, 18], we obtain the following three-field variational principle: 𝐽(Φ,𝑢,𝐸)=𝑢Φ_𝑡+𝐸Φ_𝑥+𝑎3𝑢³−𝑏𝑢²_𝑥−𝐸𝑢𝑑𝑡𝑑𝑥. (5.46) It is obvious that all variables in the obtained functional (5.46) are in first-order differentiations, leading to much convenience in numerical simulation.

It is easy to establish a variational formulation by introducing a potential function, and we can also establish a variational principle without auxiliary special function. To elucidate this, we consider the KdV equation in the form 𝑢_𝑡+𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=0, (5.47) where subscripts denote partial differentiations. If we introduce a velocity potential Φ defined as 𝑢=Φ_𝑥, then the KdV equation can be derived from the variational principle 𝛿𝐿𝑑𝑡𝑑𝑥=0, (5.48) where the Lagrangian 𝐿 can be expressed in the form 1𝐿=2Φ_𝑥Φ_𝑡+16Φ³_𝑥−12Φ²_𝑥𝑥. (5.49) Our aim is to search for a Lagrangian for (5.47). It is easy to establish a variational formulation for differential equations with even orders. The KdV equation has odd-order differentiations, and therefore, no Lagrangian for (5.47). To circumvent this obstacle, we take partial differentiation with respect to 𝑥 to both sides of KdV equation, which turns out to be the following form: 𝑢_𝑡𝑥+𝑢²_𝑥+𝑢𝑢_𝑥𝑥+𝑢_{𝑥𝑥𝑥𝑥}=0. (5.50) By the semi-inverse method, we construct a trial Lagrangian in the form 1𝐿=−2𝑢_𝑥𝑢_𝑡+12𝑢²_𝑥𝑥+𝑝𝑢𝑢²_𝑥+𝑞𝑢²𝑢_𝑥𝑥, (5.51) where 𝑝 and 𝑞 are constants to be further determined. Its Euler equation can be readily obtained as follows: 𝑢_𝑡𝑥+𝑢_{𝑥𝑥𝑥𝑥}+𝑝𝑢²_𝑥−2𝑢𝑢_𝑥𝑥+𝑞2𝑢𝑢_𝑥𝑥−𝑢²_𝑥𝑥=0, (5.52) or 𝑢_𝑡𝑥+𝑢_{𝑥𝑥𝑥𝑥}+−𝑝−2𝑞𝑢²_𝑥−2𝑝𝑢𝑢_𝑥𝑥=0. (5.53) Setting  𝑝=−1/2,𝑞=−1/4, then (5.53) turns out to be the modification version of the KdV equation (5.50).

Finally, we have the following needed Lagrangian in the form of velocity: 1𝐿=−2𝑢_𝑥𝑢_𝑡+12𝑢²_𝑥𝑥−12𝑢𝑢²_𝑥−14𝑢²𝑢_𝑥𝑥. (5.54) This approach can be extended to many nonlinear equations. Consider the modified KdV equation 𝑢_𝑡+𝑚𝑢²𝑢_𝑥+𝑛𝑢_𝑥𝑥𝑥=0. (5.55) Similarly, we change the equation so that it has even-order differentiations 𝑢_𝑡𝑥+2𝑚𝑢𝑢²_𝑥+𝑚𝑢²𝑢_𝑥𝑥+𝑛𝑢_{𝑥𝑥𝑥𝑥}=0. (5.56) By the same manipulation as illustrated above, we construct a trial Lagrange function in the form 1𝐿=−2𝑢_𝑥𝑢_𝑡+12𝑛𝑢²_𝑥𝑥+𝑝𝑢²𝑢²_𝑥+𝑞𝑢³𝑢_𝑥𝑥, (5.57) where 𝑝 and 𝑞 are constants to be further determined. Its Euler equation can be readily obtained as follows: 𝑢_𝑡𝑥+𝑛𝑢_{𝑥𝑥𝑥𝑥}+𝑝2𝑢𝑢²_𝑥−2𝑢²𝑢_𝑥𝑥+𝑞3𝑢²𝑢_𝑥𝑥−𝑢³_𝑥𝑥=0, (5.58) or 𝑢_𝑡𝑥+𝑛𝑢_{𝑥𝑥𝑥𝑥}+𝑝2𝑢𝑢²_𝑥−22𝑢𝑢²_𝑥+𝑢²𝑢_𝑥𝑥+𝑞3𝑢²𝑢_𝑥𝑥−6𝑢𝑢²_𝑥+3𝑢²𝑢_𝑥𝑥=0, (5.59) or 𝑢_𝑡𝑥+𝑛𝑢_{𝑥𝑥𝑥𝑥}+−2𝑝−6𝑞𝑢𝑢²_𝑥−2𝑝𝑢²𝑢_𝑥𝑥=0. (5.60) Setting 𝑚𝑝=−2𝑚,𝑞=−6, (5.61) (5.60) becomes (5.56), and we, therefore, obtain the following Lagrangian: 1𝐿=−2𝑢_𝑥𝑢_𝑡+12𝑛𝑢²_𝑥𝑥−𝑚2𝑢²𝑢²_𝑥−𝑚6𝑢³𝑢_𝑥𝑥. (5.62) We can also use the semi-inverse method to establish a family of variational principles for a nonlinear system. We use one-dimensional traffic flow as an example.

The research on traffic flow began at the beginning of the 20th century. Lighthill and Whitham first proposed the fluid-dynamical model for traffic flow [19, 20]. The continuum equation for unsteady one-dimensional traffic flow can be, therefore, written as 𝜕+𝜕𝜕𝑡𝜌𝐴𝜕𝑥𝜌𝑢𝐴=𝑞, (5.63) where 𝐴 is the cross-sectional area of the road, 𝑢 is the velocity, 𝜌 is the density of cars, and 𝑞 is the source. The deficiency of the model is that the traffic flow actually cannot be considered as a continuum, and to eliminate this deficiency, a fractional differential model can be introduced: 𝜕+𝐷𝜕𝑡𝜌𝐴^𝛼𝐷𝑥^𝛼𝜌𝑢𝐴=𝑞,0<𝛼<1, (5.64) where 𝐷^𝛼/𝐷𝑥^𝛼 is the fractional differential, see Section 7.

In 1994, Zheng [21] suggested the following traffic flow model: 𝜕𝜌+𝜕𝜕𝑡𝜕𝜕𝑥𝜌𝑢=0,+𝜕𝜕𝑡𝜌𝑢𝜕𝑥𝜌𝑢²+1()4𝑛𝑛−1²𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛=0, (5.65)(5.66) where 𝑢_𝑓 is the possible maximal velocity, 𝑚 is a constant, and 𝜌₀ is the minimal traffic density when the cars can travel at a maximal velocity.

In order to establish a variational principle for the system, we rewrite (5.66) in the following equivalent form: 𝜕(𝑢)+𝜕𝜕𝑡1𝜕𝑥2𝑢²+14(𝑛−1)𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−1=0. (5.67) According to (5.65) and (5.67), we can introduce two functions Φ and Ψ defined as 𝜕Φ𝜕𝑥=−𝜌,𝜕Φ𝜕𝑡=𝜌𝑢,𝜕Ψ𝜕𝑥=−𝑢,𝜕Ψ=1𝜕𝑡2𝑢²+14(𝑛−1)𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−1, (5.68)(5.69) so that (5.65) and (5.66) are automatically satisfied.

The essence of the semi-inverse method [17, 18] is to construct an energy-like functional with a certain unknown function, which can be identified step by step. An energy-like trial functional for the discussed problem can be constructed in the following form: 𝐽𝑢,𝜌,Φ=^𝑡₀^𝐿₀𝑢𝜕Φ+1𝜕𝑡2𝑢²+14(𝑛−1)𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−1𝜕Φ𝜕𝑥+𝐹𝑑𝑡𝑑𝑥, (5.70) where 𝑢, 𝜌, and Φ are considered as independent variables, and 𝐹 is an unknown function of 𝑢, 𝜌 and/or their derivatives.

There exist various approaches to the establishment of energy-like trial functionals for a physical problem, and illustrative examples can be found in [22, 23].

The advantage of the above trial functional lies on the fact that the stationary condition with respective to Φ, 𝛿_Φ𝐽(𝑢,ℎ,Φ)=^𝑡₀^𝐿₀−𝜕(𝑢)−𝜕𝜕𝑡1𝜕𝑥2𝑢²+14(𝑛−1)𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−1𝛿Φ𝑑𝑡𝑑𝑥=0, (5.71) leads to (5.67).

Calculating the functional (5.71) stationary with respect to 𝑢 and 𝜌, we obtain the following Euler equations: 𝜕Φ𝜕𝑡+𝑢𝜕Φ+𝜕𝑥𝛿𝐹1𝛿𝑢=0,4(𝑛−1)²𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−2𝜕Φ+𝜕𝑥𝛿𝐹𝛿𝜌=0. (5.72) We search for such an 𝐹, so that the above (5.72) satisfies the two-field equations. To this end, we set 𝛿𝐹𝛿𝑢=−𝜕Φ𝜕𝑡−𝑢𝜕Φ𝜕𝑥=−𝜌𝑢+𝑢𝜌=0,𝛿𝐹1𝛿𝜌=−4(𝑛−1)²𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−2𝜕Φ=1𝜕𝑥4(𝑛−1)²𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−1. (5.73) From (5.73), we can immediately identify the unknown 𝐹, which reads 1𝐹=()4𝑛𝑛−1²𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛. (5.74) So we obtain the following required variational functional: 𝐽𝑢,𝜌,Φ=^𝑡₀^𝐿₀1()4𝑛𝑛−1²𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛+𝑢𝜕Φ+1𝜕𝑡2𝑢²+14(𝑛−1)𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−1𝜕Φ𝜕𝑥𝑑𝑡𝑑𝑥. (5.75)

Proof. The Euler equations of the above functional (5.75) are 1𝛿𝜌∶4(𝑛−1)²𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−1+14(𝑛−1)²𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−2𝜕Φ𝜕𝑥=0,𝛿𝑢∶𝜕Φ𝜕𝑡+𝑢𝜕Φ𝜕𝜕𝑥=0,𝛿Φ∶−(𝑢)−𝜕𝜕𝑡1𝜕𝑥2𝑢²+14(𝑛−1)𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−1=0. (5.76)(5.77)(5.78) From (5.68), we have 𝜕Φ/𝜕𝑥=−𝜌. Substituting the result into (5.77) leads to 𝜕Φ/𝜕𝑡=𝜌𝑢.
From the above three-field variational functional, we can easily obtain two-field or one-field variational function by substituting one- or two-field equations into the functional (5.75). For example, substituting 𝜕Φ/𝜕𝑥=−𝜌 into (5.75), we obtain a two-field variational functional 𝐽₁(𝑢,Φ)=^𝑡₀^𝐿₀1()4𝑛𝑛−1²𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛+𝑢𝜕Φ−1𝜕𝑡2𝜌𝑢²+14(𝑛−1)𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛𝑑𝑡𝑑𝑥, (5.79) where the variable 𝜌 is not now an independent field. Further constraining the two-field functional (5.79) by the equation 𝜕Φ/𝜕𝑡=𝜌𝑢, we have 𝐽₂(Φ)=1()4𝑛𝑛−1²𝑢²_𝑓𝜌₀^1−𝑛𝑡₀^𝐿₀𝜌^𝑛𝑑𝑡𝑑𝑥=^𝑡₀^𝐿₀𝑃𝑑𝑡𝑑𝑥, (5.80) where 𝑃 is the traffic pressure defined as 𝑃=𝐶𝜌^𝑛=1()4𝑛𝑛−1²𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛. (5.81) The functional (5.81) has the same form of the well-known Bateman principle in fluid mechanics [18].
By a paralleling operation, we can also establish a variational functional with free fields 𝑢,  𝜌, and Ψ. A trial functional with an unknown function 𝐹 can be constructed as follows: 𝐽̃𝑢,𝜌,Ψ=^𝑡₀^𝐿₀𝜌𝜕Ψ𝜕𝑡+𝜌𝑢𝜕Ψ𝜕𝑥+𝐹𝑑𝑡𝑑𝑥. (5.82) Here, the unknown 𝐹 is free from Ψ and its derivatives. By the same manipulation as illustrated above, we set 𝛿𝐹𝛿𝑢=−𝜌𝜕Ψ𝜕𝑥=𝜌𝑢,𝛿𝐹𝛿𝜌=−𝜕Ψ𝜕𝑡−𝑢𝜕Ψ=1𝜕𝑥2𝑢²−14(𝑛−1)𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−1. (5.83) From (5.83), we can determine the unknown 𝐹 as follows: 1𝐹=2𝜌𝑢²−1()𝑢4𝑛𝑛−1²_𝑓𝜌₀^1−𝑛𝜌^𝑛. (5.84) So we obtain the following needed variational principle: 𝐽̃𝑢,𝜌,Ψ=^𝑡₀^𝐿₀12𝜌𝑢²−1()𝑢4𝑛𝑛−1²_𝑓𝜌₀^1−𝑛𝜌^𝑛+𝜌𝜕Ψ𝜕𝑡+𝜌𝑢𝜕Ψ𝜕𝑥𝑑𝑡𝑑𝑥. (5.85) It is easy to prove that the Euler equations of the above functional (5.85) satisfy the field equations (5.67) and (5.69).
Constraining the functional (5.85) by the equation 𝜕Ψ/𝜕𝑥=−𝑢, we obtain 𝐽₁̃𝜌,Ψ=^𝑡₀^𝐿₀−12𝜌𝜕Ψ𝜕𝑥²−1()𝑢4𝑛𝑛−1²_𝑓𝜌₀^1−𝑛𝜌^𝑛+𝜌𝜕Ψ𝜕𝑡𝑑𝑡𝑑𝑥, (5.86) which is under the constraint of the equation 𝜕Ψ/𝜕𝑥=−𝑢. Further substituting 𝜕Ψ=1𝜕𝑡2𝑢²+14(𝑛−1)𝑢²_𝑓𝜌₀^1−𝑛𝜌^𝑛−1 (5.87) into (5.86) results in 𝐽₂̃=^𝑡₀^𝐿₀12𝜌𝑢²−1()𝑢4𝑛𝑛−1²_𝑓𝜌₀^1−𝑛𝜌^𝑛𝑑𝑡𝑑𝑥=^𝑡₀^𝐿₀12𝜌𝑢²−𝑃𝑑𝑡𝑑𝑥, (5.88) which is similar to the well-known Hamilton principle.

5.3.2. A Possible Connection between the Uncertain Principle and the Least Action Principle

Maupertuis-Lagrange’s principle of least kinetic potential action for a particle with mass 𝑚 can be expressed as follows [18]: ^𝑡2_𝑡112𝑚𝑣²𝑑𝑡→min. (5.89) We rewrite (5.89) in the form ^𝑠2_𝑠112𝑚𝑣⋅𝑑𝑠→min, (5.90) or ^𝑠2_𝑠1𝑝⋅𝑑𝑠→min. (5.91) Equation (5.91) can be approximately written in the form 𝑝₂−𝑝₁⋅𝑑𝑠₁₂=𝐸_min, (5.92) where 𝑑𝑠₁₂=𝑠₂−𝑠₁. Equation (5.92) means that given that the particle begins at position 𝑠₁ at time 𝑡₁ and ends at position 𝑠₂ at time 𝑡₂, the physical trajectory that connects these two endpoints is an extremum of Δ𝑝⋅Δ𝑠, where Δ𝑠 is the standard deviation of the displacement, and Δ𝑝 is the deviation of the momentum.

For arbitrary Δ𝑠 or Δ𝑝, the following inequality holds: Δ𝑝⋅Δ𝑠≥𝐸_min. (5.93) This is similar to the uncertainty principle.

In optics, Fermat’s principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light 𝑡=^𝑠2_𝑠1𝑑𝑠𝜈→min. (5.94) Equation (5.94) can be approximately written in the form 1𝜈₂−1𝜈₁𝑑𝑠₁₂→min, (5.95) where 𝑑𝑠₁₂=𝑠₂−𝑠₁.

The light trajectory that connects these two endpoints 𝑠₁ and 𝑠₂ satisfies the following equation: 1𝜈₂−1𝜈₁𝑑𝑠=𝑇_min, (5.96) or −1𝜈₁𝜈₂Δ𝑣⋅Δ𝑠=𝑇_min, (5.97) where Δ𝑠=𝑠₂−𝑠₁, Δ𝜈=𝜈₂−𝜈₁.

For arbitrary Δ𝑠 or Δ𝑣, the following inequality holds: −Δ𝑣⋅Δ𝑠=𝜈₁𝜈₂𝑇_min≥𝑐²𝑇_min,Δ𝑣⋅Δ𝑠≤𝑐²𝑇_min. (5.98)

5.3.3. Variational Approach to Nonlinear Oscillators [24]

Consider a general nonlinear oscillator in the form 𝑢^′′+𝑓(𝑢)=0. (5.99) Its variational principle can be easily established as follows: 𝐽(𝑢)=₀^𝑇/4−12𝑢^′2+𝐹(𝑢)𝑑𝑡, (5.100) where 𝑇 is the period of the nonlinear oscillator, 𝜕𝐹/𝜕𝑢=𝑓.

Assume that its solution can be expressed as 𝑢(𝑡)=𝑝cos𝑞𝑡, (5.101) where 𝑝 and 𝑞 are the amplitude and frequency of the oscillator, respectively.

Substituting (5.101) into (5.100) results in 𝐽𝑝,𝑞=₀^𝑇/4−12𝑝²𝑞²sin²𝑞𝑡+𝐹𝐴cos𝑞𝑡𝑑𝑡. (5.102) Instead of setting 𝜕𝐽/𝜕𝑞=0 and 𝜕𝐽/𝜕𝑝=0, we only set [8, 24] 𝜕𝐽𝜕𝑝=0, (5.103) from which the relationship between the amplitude and frequency of the oscillator can be obtained.

Explanation of (5.103) was given in [8].

Consider a nonlinear oscillator with fractional potential [24]: 𝑢^′′+𝜀𝑢^1/3=0. (5.104) Its variational formulation can be readily obtained as follows: 𝐽(𝑢)=₀^𝑇/4−12𝑢^′2+34𝜀𝑢^4/3𝑑𝑡. (5.105) Substituting (5.101) into (5.105), we obtain 𝐽=₀^𝑇/4−12𝑝²𝑞²sin²3𝑞𝑡+4𝑝^4/3𝜀cos^4/3𝑞𝑡𝑑𝑡. (5.106) Setting 𝜕𝐽=𝜕𝐴₀^𝑇/4−𝐴𝜔²sin²𝜔𝑡+𝐴^1/3𝜀cos^4/3𝜔𝑡𝑑𝑡=0, (5.107) we have 𝜔²=∫₀^𝑇/4𝜀cos^4/3𝜔𝑡𝑑𝑡𝐴^2/3∫₀^𝑇/4sin²=∫𝜔𝑡𝑑𝑡^0𝜋/2𝜀cos^4/3𝑡𝑑𝑡𝐴^2/3∫₀^𝜋/2sin²=𝑡𝑑𝑡1.15959526696393𝜀𝐴^3/2. (5.108) The exact frequency is 𝜔=1.070451𝜀^1/2𝐴^−1/3. The 0.597% accuracy is remarkably good.

5.3.4. Variational Approach to Chemical Reactions

As an illustration, consider the following chemical reaction [25]: 𝑛𝐴→𝐶+𝐷, (5.109) which obeys the equation 𝑑𝑥()𝑑𝑡=𝑘𝑎−𝑥^𝑛,𝑥(0)=0, (5.110) where 𝑎 is the number of molecules 𝐴 at 𝑡=0, 𝑥 is the number of molecules 𝐶 (or 𝐷) after time 𝑡, and 𝑘 is a reaction constant. At the start of reaction (𝑡=0), there are no molecules 𝐶 (or 𝐷) yet formed, so that the initial condition is 𝑥(0)=0.

In order to obtain a variational model, we differentiate both sides of (5.110) with respect to time, resulting in 𝑑²𝑥𝑑𝑡²=−𝑘𝑛(𝑎−𝑥)^𝑛−1𝑑𝑥.𝑑𝑡 (5.111) Substituting (5.111) into (5.110), we obtain the following second-order differential equation: 𝑑²𝑥𝑑𝑡²=−𝑘²𝑛(𝑎−𝑥)^2𝑛−1,𝑥(0)=0,𝑥^′(0)=𝑘𝑎^𝑛, (5.112) which admits a variational expression in the form 𝐽(𝑥)=^∞₀12𝑑𝑥𝑑𝑡²+12𝑘²(𝑎−𝑥)^2𝑛𝑑𝑡. (5.113) Its Hamiltonian, therefore, can be written in the form 1𝐻=2𝑑𝑥𝑑𝑡²−12𝑘²(𝑎−𝑥)^2𝑛, (5.114) where 𝐻 is a Hamiltonian constant, and it can be determined from initial conditions: 1𝐻=2𝑥̇(0)²−12𝑘²(𝑎−𝑥(0))^2𝑛=0. (5.115) Equation (5.114) becomes 𝑑𝑥𝑑𝑡²=𝑘²(𝑎−𝑥)^2𝑛. (5.116) In view of the initial conditions 𝑥(0)=0 and 𝑥^′(0)=𝑘𝑎^𝑛, (5.116) is equivalent to (5.110). This means that the variational principle, (5.113), is exactly equivalent to its differential partner, (5.110).

Assume that the solution can be expressed in the form 𝑥=𝑎1−𝑒^−𝜂𝑡, (5.117) where 𝜂 is an unknown constant to be further determined.

Substituting (5.117) into (5.113), and setting 𝑑𝐽/𝑑𝜂=0, we obtain 1𝜂=√𝑛𝑘𝑎^𝑛−1. (5.118) So we obtain a first-order approximate solution for the discussed problem 𝑥=𝑎1−exp−𝑛^−1/2𝑘𝑎^𝑛−1𝑡. (5.119) In order to improve accuracy, we can assume that the solution can be expressed in a more general form 𝑥=𝑎1−^𝑚_𝑖=1𝑏_𝑖𝑒^{−𝜂𝑖𝑡}, (5.120) which should satisfy initial conditions 𝑥(0)=0, and this requires 1−^𝑚_𝑖=1𝑏_𝑖=0. (5.121) Substituting (5.120) into (5.113), we set 𝜕𝐽𝜕𝜂_𝑖=0(𝑖=1∼𝑚),𝜕𝐽𝜕𝑏_𝑖=0(𝑖=2∼𝑚). (5.122) Solving (5.121)-(5.122) simultaneously, we can easily determine 2𝑚 parameters. The solution procedure is similar to that illustrated in [25], and we will not discuss in details to solve space.

We can also choose the following trial function: 1𝑥=𝑎1−1+𝑝𝑡^𝑞, (5.123) where 𝑝 and 𝑞 are unknown constants to be further determined. It is obvious that (5.123) satisfies the conditions 𝑥(0)=0 and 𝑥(+∞)=𝑎. Submitting (5.123) into (5.113), and setting 𝜕𝐽𝜕𝑝=0,𝜕𝐽𝜕𝑞=0, (5.124) we obtain with ease 𝑝=(𝑛−1)𝑘𝑎^𝑛−1,1𝑞=.𝑛−1 (5.125) Thus, we obtain the solution 1𝑥=𝑎1−1+(𝑛−1)𝑘𝑎^𝑛−1𝑡^1/(𝑛−1), (5.126) which is the exact solution.

5.3.5. Variational Approach to Solitary Solution

In the review article [7], the variational approach to solitons was outlined by few lines, and now the method has been successfully applied to the search for soliton solutions [26, 27] without requirement of small parameter assumption, leading to an extremely simple and elementary but rigorous derivation of soliton solutions.

Considering the KdV equation, we seek its traveling wave solutions in the following frame: 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜈(𝑥,𝑡)=𝑉(𝜉),𝜉=𝑥−𝑐𝑡, (5.127) where 𝑐 is angular frequency. Substituting (5.127) into (4.1) yields −𝑐𝑢^′−6𝑢𝑢^′+𝑢^′′′=0, (5.128) where prime denotes the differential with respect to 𝜉.

Integrating (5.128) yields the result−𝑐𝑢−3𝑢²+𝑢^′′=𝐴, (5.129a) where 𝐴 is an integration constant, which can be determined from the initial condition. For solitary solutions or limit cycles, the solutions do not depend upon the initial condition, so we always set 𝐴=0, and this results in −𝑐𝑢−3𝑢²+𝑢^′′=0. (5.129b)By the semi-inverse method [17], the following variational formulation is established: 𝐽=^∞₀12𝑐𝑢²+𝑢³+12𝑑𝑢𝑑𝜉²𝑑𝜉. (5.130) Equation (5.130) also hints a Hamiltonian invariant, which reads12𝑐𝑢²+𝑢³−12𝑢^′2=𝐵, (5.131a) where 𝐵 is a constant, which can be determined by incorporating the initial condition. For solitary solutions or limit cycles, we set 𝐵=0, 12𝑐𝑢²+𝑢³−12𝑢^′2=0. (5.131b)Differentiating (5.131a) with respect to 𝜉, we have 𝑐𝑢𝑢^′+3𝑢²𝑢^′−𝑢^′𝑢^′′=0, (5.132) which is equivalent to (5.129b).

The Hamiltonian invariant (5.131a) and (5.131b) can also be used for construction of an asymptotic solution or used as an auxiliary function in the subequation method, see Section 5.10.

The semi-inverse method is a powerful mathematical tool to the search for variational formulae for real-life physical problems.

By Ritz method, we search for a solitary wave solution in the form 𝑢=𝑝sech²𝑞𝜉, (5.133) where 𝑝 and 𝑞 are constants to be further determined.

Substituting (5.133) into (5.130) results in 𝐽=^∞₀12𝑐𝑝²sech⁴𝑞𝜉+𝑝³sech⁶+1𝑞𝜉2(4𝑝²𝑞²sech⁴𝑞𝜉tanh²=𝑞𝜉𝑑𝜉𝑐𝑝²2𝑞^∞₀sech⁴(𝑧)𝑝𝑑𝑧+³𝑞^∞₀sech⁶(𝑧)𝑑𝑧+2𝑝²𝑞^∞₀sech⁴(𝑧)tanh⁴(𝑧=)𝑑𝑧𝑐𝑝²+3𝑞8𝑝³+15𝑞4𝑝²𝑞.15 (5.134) Making 𝐽 stationary with respect to 𝑝 and 𝑞 results in 𝜕𝐽=𝜕𝑝2𝑐𝑝+3𝑞24𝑝²+15𝑞8𝑝𝑞15=0,𝜕𝐽𝜕𝑞=−𝑐𝑝²3𝑞²−8𝑝³15𝑞²+4𝑝²15=0, (5.135) or simplifying 5𝑐+12𝑝+4𝑞²=0,−5𝑐−8𝑝+4𝑞²=0. (5.136) From (5.136), we can easily obtain the following relations: 1𝑝=−2𝑐,𝑞=𝑐4. (5.137) So the solitary wave solution can be approximated as 𝑐𝑢=−2sech²𝑐4𝑥−𝑐𝑡−𝜉₀, (5.138) which is the exact solitary wave solution of KdV equation.

For the KdV equation expressed in (5.47), we have the following variational principle (see (5.54)): 𝐽(𝑢)=^+∞_−∞−12𝑢_𝑥𝑢_𝑡+12𝑢²_𝑥𝑥−12𝑢𝑢²_𝑥−14𝑢²𝑢_𝑥𝑥𝑑𝑥𝑑𝑡. (5.139) We assume that the solitary solution can be expressed in the form 𝑢()=1𝑥,𝑡𝑎₁𝑒^{𝑎5𝑥−𝑎5𝑡}+𝑎₂𝑒^{−𝑎7𝑥+𝑎8𝑡}+𝑎₃, (5.140) where 𝑎₁,𝑎₂,…,𝑎₈ are unknown constants to be further determined.

Substituting (5.140) into (5.139), we have 𝐽(𝑢)=𝐽𝑎₁,𝑎₂,…,𝑎₈. (5.141) Making the functional, (5.139), stationary can be approximated achieved by making the function (5.141) stationary, and this requires 𝜕𝐽𝜕𝑎_𝑖=0(𝑖=1∼8). (5.142) Solving the system, (5.142), we can determine the values for 𝑎₁,𝑎₂,…,𝑎₈.

5.4. Hamiltonian Approach to Solitary Solution

5.4.1. Generalized Action Principles in Mechanics

We begin with the definition of the action functional as time integral over the Lagrangian 𝐿 of a dynamical system 𝐽𝑥_𝑖=^𝑡2_𝑡1𝐿𝑑𝑡, (5.143) and the Lagrangian is defined as follows: 1𝐿=2𝑚𝑥^′_𝑖²−𝑉𝑥_𝑖. (5.144) Here, 𝑥^′_𝑖=𝑑𝑥_𝑖/𝑑𝑡.

Newton’s motion equation can be obtained from the stationary condition of the functional (5.143), which reads 𝑚𝑥_𝑖^′′+𝜕𝑉𝜕𝑥_𝑖=0. (5.145)

We can introduce some constraints to the action functional (5.143), leading to various principles required. For example, if the total energy is a conserved quantity, that is, 𝑇+𝑉=const., which is considered as a constraint of the functional (5.143), then we obtain the Euler-Maupertuis principle (principle of least action) [28]: ^𝑡2_𝑡1(2𝑇−const.)𝑑𝑡→min, (5.146) or ^𝑡2_𝑡1𝑇𝑑𝑡→min. (5.147) In this section, we will obtain Hamiltonian and other actions from the Lagrangian (5.144) by introducing some constraints.

Now we introduce a generalized velocity 𝑝_𝑖=𝜕𝐿𝜕𝑥^′_𝑖=𝑚𝑥^′_𝑖. (5.148) We consider (5.148) as a constraint of the action functional (5.143), and accordingly the Lagrangian (5.144) can be written as follows: 𝐿𝑥_𝑖,𝑝_𝑖=1𝑝2𝑚^2𝑖−𝑉𝑥_𝑖. (5.149) By the Lagrange multiplier, we have the following generalized Lagrangian: 𝐿₁𝑥_𝑖,𝑝_𝑖,𝜆_𝑖=1𝑝2𝑚^2𝑖−𝑉𝑥_𝑖+𝜆_𝑖𝑝_𝑖−𝑚𝑥^′_𝑖. (5.150) The multiplier can be readily identified, which reads 𝜆_𝑖1=−𝑚𝑝_𝑖. (5.151) Substituting the identified multiplier into (5.150) results in 𝐿₁𝑥_𝑖,𝑝_𝑖=1𝑝2𝑚^2𝑖−𝑉𝑥_𝑖−1𝑚𝑝_𝑖𝑝_𝑖−𝑚𝑥^′_𝑖=𝑝_𝑖𝑥^′_𝑖−𝐻, (5.152) where 𝐻(𝑥_𝑖,𝑝_𝑖) is a Hamiltonian: 𝐻𝑥_𝑖,𝑝_𝑖=1𝑝2𝑚^2𝑖+𝑉𝑥_𝑖. (5.153) If we introduce a new variable 𝑢_𝑖 defined as 𝑢_𝑖=𝑥^′_𝑖, (5.154) we consider (5.154) as a constraint of the action functional (5.143), and in such case, the Lagrangian (5.144) can be rewritten as 𝐿𝑥_𝑖,𝑢_𝑖=12𝑚𝑢²_𝑖−𝑉𝑥_𝑖. (5.155) By the Lagrange multiplier, we have the following generalized Lagrangian: 𝐿₂𝑥_𝑖,𝑢_𝑖,𝜆_𝑖=12𝑚𝑢²_𝑖−𝑉𝑥_𝑖+𝜆_𝑖𝑢_𝑖−𝑥^′_𝑖. (5.156) The multiplier can be readily identified, which reads 𝜆_𝑖=−𝑚𝑢_𝑖. (5.157) Substituting the identified multiplier into (5.156) results in 𝐿₂𝑥_𝑖,𝑢_𝑖=12𝑚𝑢²_𝑖−𝑉𝑥_𝑖−𝑚𝑢_𝑖𝑢_𝑖−𝑥^′_𝑖=𝑚𝑢_𝑖𝑥^′_𝑖−𝐻𝑥_𝑖,𝑢_𝑖, (5.158) where 𝐻(𝑥_𝑖,𝑢_𝑖) is given by 𝐻𝑥_𝑖,𝑝_𝑖=12𝑚𝑢²_𝑖+𝑉𝑥_𝑖. (5.159)

From (5.157), we know that the multiplier is actually the generalized velocity 𝜆_𝑖=−𝑚𝑢_𝑖=−𝑝_𝑖. (5.160) Substituting (5.160) into (5.156), and keeping 𝑝_𝑖 an independent variable, we have 𝐿₃𝑥_𝑖,𝑢_𝑖,𝑝_𝑖=12𝑚𝑢²_𝑖−𝑉𝑥_𝑖−𝑝_𝑖𝑢_𝑖−𝑥^′_𝑖=𝑝_𝑖𝑥^′_𝑖−𝐻𝑥_𝑖,𝑢_𝑖,𝑝_𝑖, (5.161) where 𝐻𝑥_𝑖,𝑢_𝑖,𝑝_𝑖1=−2𝑚𝑢²_𝑖+𝑉𝑥_𝑖+𝑝_𝑖𝑢_𝑖. (5.162) Equation (5.161) is called the Schwinger action [28].

By the same manipulation, from (5.151), the multiplier can be also determined as 𝜆_𝑖1=−𝑚𝑝_𝑖=−𝑢_𝑖. (5.163) So we obtain another action like Schwinger’s, which reads 𝐿₄𝑥_𝑖,𝑝_𝑖,𝑢_𝑖=1𝑝2𝑚^2𝑖−𝑉𝑥_𝑖−𝑢_𝑖𝑝_𝑖−𝑚𝑥^′_𝑖=𝑚𝑢_𝑖𝑥^′_𝑖−𝐻𝑥_𝑖,𝑢_𝑖,𝑝_𝑖, (5.164) where 𝐻𝑥_𝑖,𝑢_𝑖,𝑝_𝑖1=−𝑝2𝑚^2𝑖+𝑉𝑥_𝑖+𝑢_𝑖𝑝_𝑖. (5.165)

A more generalized action can be obtained by linear combination of 𝐿₁(𝑥_𝑖,𝑢_𝑖) and 𝐿₂(𝑥_𝑖,𝑝_𝑖): 𝐿₅𝑥_𝑖,𝑢_𝑖,𝑝_𝑖=𝐿₁𝑥_𝑖,𝑢_𝑖+𝐿₂𝑥_𝑖,𝑝_𝑖1=−𝑝2𝑚^2𝑖−12𝑚𝑢²_𝑖+𝑝_𝑖+𝑚𝑢_𝑖𝑥^′_𝑖−2𝑉𝑥_𝑖=𝑝_𝑖+𝑚𝑢_𝑖𝑥^′_𝑖−𝐻_𝑒𝑥_𝑖,𝑢_𝑖,𝑝_𝑖, (5.166) where 𝐻_𝑒𝑥_𝑖,𝑢_𝑖,𝑝_𝑖=1𝑝2𝑚^2𝑖+12𝑚𝑢²_𝑖+2𝑉𝑥_𝑖. (5.167) The Euler equations can be readily obtained, which read 𝛿𝑥_𝑖∶𝑑𝑝𝑑𝑡^𝑖+𝑚𝑢_𝑖+𝜕𝐻_𝑒𝜕𝑥_𝑖=𝑑𝑝𝑑𝑡^𝑖+𝑚𝑢_𝑖+2𝜕𝑉𝜕𝑥_𝑖=0,𝛿𝑢_𝑖∶𝑚𝑥^′_𝑖−𝜕𝐻_𝑒𝜕𝑢_𝑖=𝑚𝑥^′_𝑖−𝑚𝑢_𝑖=0,𝛿𝑝_𝑖∶𝑥^′_𝑖−𝜕𝐻_𝑒𝜕𝑝_𝑖=𝑥^′_𝑖−1𝑚𝑝_𝑖=0. (5.168) In a more general form, (5.166) can be written as 𝐿₅𝑥_𝑖,𝑢_𝑖,𝑝_𝑖=𝛼𝐿₁𝑥_𝑖,𝑢_𝑖+𝛽𝐿₂𝑥_𝑖,𝑝_𝑖, (5.169) where 𝛼 and 𝛽 are constants.

Linearly combining 𝐿_𝑖(𝑖=1,2,3,4,5), we have 𝐿₆𝑥_𝑖,𝑢_𝑖,𝑝_𝑖=⁵_𝑖=1𝛼_𝑖𝐿_𝑖, (5.170) where 𝛼_𝑖 are constants, and we often set ∑⁵_𝑖=1𝛼_𝑖=1.

5.4.2. Modified Hamilton Principles for Initial Value Problems

The Hamilton principle can be written in the form 𝐽𝑢_𝑖=^𝑇₀12𝑚𝑢̇²_𝑖−𝑈𝑑𝑡, (5.171) with the special requirements 𝛿𝑢(0)=𝛿𝑢(𝑇)=0.

Here 𝑢_𝑖 is the velocity component in 𝑥_𝑖, and 𝑈 is a potential defined as 𝜕𝑈/𝜕𝑥_𝑖=−𝐹_𝑖, where 𝐹_𝑖 is the body force component in 𝑥_𝑖.

The Hamilton’s principle holds only for the conditions prescribed at the beginning and at the end of the motion and is therefore useless to deal with the usual initial condition problems, both as an analytical tool and as a basis for approximate solution methods. It is impossible for most real-life physical problems to prescribe terminal conditions.

In order to eliminate the unnecessary final condition at 𝑡=𝑇, Carini and Genna [29] obtained the following functional for the case 𝑢_𝑖(0)=𝑢̇_𝑖(0)=0: 𝐽𝑢_𝑖=^𝑇₀12𝑚𝑢̈²_𝑖−𝐹̈_𝑖𝑢_𝑖𝑑𝑡+𝐹̇_𝑖(𝑇)𝑢_𝑖(𝑇)−𝐹_𝑖(𝑇)𝑢̇_𝑖(𝑇). (5.172) We found that the natural final conditions at 𝑡=𝑇 satisfy the physical requirements.

Liu [30] considered that, in order to deal with the final condition, one term should be added to the Hamilton principle to result in the following functional: 𝐽𝑢_𝑖=^𝑇₀12𝑚𝑢̇²_𝑖−𝑈𝑑𝑡−𝑚𝑢̇_𝑖𝑢̇_{𝑖𝑡=𝑇}, (5.173) where 𝑢̇_𝑖 is a restricted variable, that is, 𝛿𝑢̇_𝑖=0. The modification, (5.172), is not a classic variational principle due to that the variable 𝑢̇_𝑖 has to be prescribed at 𝑡=𝑇, so that 𝛿𝑢̇_𝑖(𝑇)=0, (5.174) and this requirement is even more overrestricted than 𝛿𝑢_𝑖(𝑇)=0 in Hamilton principle, so no vital innovation was made due to its inconsistent at the final condition. However, Liu [30] obtained some functionals that can successfully deal with the initial conditions, but the final condition still keeps an issue of polemics.

It would be a landmark in the history of calculus of variations after Hamilton if we can extend the principle to all initial-value problems without prescribing both initial and final conditions. In order to deal with the final condition, we consider the conserved Hamiltonian: 12𝑚³_𝑖=1𝑢̇²_𝑖1+𝑈=2𝑚³_𝑖=1𝑢̇²_𝑖0+𝑈₀. (5.175) Here 𝑈₀=𝑈|_𝑡=0, so we have the following identity: 𝑢̇_𝑖=³_𝑖=1𝑢̇²_𝑖0−³_𝑗=1𝑢̇²_𝑗+2𝑈₀−𝑈𝑚,𝑖≠𝑗, (5.176) for all 𝑡≥0.

In order to convert the initial conditions 𝑢_𝑖(0)=𝑢_𝑖0 and 𝑢̇_𝑖(0)=𝑢̇_𝑖0 into natural initial conditions and make the natural final condition satisfy the physical requirement, (5.176), we assume that its modified Hamilton principle can be written in the form 𝐽𝑢_𝑖=^𝑇₀12𝑚𝑢̇²_𝑖−𝑈𝑑𝑡+𝑓𝑢_𝑖,𝑢̇_𝑖^𝑡=𝑇_𝑡=0, (5.177) where 𝑓 is an unknown function of 𝑢_𝑖 and/or its derivatives.

Making the above functional stationary, we obtain the following stationary conditions at 𝑡=0 and 𝑡=𝑇: 𝑚𝑢̇_𝑖+𝜕𝑓𝜕𝑢_𝑖^𝑡=𝑇_𝑡=0=0,𝜕𝑓𝜕𝑢̇_𝑖^𝑡=𝑇_𝑡=0=0. (5.178) We search such an 𝑓 so that the above 4 equations satisfy all initial conditions at 𝑡=0 and meet exactly the physical requirement, for example, (5.176), at 𝑡=𝑇.

Accordingly, we can assume that 𝜕𝑓𝜕𝑢_𝑖𝑡=0=𝑚𝑢̇_𝑖0𝑡=0,𝜕𝑓𝜕𝑢̇_𝑖𝑡=0=𝑘𝑢_𝑖−𝑢_𝑖0𝑡=0,𝜕𝑓𝜕𝑢_{𝑖𝑡=𝑇}=−𝑚𝑢̇_{𝑖𝑡=𝑇}=−𝑚³_𝑖=1𝑢̇²_𝑖0−³_𝑗=1𝑢̇²_𝑗+2𝑈₀−𝑈𝑚_𝑡=𝑇,𝑖≠𝑗. (5.179) From the above relations, we can identify 𝑓 in the form 𝑓=𝑚𝑢̇_𝑖0𝑢_𝑖1+𝑘2𝑢̇²_𝑖−𝑢̇_𝑖0𝑢̇_𝑖𝑡=0+He|_𝑡=𝑇, (5.180) where 𝑘 is a nonzero constant; He is defined as 𝑚He=²3𝐹_𝑖³_𝑖=1𝑢̇²_𝑖0−³_𝑗=1𝑢̇²_𝑗+2𝑈₀−𝑈𝑚^3/2. (5.181) So we obtain the following modified Hamilton principle: 𝐽𝑢_𝑖=^𝑇₀12𝑚𝑢̇²_𝑖−𝑈𝑑𝑡+𝑚𝑢̇_𝑖0𝑢_𝑖1+𝑘2𝑢̇²_𝑖−𝑢̇_𝑖0𝑢̇_𝑖𝑡=0+He|_𝑡=𝑇, (5.182) which holds for all initial value problems.

Proof. Making the obtained functional (5.182) stationary, we obtain (1)in the solution domain (0<𝑡<𝑇): −𝑚𝑢̈_𝑖+𝐹_𝑖=0, (5.183) which is Newton’s motion equation; (2)natural initial conditions (𝑡=0): −𝑚𝑢̇_𝑖𝑡=0+𝑚𝑢̇_𝑖0𝑡=0𝑘𝑢̇=0,_𝑖−𝑢̇_𝑖0𝑡=0=0, (5.184) which satisfy obviously the initial conditions 𝑢_𝑖(0)=𝑢_𝑖0 and 𝑢̇_𝑖(0)=𝑢̇_𝑖0, respectively; (3)natural final condition (𝑡=𝑇): 𝑚𝑢̇_{𝑖𝑡=𝑇}−𝑚³_𝑖=1𝑢̇²_𝑖0−³_𝑗=1𝑢̇²_𝑗+2𝑈₀−𝑈𝑚_𝑡=𝑇=0, (5.185) which meet the physical requirement of (5.176) at 𝑡=𝑇.
We have alternative approaches to identifying 𝑓 by different assumptions in (5.178), leading to various new modifications. We write here few modified Hamilton principles for reference 𝐽𝑢_𝑖=^𝑇₀12𝑚𝑢̇²_𝑖−𝑈𝑑𝑡+𝑚𝑢̇_𝑖0𝑢_𝑖+𝑘₁12𝑢²_𝑖−𝑢_𝑖0𝑢_𝑖+𝑘₂12𝑢̇²_𝑖−𝑢̇_𝑖0𝑢̇_𝑖𝑡=0+He|_𝑡=𝑇,𝐽𝑢_𝑖=^𝑇₀12𝑚𝑢̇²_𝑖+−𝑈𝑑𝑡𝑚−𝑘₄𝑢_𝑖0𝑢_𝑖−𝑘₃𝑢_𝑖0𝑢̇_𝑖+𝑘₃𝑢_𝑖𝑢̇_𝑖−𝑢̇_𝑖0+𝑘₄𝑢̇_𝑖𝑢_𝑖−𝑢_𝑖0𝑡=0+He|_𝑡=𝑇, (5.186) where 𝑘’s are nonzero constants, and it requires that 𝑘₁−𝑚≠0 and 𝑘₃+𝑘₄=0.
To summarize, we can conclude from the above derivation and strict proof that the obtained modified Hamilton principles, which are first deduced in the history, and valid for all initial-value problems, are extremely important in both pure and applied sciences due to complete elimination of the long-existing shortcomings in Hamilton principle. The stationary conditions of the obtained variational principle satisfy the Newton’s motion equation, and all initial conditions, furthermore, the natural final condition (𝑡=𝑇), satisfy automatically the physical requirement, making a vital innovation of Hamilton principle.

5.4.3. Hamiltonian Approach to Nonlinear Oscillators [31, 32]

In this paper, we consider the following general oscillator: 𝑢^′′+𝑓(𝑢)=0, (5.187) with initial conditions 𝑢(0)=𝐴 and 𝑢^′(0)=0.

It is easy to establish a variational principle for (5.187), which reads [31] 𝐽(𝑢)=₀^𝑇/412𝑢^′2−𝐹(𝑢)𝑑𝑡, (5.188) where 𝑇 is the period of the oscillator, 𝜕𝐹/𝜕𝑢=𝑓(𝑢).

In the functional (5.188), (1/2)𝑢^′2 is kinetic energy, and 𝐹(𝑢) is potential energy, so the functional (5.188) is the least Lagrangian action, from which we can immediately obtain its Hamiltonian, which reads 1𝐻=2𝑢^′2+𝐹(𝑢)=constant=𝐻₀, (5.189) or 12𝑢^′2+𝐹(𝑢)−𝐻₀=0. (5.190) Equation (5.189) replies that the total energy keeps unchanged during the oscillation.

Assume that the solution can be expressed as 𝑢=𝐴cos𝜔𝑡, (5.191) where 𝜔 is the frequency.

Submitting (5.191) into (5.190) results in a residual 𝑅(𝑡)=12𝐴²𝜔²sin²𝜔𝑡+𝐹(𝐴cos𝜔𝑡)−𝐻₀. (5.192) According to the energy balance method proposed in [33], locating at some a special point, that is, 𝜔𝑡=𝜋/4, and setting 𝑅(𝑡=𝜋/4𝜔)=0, we can obtain an approximate frequency-amplitude relationship of the studied nonlinear oscillator. Such treatment is much simple and has been widely used by engineers [34–36]. The accuracy of such location method, however, strongly depends upon the chosen location point. To overcome the shortcoming of the energy balance method, in this paper, we suggest a new approach based on Hamiltonian.

From (5.192), we have 𝜕𝐻𝜕𝐴=0. (5.193) Introducing a new function, 𝐻(𝑢), defined as 𝐻(𝑢)=₀^𝑇/412𝑢^′2(𝑢1+𝐹)𝑑𝑡=4𝑇𝐻, (5.194) it is obvious that 𝜕𝐻=1𝜕𝑇4𝐻. (5.195) Equation (5.195) is, then, equivalent to the following one: 𝜕𝜕𝜕𝐴𝐻𝜕𝑇=0, (5.196) or 𝜕𝜕𝜕𝐴𝐻𝜕(1/𝜔)=0. (5.197) From (5.197), we can obtain approximate frequency-amplitude relationship of a nonlinear oscillator.

Consider the Duffing equation 𝑢^′′+𝑢+𝜀𝑢³=0,𝑢(0)=𝐴,𝑢^′(0)=0. (5.198) Its Hamiltonian can be easily obtained, which reads 1𝐻=2𝑢^′2+12𝑢²+14𝜀𝑢⁴. (5.199) Integrating (5.199) with respect to 𝑡 from 0 to 𝑇/4, we have 𝐻(𝑢)=₀^𝑇/412𝑢^′2+12𝑢²+14𝜀𝑢⁴𝑑𝑡. (5.200) Assuming that the solution can be expressed as 𝑢=𝐴cos𝜔𝑡 and substituting it to (5.200), we obtain 𝐻(𝑢)=₀^𝑇/412𝐴²𝜔²sin²1𝜔𝑡+2𝐴²cos²1𝜔𝑡+4𝜀𝐴⁴cos⁴𝜔𝑡𝑑𝑡=₀^𝜋/212𝐴²𝜔sin²1𝑡+𝜔12𝐴²cos²1𝑡+4𝜀𝐴⁴cos⁴𝑡=1𝑑𝑡2𝐴²𝜋𝜔⋅4+1𝜔12𝐴²⋅𝜋4+14𝜀𝐴⁴⋅34⋅𝜋4. (5.201) Setting 𝜕𝜕𝜕𝐴𝐻𝜕(1/𝜔)=−𝐴𝜔²⋅𝜋4+𝜋𝐴⋅4+𝜀𝐴³⋅34⋅𝜋4=0, (5.202) we obtain the following frequency-amplitude relationship: 𝜔=31+4𝜀𝐴³. (5.203) Now we consider another nonlinear oscillator with discontinuity: 𝑑²𝑢𝑑𝑡²+sgn(𝑢)=0,𝑢(0)=𝐴,𝑢^′(0)=0. (5.204)sgn(𝑢) is +1 and −1 for 𝑢>0 and 𝑢<0, respectively.

Its variational formulation can be written as 𝐽(𝑢)=₀^𝑇/4−12𝑢^′2+𝑢𝑑𝑡+^𝑇/2_𝑇/4−12𝑢^′2−𝑢𝑑𝑡, (5.205) and 𝐻(𝑢) can be written in the form 𝐻(𝑢)=₀^𝑇/412𝑢^′2+𝑢𝑑𝑡+^𝑇/2_𝑇/412𝑢^′2−𝑢𝑑𝑡. (5.206) Using 𝑢=𝐴cos𝜔𝑡 as an approximate solution, we have 𝐻=₀^𝑇/412𝐴²𝜔²sin²𝜔𝑡+𝐴cos𝜔𝑡𝑑𝑡+^𝑇/2_𝑇/412𝐴²𝜔²sin²=𝜔𝑡−𝐴cos𝜔𝑡𝑑𝑡_0𝜋/212𝐴²𝜔sin²1𝑡+𝜔𝐴cos𝑡𝑑𝑡+^𝜋_𝜋/212𝐴²𝜔sin²1𝑡−𝜔=1𝐴cos𝑡𝑑𝑡2𝐴²𝜋𝜔⋅4+1𝜔1𝐴+2𝐴²𝜋𝜔⋅4+1𝜔𝐴=𝐴²𝜋𝜔⋅4+2𝜔𝐴. (5.207) Setting 𝜕𝜕𝜕𝐴𝐻𝜕(1/𝜔)=−𝐴𝜔²⋅𝜋2+2=0, (5.208) we obtain 2𝜔=√=𝜋𝐴1.128379√𝐴, (5.209) which is very close to the exact one 𝜔_exact(𝐴)=𝜋2√=2𝐴1.110721√𝐴. (5.210) The 1.59% accuracy is acceptable considering the simple solution procedure.

5.4.4. Hamilton Approach to Solitary Solutions

The Hamiltonian approach to nonlinear oscillators has been now widely used [37–40]; in this section, we will extend the technology to nonlinear wave equations.

The Euler-Lagrange equation of (5.130) reads 𝑑𝐿−𝑑𝑑𝑢𝑑𝜉𝑑𝐿𝑑𝑢_𝜉=0, (5.211) where 𝐿=(1/2)𝑐𝑢²+𝑢³+(1/2)(𝑑𝑢/𝑑𝜉)² and 𝑢_𝜉=𝑑𝑢/𝑑𝜉.

It is obvious that the above Euler-Lagrange equation is the KdV equation.

We write the Lagrange function in the form 𝐿=𝑇−𝑉, where 𝑇 is the kinetic energy defined as 1𝑇=2𝑑𝑢𝑑𝜉², (5.212) and 𝑉 is potential energy defined as 1𝑉=−2𝑐𝑢²−𝑢³. (5.213) The Hamiltonian invariant reads 𝑇+𝑉=𝐵, that is, 12𝑢^′2−12𝑐𝑢²−𝑢³=𝐵, (5.214) where 𝐵 is a constant, which can be determined by incorporating the initial condition. For solitary solutions or limit cycles, we set 𝐵=0. Equation (5.214) becomes 12𝑐𝑢²+𝑢³−12𝑢^′2=0. (5.215) Differentiating (5.215) with respect to 𝜉, we have 𝑐𝑢𝑢^′+3𝑢²𝑢^′−𝑢^′𝑢^′′=0, (5.216) which is equivalent to (5.129b) in case 𝑢^′≠0.

If a solitary solution is solved, considering tail property of a soliton, we assume that 𝑢(𝜉)=∑_𝑛𝑎_𝑛exp𝑞_𝑛𝜉∑_𝑚𝑏_𝑚exp𝑝_𝑚𝜉, (5.217) where 𝑎_𝑖,𝑏_𝑖,𝑝_𝑖,and𝑞_𝑖 are constants to be further determined asymptotically. Equation (5.217) should satisfy (2.2).

The assumption, (5.217), is also used in the exp-function method (see Section 5.9), where the unknowns are identified exactly, whereas we will determine their values approximated by the weighted residual method.

Substituting (5.217) into (5.216) results in the following residual: 𝑅(𝜉)=12𝑐𝑢²+𝑢³−12𝑢^′2. (5.218) We can use the method of the least squares to determine the unknown constants involved in (5.217) ^+∞_−∞𝑅²(𝜉)𝑑𝜉→min. (5.219) We can also use the location method to simply identify the unknowns. To elucidate the solution procedure, we assume its solitary solution has a simple symmetrical form 𝑢(𝜉)=1𝑎𝑒^𝑘𝜉+𝑒^−𝑘𝜉,+𝑏 (5.220) where 𝑎,𝑏, and 𝑘 are unknown constants.

The residual equation is 𝑅(𝜉)=𝑐2𝑎𝑒^𝑘𝜉+𝑒^−𝑘𝜉+𝑏²+1𝑎𝑒^𝑘𝜉+𝑒^−𝑘𝜉+𝑏³−𝑎𝑘𝑒^𝑘𝜉−𝑒^−𝑘𝜉2𝑎𝑒^𝑘𝜉+𝑒^−𝑘𝜉+𝑏⁴. (5.221) To determine the values of the unknowns in (5.220), we set 𝑅(0)𝑅=0,_′′(0)𝑅=0,₍₄₎(0)=0. (5.222) Note that 𝑅^′(0)=𝑅^′′′(0)≡0.

Simplifying (5.222), we obtain 2𝑎𝑐+𝑏𝑐+2=0,2𝑎𝑐+𝑏𝑐+3+2𝑎𝑘²=0,32𝑎²𝑐+14𝑎𝑏𝑐−𝑏²𝑐+66𝑎+80𝑎²𝑏𝑘²−3𝑏=0. (5.223) Solving (5.223) simultaneously, we have 1𝑎=−2𝑘²,1𝑏=−𝑘²,𝑐=𝑘². (5.224) We, therefore, obtain the following approximate solitary solution: 𝑢()=1𝑥,𝑡−1/2𝑘²exp𝑘𝑥−𝑘²𝑡+exp−𝑘𝑥−𝑘²𝑡−1/𝑘²=−2𝑘²exp𝑘𝑥−𝑘²𝑡+exp−𝑘𝑥−𝑘²𝑡.+2 (5.225) Submitting the obtained solution into (4.1), we find it as an exact solution!

Consider the following generalized KdV equation: 𝑢_𝑡+6𝑢^𝑛𝑢_𝑥+𝑢_𝑥𝑥𝑥=0. (5.226) This equation admits no variational formulation. If a traveling wave solution is searched for, we can make a transformation 𝜉=𝑥−𝑐𝑡, and then (5.226) reduces to an ordinary differential equation −𝑐𝑢^′+6𝑢^𝑛𝑢^′+𝑢^′′′=0. (5.227) Integrating the above equation results in 6−𝑐𝑢+𝑢𝑛+1^𝑛+1+𝑢^′′=𝐴, (5.228) which admits a variational formulation 𝐽(𝑢)=^+∞_−∞−12𝑐𝑢²+6()𝑢𝑛+1)(𝑛+2^𝑛+21−𝐴𝑢−2𝑢^′2𝑑𝜂. (5.229) We write 𝑇=(1/2)𝑢^′2 and 𝐸=−(1/2)𝑐𝑢²+(6/(𝑛+1)(𝑛+2))𝑢^𝑛+2−𝐴𝑢, and then 𝑇 can be interpreted as “kinetic energy” and 𝐸 “potential energy.” The variational functional, (5.229), can be rewritten in the form of Lagrangian action 𝐽(𝑢)=−^+∞_−∞{𝑇−𝐸}𝑑𝜂, (5.230) and the Hamiltonian invariant implies the conversation of energy, which requires −12𝑐𝑢²+6()𝑢𝑛+1)(𝑛+2^𝑛+21−𝐴𝑢+2𝑢^′2=𝐵, (5.231) or 𝑢^′2=𝑐𝑢²−12()𝑢𝑛+1)(𝑛+2^𝑛+2+2𝐴𝑢+2𝐵. (5.232) For solitary solutions, 𝐴=𝐵=0, and (5.232) reduces to 𝑢^′2=𝑐𝑢²−12()𝑢𝑛+1)(𝑛+2^𝑛+2. (5.233) If a solitary solution is solved, we assume that 𝑢(𝜉)=1𝑎𝑒^𝑘𝜉+𝑏𝑒^−𝑘𝜉.+𝑑 (5.234) Substituting (5.234) into (5.233) results in the following residual: 𝑅(𝜉)=𝑢^′2−𝑐𝑢²+12()𝑢𝑛+1)(𝑛+2^𝑛+2. (5.235) Proceeding a similar way as that for (5.222), we can identify 𝑎,𝑏, and 𝑑 in (5.234), and finally, we obtain 𝑢(𝑥,𝑡)=−2𝑘²exp𝑘𝑥−𝑘³𝑡+𝐷+exp−𝑘𝑥−𝑘³.𝑡+𝐷−2 (5.236) For a compacton solution, we assume that 𝑢(𝜉)=𝑎cos²𝜉𝑏+𝑐cos²𝜉. (5.237) By a similar solution procedure as above, we can determine 𝑎, 𝑏, and 𝑐 with ease.

5.5. Variational Principles for Fluids

5.5.1. A Bernoulli-Like Equation for Shallow Wave Propagation

Strong storms and cyclones, underwater earthquakes, high-speed ferries, and aerial and submarine landslides can cause giant surface waves approaching the coast and frequently cause extensive coastal flooding, destruction of coastal constructions, and loss of lives. A fast but reliable prediction of a tsunami pulse is of critical importance, and a simple equation like Bernoulli equation is, therefore, much needed.

To this end, we use a one-dimensional nonlinear shallow wave propagation for describing runup of irregular waves on a beach. The basic equations are [41] 𝜕𝑢𝜕𝑡+𝑢𝜕𝑢𝜕𝑥+𝑔𝜕𝜂𝜕𝑥=0,𝜕𝜂+𝜕𝜕𝑡𝑢𝜕𝑥ℎ+𝜂=0, (5.238)(5.239) where 𝜂 is water displacement, 𝑢 is the depth-averaged velocity, ℎ is the unperturbed water depth, and 𝑔 is the gravity acceleration.

We will establish a variational principle for the system of (5.238) and (5.239) to search for a principle of conservation of energy for the present problem.

In order to establish a variational formulation for the system of (5.238) and (5.239), we introduce a function Φ defined as 𝜕Φ𝜕𝑥=𝑢,𝜕Φ1𝜕𝑡=−2𝑢²+𝑔𝜂. (5.240) Equation (5.240) is equivalent to (5.238).

Using the semi-inverse method [17], we construct a trial functional in the form 𝐽Φ,𝑢,𝜂=𝜂𝜕Φ+𝑢𝜕𝑡ℎ+𝜂𝜕Φ𝜕𝑥+𝐹𝑢,𝜂𝑑𝑡𝑑𝑥, (5.241) where 𝐹 is an unknown function of 𝑢 and 𝜂.

It is obvious that the stationary condition of (5.241) with respect to Φ is (5.239). Now making the functional, (5.241), stationary with respect to 𝜂 and 𝑢, we obtain the following Euler-Lagrange equations: 𝜕Φ𝜕𝑡+𝑢𝜕Φ+𝜕𝑥𝜕𝐹𝜕𝜂=0,ℎ+𝜂𝜕Φ+𝜕𝑥𝜕𝐹𝜕𝑢=0. (5.242) The above equations should satisfy (5.240). To this end, using (5.240), we set 𝜕𝐹𝜕𝜂=−𝜕Φ𝜕𝑡−𝑢𝜕Φ=1𝜕𝑥2𝑢²+𝑔𝜂−𝑢²1=−2𝑢²+𝑔𝜂,𝜕𝐹𝜕𝑢=−ℎ+𝜂𝜕Φ𝜕𝑥=−ℎ+𝜂𝑢. (5.243) From (5.243), the unknown 𝐹 can be identified, which reads 1𝐹=−2𝑢²1𝜂+2𝑔𝜂²−12ℎ𝑢². (5.244) We, therefore, obtain the needed variational formulation, which is 𝐽Φ,𝑢,𝜂=𝜂𝜕Φ+𝑢𝜕𝑡ℎ+𝜂𝜕Φ−1𝜕𝑥2𝜂+ℎ𝑢²+12𝑔𝜂²𝑑𝑡𝑑𝑥. (5.245) Submitting (5.240) into (5.245), we obtain a constrained variational formulation 𝐽(Φ)=12ℎ𝑢²−12𝑔𝜂²𝑑𝑡𝑑𝑥, (5.246) which is under constraints of (5.240).

Proof. Making the functional, (5.246), stationary, we have 𝛿𝐽(Φ)=ℎ𝑢𝛿𝑢−𝑔𝜂𝛿𝜂𝑑𝑡𝑑𝑥. (5.247) According to the constraints, (5.240), we have 𝛿𝜕Φ𝛿𝜕𝑥=𝛿𝑢,𝜕Φ1𝜕𝑡=−𝛿2𝑢²+𝑔𝜂=−𝑢𝛿𝑢+𝑔𝛿𝜂. (5.248)(5.249) Replacing 𝛿𝑢 in (5.249) by 𝛿(𝜕Φ/𝜕𝑥) results in 𝑔𝜂𝛿𝜂=−𝜂𝛿𝜕Φ𝜕𝑡−𝜂𝑢𝛿𝜕Φ𝜕𝑥. (5.250) Submitting (5.248) and (5.250) into (5.247), we obtain 𝛿𝐽(Φ)=ℎ𝑢𝛿𝜕Φ𝜕𝑥+𝜂𝛿𝜕Φ𝜕𝑡+𝜂𝑢𝛿𝜕Φ=𝑢𝜕𝜕𝑥𝑑𝑡𝑑𝑥ℎ+𝜂()𝜕𝜕𝑥𝛿Φ+𝜂(𝜕𝑢𝜕𝑡𝛿Φ)𝑑𝑡𝑑𝑥=−ℎ+𝜂+𝜕𝑥𝜕𝜂𝜕𝜕𝑡𝛿Φ𝑑𝑡𝑑𝑥++𝜕𝜕𝑥ℎ+𝜂𝑢𝛿Φ𝜕𝑢𝜕𝑡𝜂𝛿Φ𝑑𝑡𝑑𝑥=−ℎ+𝜂+𝜕𝑥𝜕𝜂𝜕𝑡𝛿Φ𝑑𝑡𝑑𝑥+_𝑡ℎ+𝜂𝑢𝛿Φ𝑑𝑡+_𝑥𝜂𝛿Φ𝑑𝑥. (5.251) Setting 𝛿𝐽(Φ)=0, and ignoring the boundary terms, for arbitrary 𝛿Φ, we can obtain (5.239) as the needed stationary condition.
We can rewrite (5.246) in the form 𝐽(Φ)={𝑇−𝑉}𝑑𝑡𝑑𝑥, (5.252) where 𝑇=(1/2)ℎ𝑢² is a generalized kinetic energy of water wave, is 𝑉=(1/2)𝑔𝜂² and a generalized gravitational potential.
Equation (5.252) is a principle of the Lagrange action for the shallow wave propagation, and it also implies conservation of energy 𝑇+𝑉=𝐻: 12ℎ𝑢²+12𝑔𝜂²=𝐻, (5.253) where 𝐻 is a constant.
We call (5.253) a Bernoulli-like equation for shallow wave propagation.

Tsunami prediction, for example, after an earthquake, is of critical importance to save life and property. The extent of catastrophe depends mainly on the height of the Tsunami pulses, that is, the value of 𝜂. It is easy to observe the velocity of a wave in the sea, and we, therefore, can make easy but reliable tsunami prediction using the obtained Bernoulli-like equation.

5.5.2. Lin’s Variational Principle for Ideal Flows

It is well known that the Hamilton’s principle can be applied to a single fluid particle or a closed system by the involutory transformation. For an isoentropy rotational flow, a variational principle can be established using Lin’s constraints [42]. But the essence of Lin’s constraints is not clear yet, and this paper concludes that the functional with Lin’s constraints is not a genuine variational principle, but an approximate one. In addition, a new generalized variational principle with only 6 independent variables (including only one Lin’s constraint) for three-dimensional unsteady compressible rotational flow is established.

In 1950s, great progress had been made on the research of variational principle in solid mechanics, with Hu-Washizu variational principle [43] as milestone, and in fluid mechanics, with Lin’s constraints [42] as milestone.

Hamilton’s principle was so successfully and powerfully applied to particle mechanics that many attempts have been made to obtain the momentum equations from a variational principle patterned after Hamilton’s principle. These attempts have not all been successful except in the case of isoentropy irrotational flow. For a more general one, Lin’s constraints [44] must be added.

Considering a 3D unsteady inviscid compressible rotational flow, we have the following equations [45].(1)Momentum equation: 𝜕𝑢_𝑖𝜕𝑡+𝑢_𝑖𝜕𝑢_𝑖𝜕𝑥_𝑖1=−𝜌𝜕𝑃𝜕𝑥_𝑖, (5.254) where 𝑢_𝑖 is the flow velocity in 𝑥_𝑖 indirect, 𝑃 is the pressure, and 𝜌 is the density. (2)Equation of state: 𝑃=𝜌^𝜅exp𝑆−𝑆₀𝑚, (5.255) where 𝜅 is a specific heat ratio, 𝑚=(𝜅−1)⁻¹, and 𝑆 is entropy.(3)Continuity equation: 𝜕𝜌+𝜕𝜕𝑡𝜌𝑢_𝑗𝜕𝑥_𝑗=0. (5.256)(4)Isoentropic equation: 𝜕𝑆𝜕𝑡+𝑢_𝑗𝜕𝑆𝜕𝑥_𝑗=0. (5.257)(5)Steady Bernoulli’s equation: 12𝜈²+Π=𝐵. (5.258)

For unsteady flow, we should use the following momentum equation: 𝜕^→𝜈1𝜕𝑡+∇2𝜈²−^→𝜈×∇×^→𝜈+∇Π=0, (5.259) where Π=∫(𝑑𝑃/𝜌) and ^→𝜈=𝑢₁^→𝑖+𝑢₂^→𝑗+𝑢₃^→𝑘.

From the equation of state, we can deduce the following equation: 𝜅Π=𝜌𝜅−1^𝜅−1exp𝑆−𝑆₀𝑚. (5.260) In 1955, Herivel first applied Hamilton’s principle to fluid mechanics and deduced the following functional [46]: 𝐽𝑥_𝑖=_𝑡CV12𝜌𝑢²_𝑖−𝜌𝐸𝜌,𝑆𝑑𝑉𝑑𝑡, (5.261) where 𝐸 is specific internal energy. CV is the control volume in Eulerian open space. Hamiltonian fluid mechanics becomes a branch of fluid mechanics and applied mathematics as well [47–56].

Applying the Lagrange multipliers to remove the constraint equations (5.257) and (5.256) yields 𝐽=_𝑡CV12𝜌𝑢²_𝑖−𝜌𝐸𝜌,𝑆−𝜌𝜂𝜕𝑆𝜕𝑡+𝑢_𝑗𝜕𝑆𝜕𝑥_𝑗+Φ𝜕𝜌+𝜕𝜕𝑡𝜌𝑢_𝑗𝜕𝑥_𝑗𝑑𝑉𝑑𝑡. (5.262) Making the above functional stationary with respect to 𝑢_𝑖, we have ^→𝜈=∇Φ+𝜂∇𝑆. (5.263) However, according to the Crocco equation [45], ∇×^→𝜈×^→𝜈=𝑇∇𝑆−∇𝑖^•, (5.264) where 𝑖^• is enthalpy, 𝑇 is temperature, and the homentropic flow may be a rotational one. Equation (5.263) implies, however, that homentropic flow (∇𝑆=0) must be a potential flow or irrotational flow (^→𝜈=∇Φ), confiliting with (5.264), and therefore, the functional, equation (5.261) or (5.262), is not reasonable, which was just pointed out by Lin [44].

According to Clebsch [57], an arbitrary velocity vector can be expressed as follows: ^→𝜈=∇Φ+𝜆_𝑖+∇𝜏_𝑖, (5.265) whereΦ, 𝜆_𝑖, and 𝜏_𝑖 are Clebsch variables.

Lin introduced three-additional constraint equations (Lin’s constraint equations) to the functional (5.262) [44]: 𝐷𝛼_𝑖(),𝐷𝑡=0𝑖=1,2,3 (5.266) where 𝛼_𝑖 is Lagrange variables in Lagrange space.

Introducing the Lagrange multipliers to remove Lin’s constraints, we have the following general functional: 𝐽=_𝑡𝑉12𝜌𝑢²_𝑖−𝜌𝐸𝜌,𝑆−𝜌𝜂𝜕𝑆𝜕𝑡+𝑢_𝑗𝜕𝑆𝜕𝑥_𝑗+Φ𝜕𝜌+𝜕𝜕𝑡𝜌𝑢_𝑗𝜕𝑥_𝑗−𝜌𝛽_𝑖𝜕𝛼_𝑖𝜕𝑡+𝑢_𝑗𝜕𝛼_𝑖𝜕𝑥_𝑗𝑑𝑉𝑑𝑡. (5.267) Making the above functional (5.267) stationary with respect to 𝑢_𝑖, we have ^→𝜈=∇Φ+𝜂∇𝑆+𝛽_𝑖∇𝛼_𝑖. (5.268) This expression says that a homentropic flow (∇𝑆=0) can also be an rotational flow, agreeing with (5.264). So the functional (5.267) with Lin’s constraints is acceptable for practical applications. But the essence of the Lin’s constraints has been bewildered for more than half century.

Now considering a fluid particle, and applying Hamilton’s principle to construct the following functional: 𝐽𝑥_𝑖=_𝑡12𝑥̇²_𝑖−Π𝑑𝑡, (5.269) where dot means the partial derivation to time alone pathline, that is, material derivation, its Eulerian equation can be easily deduced 𝑥̈_𝑖=−𝜕Π𝜕𝑥_𝑖1=−𝜌𝜕𝑃𝜕𝑥_𝑖 (5.270) that is, the momentum equation for a perfect flow, and it is easly to prove [46] that Π=𝐸+𝑃/𝜌, so Π represents enthalpy.

The involutory transformation is to convert Lagrange space into Euler space: 𝑥̇_𝑖=𝑢_𝑖(𝑖=1,2,3), (5.271) which are the pathline equations in the Eulerian form and are equivalent to (5.266) in Lagrangian form, that is, Lin’s constraint equations. So it is very clear that Lin’s constraints are actually the variational constraints introduced by the space exchange. The Hamilton principle is valid for Lagrange space, while all equations in fluid mechanics are derived through Euler space, and Lin’s constraints arise in the space change.

Using the Lagrange multipliers to remove the constraints, (5.271), we have 𝐽𝑥_𝑖,𝑢_𝑖,𝜆_𝑖=_𝑡12𝜈²−Π+𝜆_𝑖𝑥̇_𝑖−𝑢_𝑖𝑑𝑡. (5.272) Identifying the Lagrange multipliers 𝜆_𝑖=𝑢_𝑖 (5.273) results in the following variational principle: 𝐽𝑥_𝑖,𝑢_𝑖=_𝑡12𝜈²−𝑢_𝑖𝑥̇_𝑖+Π𝑑𝑡. (5.274) Applying Hamilton’s principle to a closed system, we have 𝐽𝑥_𝑖=_{𝑡𝑉(𝑡)}𝜌₀12𝑥̇²_𝑖−𝐸𝑑𝑉(𝑡)𝑑𝑡, (5.275) where 𝑉(𝑡) is the volume of the closed system in Lagrange space and keeps unchanged, and ∫_𝑉(𝑡)𝜌₀𝑑𝑉 is the total mass in the closed system.

Making the involutory transformation, we obtain the following functional via Lagrange multiplier method 𝐽𝑥_𝑖,𝑢_𝑖=_{𝑡𝑉(𝑡)}𝜌₀𝑢_𝑖𝑥̇_𝑖−12𝑢²_𝑖−𝐸𝑑𝑉(𝑡)𝑑𝑡. (5.276) It is easy to see that the functional (5.275) or (5.276) ceases to be a stationary variational principle if we transform the closed system in Lagrange space to an open system is Euler space ∫_𝑡∫_𝑉(𝑡)𝜌₀𝑑𝑉(𝑡)𝑑𝑡→∫_𝑡∫_CV𝜌𝑑𝑉𝑑𝑡 where a new variable 𝜌 in introduced.

In order to guarantee functional (5.276) to be still a stationary principle during space transformation, according to the semi-inverse method [14], let 𝜌 be an independent variable and include an unknown function 𝐹 in the functional: 𝐽𝑢_𝑖,𝜌=_𝑡CV𝜌12𝑢²_𝑖−𝐸+𝐹𝑢_𝑖,𝜌𝑑𝑉𝑑𝑡, (5.277) where 𝐹 is an unknown functional of 𝑥_𝑖,𝜌, and CV is the control volume. In the next section the identification of 𝐹 will be given in details.

Rotational equation with Clebsch variables can be expressed as follows: ^→𝜈=∇Φ+𝛼∇𝛽. (5.278) Assuming that 𝜕𝛼/𝜕𝑡=0 and 𝐷𝛼/𝐷𝑡=0, we can deduce the following energy equation: 𝜕Φ𝜕𝑡+𝛼𝜕𝛽+1𝜕𝑡2𝜈²+𝜅𝑃𝜅−1𝜌=𝑅(𝑡). (5.279) According to the semi-inverse method [17], we construct a trial functional in the form 𝐽^→𝜈,𝜌,Π,Φ,𝛼,𝛽=_𝑡CV𝜌12𝜈²−Π+𝐹𝑑𝑉𝑑𝑡, (5.280) where 𝐹 is an unknown function of ^→𝜈,𝜌,Π,Φ,𝛼,  and  𝛽.

Making the functional (5.280) stationary with respect to ^→𝜈, we have 𝜌^→𝜈+𝜕𝐹𝜕^→𝜈=0. (5.281) Assuming that (5.281) satisfies (5.278), we set 𝜕𝐹𝜕^→𝜈=−𝜌^→𝜈=−𝜌∇Φ+𝛼∇𝛽. (5.282) The unknown function can be identified as follows: 𝐹=−𝜌^→𝜈⋅∇Φ+𝛼∇𝛽+𝑓, (5.283) where 𝑓 is an unknown function of 𝜌,Π,Φ,𝛼,  and  𝛽. Such and trial functional can be updated as follows: 𝐽^→𝜈,𝜌,Π,Φ,𝛼,𝛽=_𝑡CV𝜌12𝜈²−^→𝜈⋅∇Φ+𝛼∇𝛽−Π+𝑓𝑑𝑉𝑑𝑡. (5.284) Now the stationary condition with respect to 𝛿Φ and continuity equation yield 𝜕𝑓𝜌𝜕Φ=−∇⋅^→𝜈=𝜕𝜌.𝜕𝑡 (5.285) From (5.285), 𝑓 can be determined in the form 𝑓=Φ𝜕𝜌𝜕𝑡+ℎ, (5.286) where ℎ is an unknown function of 𝜌,Π,𝛼,  and  𝛽. Substituting (5.286) to the trial functional and then taking by part to yield the following new trial functional: 𝐽^→𝑣,𝜌,Π,Φ,𝛼,𝛽=_𝑡CV𝜌12𝑣²−^→𝑣⋅∇Φ+𝛼∇𝛽−Π−𝜕Φ𝜕𝑡+ℎ𝑑𝑉𝑑𝑡. (5.287) Doing the same as before, 𝛿𝜌∶𝜕ℎ1𝜕𝜌=−2𝑣²+^→𝑣⋅∇Φ+𝛼∇𝛽+Π+𝜕Φ=1𝜕𝑡2𝑣²+Π+𝜕Φ𝜕𝑡=𝑅−𝛼𝜕𝛽.𝜕𝑡 (5.288) We identifying the unknown function ℎ to yield ℎ=𝜌𝑅−𝛼𝜕𝛽𝜕𝑡+𝑔, (5.289) where 𝑔 is an unknown function of Π,𝛼,  and  𝛽, doing the same way as before, 𝛿Π∶𝜕𝑔−𝜕Π=𝜌=exp𝑆−𝑆₀𝑚𝜅−1𝜅Π^𝑚. (5.290) Identifying the unknown function 𝑔 to get 𝑔=𝜅−1𝜅exp−𝑆−𝑆₀𝑚^𝑚𝜅−1𝜅Π^{𝜅/(𝜅−1)}, (5.291) we get the following generalized variational principle: 𝐽^→𝑣,𝜌,Π,Φ,𝛼,𝛽=_𝑡CV𝐿𝑑𝑉𝑑𝑡, (5.292) where 1𝐿=𝜌2𝜈²−^→𝜈⋅∇Φ+𝛼∇𝛽−Π−𝜕Φ𝜕𝑡−𝛼𝜕𝛽+𝜕𝑡+𝑅𝜅−1𝜅exp−𝑆−𝑆₀𝑚^𝑚𝜅−1𝜅Π^{𝜅/(𝜅−1)}. (5.293) The deduced generalized variational principle (GVP) with only 6 independent variables, (5.292), is unknown to the present time and more general and concise than any other known generalized variationals such as the correspondent GVP with Lin’s constraints which needs 11 independent variables (^→𝑣,𝑃,𝜌 and extra three Lin’s constraints together with three Lagrange multipliers, and one Lagrange multiplier for continuity equation, one Lagrange multiplier for isoentropy equation) and actually is of little utility solving the 3D unsteady rotational flow, contrarily the obtained GVP of (5.292) is so beautiful and concise that the theorem makes it possible to deal with the problem via VP-based FEM; and have wide applicability of the solution (𝐷𝑆/𝐷𝑡≠0). Making the above functional (5.292) stationary, we can get all the Eulerian equations and following additional two equations: 𝐷𝛼𝐷𝑡=0,𝐷𝛽=1𝐷𝑡𝜌𝜅−1𝜅^𝑚1𝜅Π^{𝜅/(𝜅−1)}exp−𝑆−𝑆₀𝜕𝑆,=1𝜕𝛼𝜌𝜅−1^𝜅−1exp−𝑆−𝑆₀𝑚𝜕𝑆=1𝜕𝛼𝜅𝑃𝜌𝜕𝑆.𝜕𝛼 (5.294) The first equation of (5.294) is actually Lin’s constraint equation. So via semi-inverse method, the Lin’s constraints equations can reduce to only one, which is the best way to deal with Lin’s constraints in the history.

Hereby we have successfully explained the phenomenon of Lin’s constraints via involutory transformation and deduced a generalized variational principle with only 6 independent variables via semi-inverse method, which makes it possible to use FEM to calculate the 3D unsteady compressible rotational flow. The corresponding generalized variational principle with Lin’s constraints is actually an approximate one (for Herrivel’s principle actually is a wrong principle), which, however, can find some application, especially in 2D problems, due to the fact that we can deduce all the Eulerian equations from the functional when making it stationary.

5.6. Variational Iteration Method

The variational iteration method [58–61] is an effective method for searching for various wave solutions including periodic solutions, solitons, and compacton solutions without linearization or weak nonlinearity assumptions, see, for example, [62–64].

5.6.1. The Lagrange Multiplier

The variational iteration method has been shown to solve a large class of nonlinear problems effectively, easily, and accurately with the approximations converging rapidly to accurate solutions.

To illustrate the basic idea of the technique, we consider the following general nonlinear system: 𝐿[𝑢(𝑡)]+𝑁[𝑢(𝑡)]=0, (5.295) where 𝐿 is a linear operator, and 𝑁 is a nonlinear operator.

The basic concept of the method is to construct a correction functional for the system (5.295), which reads 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡_𝑡0𝜆𝐿𝑢_𝑛(𝑠)+𝑁𝑢_𝑛(𝑠)𝑑𝑠, (5.296) where 𝜆 is a general Lagrange multiplier that can be identified optimally via variational theory, 𝑢_𝑛 is the 𝑛th approximate solution, and 𝑢_𝑛 denotes a restricted variation, that is, 𝛿𝑢_𝑛=0. To illustrate how restricted variation works in the variational iteration method, we consider a simple algebraic equation 𝑥²−3𝑥+2=0. (5.297) We rewrite (5.297) in the form 𝑥⋅𝑥−3𝑥+2=0, (5.298) where 𝑥 is called a restricted variable whose value is assumed to be known (5.6.1. the initial guess). Solving 𝑥 from (5.298) leads to the result 2𝑥=𝑥,3− (5.299) or in iteration form, 𝑥_𝑛+1=23−𝑥_𝑛. (5.300) After identifying the multiplier in (5.296), we have the following iteration algorithm: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡_𝑡0𝜆𝐿𝑢_𝑛(𝑠)+𝑁𝑢_𝑛(𝑠)𝑑𝑠. (5.301) Equation (5.301) is called the variational iteration algorithm-I.

Consider the following nonlinear equation of 𝑘th order: 𝑢^(𝑘)+𝑓𝑢,𝑢^′,𝑢^′′,…,𝑢^(𝑘)=0. (5.302) The variational iteration formulation is constructed as follows: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡_𝑡0𝜆𝑢_𝑛^(𝑘)+𝑓_𝑛𝑑𝑠, (5.303) where 𝛿𝑓_𝑛=0, 𝑓_𝑛=𝑓(𝑢_𝑛,𝑢^′_𝑛,𝑢_𝑛^′′,…). After identifying the multiplier, we have 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+(−1)^𝑛𝑡_𝑡01()!()𝑛−1𝑠−𝑡^𝑛−1𝑢_𝑛^(𝑘)(𝑠)+𝑓_𝑛𝑑𝑠. (5.304) This formulation is generally called the variational iteration algorithm-I [11, 61].

The main merit of this iteration formula lies in the fact that 𝑢₀(𝑡), the initial solution, can be freely chosen, with even unknown parameters contained. However, some repeated and unnecessary iterations are involved in this iteration algorithm at each step.

For initial value problems, we can begin with 𝑢₀(𝑡)=𝑢(0)+𝑡𝑢^′(0)+1𝑡2!²𝑢^′′(0)1+⋯𝑡𝑘!^𝑘𝑢^(𝑘)(0). (5.305) This leads to a series solution converging to the exact solution.

For boundary value problems, the initial guess can be expressed in the form 𝑢₀(𝑡)=𝑎₁𝑔₁(𝑡)+𝑎₂𝑔₂(𝑡)+⋯+𝑎_𝑘𝑔_𝑘(𝑡), (5.306) where 𝑔_𝑘(𝑡) are known functions, and 𝑎_𝑘 are unknowns to be further determined after a few iterations by the boundary conditions.

After identifying the Lagrange multiplier 𝜆 in (5.296), we can construct the iteration formula 𝑢_𝑛+1(𝑡)=𝑢₀(𝑡)+(−1)^𝑛𝑡_𝑡01()!()𝑛−1𝑠−𝑡^𝑛−1𝑓_𝑛𝑑𝑠, (5.307) instead of the iteration algorithm (5.301). We call (5.307) the variational iteration algorithm-II [11, 61].

Note: 𝑢₀ must satisfy the initial/boundary conditions. This is the main shortcoming of the algorithm.

From (5.307), we obtain the following variational iteration algorithm-III 𝑢_𝑛+2(𝑡)=𝑢_𝑛+1(𝑡)+^𝑡_𝑡0𝜆𝑓_𝑛+1(𝑠)−𝑓_𝑛(𝑠)𝑑𝑠. (5.308) One common property of both the variational iteration algorithm-I and the variational iteration algorithm-III is the allowed dependence of the initial guess on unknown parameters whose values could be identified after a few iterations by using the initial/boundary conditions. the variational iteration algorithm-III, in particular, is highly suitable for boundary value problems of high orders.

5.6.2. Laplace Transform for Identification of the Lagrange Multiplier [65]

We can also apply Laplace transform to identify the Lagrange multiplier. By Laplace transform, we have 𝑠^𝑘𝑈(𝑠)+ℓ𝑓𝑢,𝑢^′,…,𝑢^(𝑘)=0, (5.309) or 𝑈(𝑠)=−ℓ𝑓𝑢,𝑢^′,…,𝑢^(𝑘)𝑠^𝑘. (5.310) The inverse Laplace transform reads 𝑢(𝑡)=(−1)^𝑘𝑡₀𝜂−𝑡^𝑘−1()!𝑓𝑢𝑘−1^𝑛𝜂,𝑢^′_𝑛𝜂,…,𝑢_𝑛^(𝑘)𝜂𝑑𝜂. (5.311) Hence, the following iteration algorithm is derived: 𝑢_𝑛+1(𝑡)=𝑢₀(𝑡)+(−1)^𝑘𝑡₀𝜂−𝑡^𝑘−1()!𝑓𝑢𝑘−1^𝑛𝜂,𝑢^′_𝑛𝜂,…,𝑢_𝑛^(𝑘)𝜂𝑑𝜂. (5.312)

5.6.3. Variational Iteration Method for Solitary Solutions

Here is an incomplete list for variational iteration formulas for various differential equations [8, 9]: 𝑢^′+𝑓𝑢,𝑢^′𝑢=0,_𝑛+1(𝑡)=𝑢₀(𝑡)−^𝑡₀𝑓𝑢_𝑛,𝑢^′_𝑛𝑢𝑑𝑠,_′+𝛼𝑢+𝑓𝑢,𝑢^′𝑢=0,_𝑛+1(𝑡)=𝑢₀(𝑡)−^𝑡₀𝑒^{𝛼(𝑠−𝑡)}𝑓𝑢_𝑛,𝑢^′_𝑛𝑢𝑑𝑠,_′′+𝑓𝑢,𝑢^′,𝑢^′′𝑢=0,_𝑛+1(𝑡)=𝑢₀(𝑡)+^𝑡₀(𝑠−𝑡)𝑓𝑢_𝑛,𝑢^′_𝑛,𝑢_𝑛^′′𝑢𝑑𝑠,_′′+𝜔²𝑢+𝑓𝑢,𝑢^′,𝑢^′′𝑢=0,_𝑛+1(𝑡)=𝑢₀(𝑡)+1𝜔^𝑡₀sin𝜔(𝑠−𝑡)𝑓𝑢_𝑛,𝑢^′_𝑛,𝑢_𝑛^′′𝑢𝑑𝑠,_′′−𝛼²𝑢+𝑓𝑢,𝑢^′,𝑢^′′𝑢=0,_𝑛+1(𝑡)=𝑢₀(𝑡)+^𝑡₀1𝑒2𝛼^{𝛼(𝑠−𝑡)}−𝑒^{𝛼(𝑡−𝑠)}𝑓𝑢_𝑛,𝑢^′_𝑛,𝑢_𝑛^′′𝑢𝑑𝑠,_′′′+𝑓𝑢,𝑢^′,𝑢^′′,𝑢^′′′𝑢=0,_𝑛+1(𝑡)=𝑢₀(𝑡)−^𝑡₀12(𝑠−𝑡)²𝑓𝑢_𝑛,𝑢^′_𝑛,𝑢_𝑛^′′,𝑢_𝑛^′′′𝑢𝑑𝑠,₍₄₎+𝑓𝑢,𝑢^′,𝑢^′′,𝑢^′′′,𝑢⁽⁴⁾𝑢=0,_𝑛+1(𝑡)=𝑢₀(𝑡)+^𝑡₀16(𝑠−𝑡)³𝑓𝑢_𝑛,𝑢^′_𝑛,𝑢_𝑛^′′,𝑢_𝑛^′′′,𝑢_𝑛⁽⁴⁾𝑢𝑑𝑠,_(𝑘)+𝑓𝑢,𝑢^′,𝑢^′′,…,𝑢^(𝑘)𝑢=0,_𝑛+1(𝑡)=𝑢₀(𝑡)+(−1)^𝑛𝑡₀1()!()𝑛−1𝑠−𝑡^𝑛−1𝑓𝑢_𝑛,𝑢^′_𝑛,𝑢_𝑛^′′,…,𝑢_𝑛^(𝑘)𝑑𝑠. (5.313) The main feature of the method is that the initial solution can be chosen with some unknown parameters in the form of the searched solution. For example, for solitons, we begin with 𝑢₀=𝑝sech²𝑞𝜉,𝜉=𝑥+𝑐𝑡, (5.314) where 𝑝 and 𝑞 are the unknown parameters to be further identified after one or few iterations. For a more general form for solitary solutions, we assume that 𝑢₀(𝜉)=∑^𝑚_𝑖=−𝑛𝑐_𝑖𝑒^𝑖𝜉∑^𝑞_𝑖=−𝑝𝑏_𝑖𝑒^𝑖𝜉,𝜉=𝑥+𝑐𝑡, (5.315) where 𝑐_𝑖 and 𝑏_𝑖 are constants to be further determined.

For discontinuous solitons, we can assume, for example, the following form: 𝑢(𝜉)=𝑝exp−𝑞𝜉, (5.316) or 𝑢₀(𝜉)=∑^𝑚_𝑖=−𝑛𝑐_𝑖𝑒^𝑖|𝜉|∑^𝑞_𝑖=−𝑝𝑏_𝑖𝑒^𝑖|𝜉|, (5.317) where 𝑝 and 𝑞 are the unknown parameters to be further identified.

For compacton-like solution, we assume that the solution has the form 𝑢₀(𝑥,𝑡)=𝑎sin²(𝑘𝑥+𝜔𝑡)𝑏+𝑐sin²(),𝑘𝑥+𝜔𝑡 (5.318) where 𝑎,𝑏,𝑘, and 𝜔 are unknown constants further to be determined 𝑢₀()=∑𝑥,𝑡^𝑖𝑎_𝑖sin²𝑘_𝑖𝑥+𝜔_𝑖𝑡∑_𝑗𝑏_𝑗+𝑐_𝑗sin²𝑘_𝑗𝑥+𝜔_𝑗𝑡. (5.319)

As an illustrating example, we consider the following modified KdV equation 𝑢_𝑡+𝑢²𝑢_𝑥+𝑢_𝑥𝑥𝑥=0. (5.320) Its iteration formulation can be constructed as follows: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)−^𝑡₀𝑢_𝑛𝑡+𝑢²_𝑛𝑢_𝑛𝑥+𝑢_{𝑛𝑥𝑥𝑥}𝑑𝑡. (5.321) To search for its compacton-like solution, we assume that the solution has the form 𝑢₀(𝑥,𝑡)=𝑎sin²(𝑘𝑥+𝑤𝑡)𝑏+𝑐sin²(),𝑘𝑥+𝑤𝑡 (5.322) where 𝑎, 𝑏, 𝑘,and 𝑤 are unknown constants further to be determined.

Substituting (5.322) into (5.321), we can calculate 𝑢₁ and 𝑢₂ with ease. In order to identify the constants in the initial solution, we can set 𝜕^𝑘𝜕𝑡^𝑘𝑢_𝑛()=𝜕𝑥,𝑡^𝑘𝜕𝑡^𝑘𝑢_𝑛+1(𝑥,𝑡), (5.323) setting 𝜕𝑢𝜕𝑡⁰()=𝜕𝑥,𝑡𝑢𝜕𝑡¹(𝑥,𝑡). (5.324) By a simple calculation (see [60] for details), the compacton-like solution is obtained, which reads 𝑢()=√𝑥,𝑡±22𝑘𝑐sin²𝑘𝑥−4𝑘³𝑡−(3/2)𝑐+𝑐sin²𝑘𝑥−4𝑘³𝑡, (5.325) or 𝑢()=√𝑥,𝑡±42𝑘sin²𝑘𝑥−4𝑘³𝑡−3+2sin²𝑘𝑥−4𝑘³𝑡. (5.326) We can also begin with a more general initial solution in the form 𝑢₀()1𝑥,𝑡=𝑎+(),𝑐+𝑑cos𝑘𝑥+𝑤𝑡 (5.327) where 𝑎, 𝑏, 𝑐, 𝑘, and 𝑤 are unknown constants.

Setting 𝜕𝑢₀/𝜕𝑡=𝜕𝑢₁/𝜕𝑡, we, therefore, obtain the following new compacton-like solution: 𝑢()1𝑥,𝑡=𝑎+−𝑎/3𝑘²±1/6𝑘²4𝑎²−6𝑘²cos𝑘𝑥+𝑘³−𝑘𝑎²𝑡. (5.328) If we choose 𝑘=𝑖𝐾, where 𝐾 is a constant, then (5.328) becomes 𝑢()1𝑥,𝑡=𝑎+𝑎/3𝐾²±1/6𝐾²4𝑎²+6𝐾²cosh𝐾𝑥+−𝐾³−𝐾𝑎²𝑡1=𝑎+𝑎/3𝐾²±1/12𝐾²4𝑎²+6𝐾²exp𝐾𝑥+−𝐾³−𝐾𝑎²𝑡+exp−𝐾𝑥−−𝐾³−𝐾𝑎²𝑡, (5.329) which is a solitary solution. It is interesting that we can convert the compacton-like solution to solitary solution.

The initial solution (trial function) can be also constructed in a solitary form. Now we begin with 𝑢₀1=𝑎+()().𝑏+𝑐exp𝑘𝑥+𝑤𝑡+𝑑exp−𝑘𝑥−𝑤𝑡 (5.330) By the same manipulation as illustrated above, we set 𝜕𝑢₀/𝜕𝑡=𝜕𝑢₁/𝜕𝑡.

So we obtain the following needed solitary solution: 𝑢()1𝑥,𝑡=𝑎+𝑎/3𝑘²+𝑐exp𝑘𝑥+−𝑘³−𝑘𝑎²𝑡+3𝑘²+2𝑎²/72𝑘⁴(1/𝑐)exp−𝑘𝑥−−𝑘³−𝑘𝑎²𝑡. (5.331) Hereby 𝑎 and 𝑐 are free parameters.

It is also interesting to note that the solitary solution can be converted into a compacton-like solution if we choose 𝑘=𝑖𝐾, where 𝐾 is a constant. If 𝑘=𝑖𝐾, (5.331) becomes 𝑢()1𝑥,𝑡=𝑎+−𝑎/3𝐾²+(𝑐+𝒜)cos𝐾𝑥+𝐾³−𝐾𝑎²𝑡+(𝑐−𝒜)𝑖sin𝐾𝑥+𝐾³−𝐾𝑎²𝑡, (5.332) where  𝒜 donates (−3𝐾²+2𝑎²)/72𝐾⁴𝑐. In the above derivation, we use the relations ()()(),()()().exp𝑖𝑘𝑥+𝑖𝑤𝑡=cos𝑘𝑥+𝑤𝑡+𝑖sin𝑘𝑥+𝑤𝑡exp−𝑖𝑘𝑥−𝑖𝑤𝑡=cos𝑘𝑥+𝑤𝑡−𝑖sin𝑘𝑥+𝑤𝑡 (5.333) In order to convert (5.332) into a compact form, the last term in denominator must be vanished, which requires 𝑐−−3𝐾²+2𝑎²72𝐾⁴𝑐=0. (5.334)

Solving 𝑐 from (5.334) results in 1𝑐=±12−6𝐾²+4𝑎²𝐾². (5.335)

Substituting (5.335) into (5.332) yields a compacton-like solution 𝑢()1𝑥,𝑡=𝑎+−𝑎/3𝐾²±1/6𝐾²4𝑎²−6𝐾²cos𝐾𝑥+𝐾³−𝑘𝑎²𝑡. (5.336)

5.7. Homotopy Perturbation Method

5.7.1. Homotopy Technology

Before proceeding the method, we give an interesting application of the homotopy technology to an asymptotic match of Rayleigh grain size distribution and Hillert grain size distribution. Grain growth is a well-known phenomenon in the evolution of crystalline microstructures. A stochastic continuity or Fokker-Planck continuity equation was proposed to accurately predict properties of grain growth [66]. The equation can be expressed in the form [66] 𝜀𝑑²𝑓𝑑𝑥²+𝛼+𝜀𝑥−𝛼+2𝑥𝑑𝑓+𝑑𝑥6−𝛼+𝜀𝑥²𝑓=0. (5.337) Let us consider two limiting cases where the driving force is either due to diffusion (Rayleigh) or to the drift velocity (Hillert). In the first case, the Rayleigh grain size distribution occurs when 𝛼=0 and 𝜀≠0 [66]: 𝑓_𝑟(𝑥)=𝜋2−𝜋𝑥exp4𝑥². (5.338) The other limiting case, 𝛼=8 and 𝜀=0, leads to Hillert grain size distribution [66]: 𝑓_ℎ(𝑥)=8exp(2𝑥/(𝑥−2))(𝑥−2)². (5.339) The actual solution to (5.337) falls between (5.338) and (5.339); in the work of Pande and Cooper [66], a very good but approximate solution was given (see (18) of Pande and Cooper’s publication [66]). Hereby we will suggest a simple matching technology to bridge the limiting cases.

An asymptotic match of (5.338) and (5.339) can be expressed in the form [67] 𝑓(𝑥,𝛼,𝜀)=𝐶(𝛼,𝜀)𝑓_𝑟(𝑥)+𝐷(𝛼,𝜀)𝑓_ℎ(𝑥)()𝜋=𝐶𝛼,𝜀2−𝜋𝑥exp4𝑥²()((+𝐷𝛼,𝜀8exp2𝑥/𝑥−2))(𝑥−2)², (5.340) where 𝐶 and 𝐷 are matching parameters.

Equation (5.340) should turn out to be exactly (5.338) and (5.339), respectively, for the above two limiting cases. This requires 𝐶()()𝐶()()0,𝜀=1,𝐷0,𝜀=0,8,0=0,𝐷8,0=1. (5.341) We can freely choose 𝐶 and 𝐷 satisfying (5.341) and the following normalization condition 𝐶+𝐷=1. (5.342) Hereby we give simple expressions for 𝐶 and 𝐷 in the forms 𝐶()1𝛼,𝜀=1−8𝛼(),𝐷()=11−𝜀𝛼,𝜀8𝛼(1−𝜀). (5.343) We, therefore, obtain the following approximate solution to (5.337) [67]: 𝑓(𝑥)=𝜋211−8𝛼(−𝜋1−𝜀)𝑥exp4𝑥²+18𝛼()((1−𝜀8exp2𝑥/𝑥−2))(𝑥−2)². (5.344) This equation is simpler than that given by Pande and Cooper [66]. There are many alternative ways for determination of 𝐶 and 𝐷 satisfying (5.341) and (5.342).

In this short paper we suggest an asymptotic match to bridge the two limited cases, and the obtained result is valid for the whole case. The asymptotic match is a simple and useful mathematical tool in engineering for reliable treatment of a nonlinear problem whose analytical solution can be easily obtained for two limited cases (e.g., 𝜀→0 and 𝜀→∞, resp.) by some analytical methods.

Consider another example of the relativistic oscillator 𝑢^′′+𝑢√1+𝑢²=0, (5.345) with initial conditions 𝑢(0)=𝐴,𝑢^′(0)=0.

It is easy to obtain the following approximate frequency [68]: 2√𝜔=2𝜋41+𝜋²⋅𝐴^2−1/41,for𝐴≫1,𝜔=1+2𝐴^2−1/4,for𝐴≪1. (5.346)

In order to match both the cases 𝐴→0 and 𝐴→∞, we construct the following homotopy [68]: 𝜔=𝑒^−𝛼𝐴𝑓(𝐴)+1−𝑒^−𝛼𝐴𝑔(𝐴), (5.347) where 𝑓(𝐴)=(2√2/𝜋)(1+(4/𝜋²)⋅𝐴²)^−1/4, 𝑔(𝐴)=(1+(1/2)𝐴²)^−1/4, and 𝛼 is a free parameter. Now considering the case when 𝐴=1, we have exact frequency, which is 𝜔=0.8736. From this relationship, we can identify 𝛼 as follows: 𝛼=0.4962. (5.348) Finally, we obtain the following result: 𝜔=𝑒^{−0.4962𝐴}⋅2√2𝜋41+𝜋²⋅𝐴^2−1/4+1−𝑒^−0.496𝐴⋅11+2𝐴^2−1/4. (5.349)

5.7.2. Homotopy Perturbation Method with an Auxiliary Term [69]

The two most important steps in application of the homotopy perturbation method [69–72] are to construct a suitable homotopy equation and to choose a suitable initial guess. The homotopy equation should be constructed such that, when the homotopy parameter is zero, it can approximately describe the solution property, and the initial solution can be chosen with an unknown parameter, which is determined after one or two iterations. This paper suggests an alternative approach for construction of the homotopy equation with an auxiliary term Duffing equation is used as example to illustrate the solution procedure.

Consider a general nonlinear equation 𝐿𝑢+𝑁𝑢=0, (5.350) where 𝐿 and 𝑁 are, respectively, the linear operator and nonlinear operator.

The first step for the method is to construct a homotopy equation in the form 𝐿𝑢+𝑝𝐿𝑢−𝐿𝑢+𝑁𝑢=0, (5.351) where 𝐿 is a linear operator with a possible unknown constant, and 𝐿𝑢=0 can best describe the solution property. The embedding parameter 𝑝 monotonically increases from zero to unit as the trivial problem, 𝐿𝑢=0, is continuously deformed to the original one.

For example, consider a nonlinear oscillator [9] 𝑢^′′+𝜀𝑢³=0,𝑢(0)=𝐴,𝑢^′(0)=0. (5.352) For an oscillator, we can use sine or cosine function. We assume that the approximate solution of (5.352) is 𝑢(𝑡)=𝐴cos𝜔𝑡, (5.353) where 𝜔 is the frequency to be determined later. We, therefore, can choose 𝐿𝑢=𝑢̈+𝜔²𝑢. (5.354) Accordingly, we can construct a homotopy equation in the form 𝑢̈+𝜔²𝑢+𝑝𝑢³−𝜔²𝑢=0. (5.355) When 𝑝=0, we have 𝑢̈+𝜔²𝑢=0,𝑢(0)=𝐴,𝑢^′(0)=0, (5.356) which describes the basic solution property of the original nonlinear equation (5.352).

When 𝑝=1, (5.355) becomes the original one. So the solution procedure is to deform from the initial solution, (5.353), to the real one. Due to one unknown parameter in the initial solution, only one iteration is enough. For detailed solution procedure, refer to [9].

If a higher-order approximate solution is searched for, we can construct a homotopy equation in the form 𝑢̈+0⋅𝑢+𝑝𝑢³=0. (5.357) We expand the solution and the coefficient, zero, of the linear term into a series in 𝑝: 𝑢=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯,0=𝜔²+𝑝𝑎₁+𝑝²𝑎₂+⋯, (5.358)(5.359) where the unknown constant, 𝑎_𝑖, is determined in the (𝑖+1)th iteration. The solution procedure is given in [9].

In this paper, we suggest an alternative approach for construction of homotopy equation, which is 𝐿𝑢+𝑝𝐿𝑢−𝐿𝑢+𝑁𝑢+𝛼𝑝1−𝑝𝑢=0, (5.360) where 𝛼 is an auxiliary parameter. When 𝛼=0, (5.360) turns out to be that of the classical one, expressed in (5.351). The auxiliary term, 𝛼𝑝(1−𝑝)𝑢, vanishes completely when 𝑝=0 or 𝑝=1, so the auxiliary term will affect neither the initial solution (𝑝=0), nor the real solution (𝑝=1).

To illustrate the solution procedure, we consider a nonlinear oscillator in the form 𝑑²𝑢𝑑𝑡²+𝑏𝑢+𝑐𝑢³=0,𝑢(0)=𝐴,𝑢^′(0)=0, (5.361) where 𝑏 and 𝑐 are positive constants.

Equation (5.361) admits a periodic solution, and the linearized equation of (5.361) is 𝑢^′′+𝜔²𝑢=0,𝑢(0)=𝐴,𝑢^′(0)=0, (5.362) where 𝜔 is the frequency of (5.361).

We construct the following homotopy equation with an auxiliary term [69]: 𝑢^′′+𝜔²𝑢+𝑝𝑏−𝜔²𝑢+𝑐𝑢³+𝛼𝑝1−𝑝𝑢=0. (5.363) Assume that the solution can be expressed in a power series in 𝑝 as shown in (5.358). Substituting (5.358) into (5.363), and processing as the standard perturbation method, we have 𝑢₀^′′+𝜔²𝑢₀=0,𝑢₀(0)=𝐴,𝑢^′₀(0)𝑢=0,_1′′+𝜔²𝑢₁+𝑏−𝜔²𝑢₀+𝑐𝑢³₀+𝛼𝑢₀=0,𝑢₁(0)=0,𝑢^′₁(0)𝑢=0,_2′′+𝜔²𝑢₂+𝑏−𝜔²𝑢₁+3𝑐𝑢²₀𝑢₁+𝛼𝑢₁−𝑢₀=0, (5.364)(5.365)(5.366) with initial conditions _𝑖=0𝑢_𝑖(0)=𝐴,_𝑖=0𝑢^′_𝑖(0)=0. (5.367) Solving (5.364), we have 𝑢₀=𝐴cos𝜔𝑡. (5.368)

Substituting 𝑢₀ into (5.365) results in 𝑢₁^′′+𝜔²𝑢₁+𝐴𝛼+𝑏−𝜔²+34𝑐𝐴²1cos𝜔𝑡+4𝑐𝐴³cos3𝜔𝑡=0. (5.369)

Eliminating the secular term needs 𝛼+𝑏−𝜔²+34𝑐𝐴²=0. (5.370) A special solution of (5.369) is 𝑢₁=−𝑐𝐴³32𝜔²cos3𝜔𝑡. (5.371) If only a first-order approximate solution is enough, we just set 𝛼=0, and this results in 𝜔=3𝑏+4𝑐𝐴². (5.372) The accuracy reaches 7.6% even for the case 𝑐𝐴²→∞.

The solution procedure continues by submitting 𝑢₁ into (5.366), and after some simple calculation, we obtain 𝑢₂^′′+𝜔²𝑢₂−𝛼𝐴+3𝑐²𝐴⁵128𝜔²cos𝜔𝑡−𝑐𝐴³𝑏−𝜔²32𝜔²+3𝑐²𝐴⁵64𝜔²+𝛼𝑐𝐴³32𝜔²−cos3𝜔𝑡3𝑐²𝐴⁵128𝜔²cos5𝜔=0. (5.373)

No secular term in 𝑢₂ requires 𝛼𝐴+3𝑐²𝐴⁵128𝜔²=0. (5.374)

Solving (5.370) and (5.374) simultaneously, we obtain 𝜔=𝑏+(3/4)𝑐𝐴²+𝑏+(3/4)𝑐𝐴²²+(3/32)𝑐²𝐴⁴2, (5.375) and the approximate solution is 𝑢(𝑡)=𝐴cos𝜔𝑡, where 𝜔 is given in (5.375).

In order to compare with the perturbation solution and the exact solution, we set 𝑏=1. In case 𝑐≪1, (5.375) agrees with that obtained by the classical perturbation method; when 𝑐→∞, we have lim_𝑐→∞𝜔=3/4+(3/4)²+3/322𝑐𝐴²=0.8832𝑐𝐴². (5.376) The exact period reads 𝑇_ex=41+𝑐𝐴^20𝜋/2𝑑𝑥1−𝑘sin²𝑥, (5.377) where 𝑘=𝑐𝐴²/2(1+𝑐𝐴²).

In case 𝑐→∞, we have lim_𝑐→∞𝑇_ex=6.743𝑐𝐴²,𝜔_ex≈2𝜋6.743𝑐𝐴². (5.378)(5.379) Comparing between (5.376) and (5.378), we find that the accuracy reaches 5.5%, while accuracy of the first-order approximate frequency is 7.6%.

If a higher-order approximate solution is needed, we rewrite the homotopy equation in the form 𝑢^′′+𝜔²𝑢+𝑝𝑏−𝜔²𝑢+𝑐𝑢³+1⋅𝑝1−𝑝𝑢=0. (5.380) The coefficient, 1, in the auxiliary term is also expanded in a series in 𝑝 in the form 1=𝛼₀+𝑝𝛼₁+𝑝²𝛼₂+⋯, (5.381) where 𝛼_𝑖 is identified in the (𝑖+2)th iteration. The solution procedure is similar to that illustrated above.

Generally, the homotopy equation can be constructed in the form 𝐿𝑢+𝑝𝐿𝑢−𝐿𝑢+𝑁𝑢+1⋅𝑓𝑝𝑔𝑝ℎ𝑢,𝑢^′,𝑢^′′,…=0, (5.382) where 𝑓 and 𝑔 are functions of 𝑝, satisfying 𝑓(0)=0 and 𝑔(1)=0, and ℎ can be generally expressed in the form ℎ=𝑢+𝛽₁𝑢^′+𝛽₂𝑢^′′+⋯. (5.383)

5.7.3. Homotopy Perturbation Method for Solitary Solutions

Homotopy perturbation method [69–72] provides a simple mathematical tool for searching for soliton solutions without any small perturbation. Considering the following nonlinear equation: 𝜕𝑢𝜕𝑡+𝑎𝑢𝜕𝑢𝜕𝜕𝑥+𝑏³𝑢𝜕𝑥³+𝑁(𝑢)=0,𝑎>0,𝑏>0, (5.384) we can construct a homotopy in the form 1−𝑝𝜕𝑢𝜕𝑡+6𝑢𝜕𝑢+𝜕𝜕𝑥³𝑢𝜕𝑥³+𝑝𝜕𝑢𝜕𝑡+𝑎𝑢𝜕𝑢𝜕𝜕𝑥+𝑏³𝑢𝜕𝑥³+𝑁(𝑢)=0. (5.385) When 𝑝=0, we have 𝜕𝑢𝜕𝑡+6𝑢𝜕𝑢+𝜕𝜕𝑥³𝑢𝜕𝑥³=0, (5.386) a well-known KdV equation, whose soliton solution is known. When 𝑝=1, (5.385) turns out to be the original equation. According to the homotopy perturbation method, we assume that 𝑢=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯. (5.387) Substituting (5.387) into (5.385), and proceeding the same process as the traditional perturbation method does, we can easily solve 𝑢₀, 𝑢₁, and other components. The solution can be expressed finally in the form 𝑢=𝑢₀+𝑢₁+𝑢₂+⋯. (5.388) The homotopy perturbation method always stops before the second iteration, so the solution can be expressed as 𝑢=𝑢₀+𝑢₁, (5.389) for most cases.

For traveling wave solutions, we can use the following transformation: 𝑢(𝑡,𝑥)=𝑢(𝜉),𝜉=𝑥−𝐷𝑡. (5.390) Equation (5.384) becomes −𝐷𝑢_𝜉+𝑎𝑢𝑢_𝜉+𝑏𝑢_𝜉𝜉𝜉+𝑁(𝑢)=0. (5.391) Integrating the equation with respect to 𝜉, we have 1−𝐷𝑢+2𝑎𝑢²+𝑏𝑢_𝜉𝜉+𝐹(𝑢)=0, (5.392) where 𝜕𝐹/𝜕𝑢=𝑁(𝑢)+𝐶, and 𝐶 is an integral constant.

We begin with a soliton in the form 𝑢₀=1𝑒^−𝛼𝜉+𝑒^𝛼𝜉, (5.393) which is the solution of the following equation: 𝑢_𝜉𝜉−𝛼²𝑢+8𝛼²𝑢³=0. (5.394) Accordingly, we construct the following homotopy equation: 𝑢_𝜉𝜉−𝐷𝑏𝑢+0⋅𝑢³𝑎+𝑝𝑢2𝑏²+1𝑏𝐹(𝑢)=0. (5.395) The solution and the parameters 𝐷/𝑏 and 0 are expanded in the forms 𝑢=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂𝐷+⋯,𝑏=𝛼²+𝑝𝛼₁+𝑝²𝛼₂+⋯,0=8𝛼²+𝑝𝛽₁+𝑝²𝛽₂+⋯. (5.396) Substituting (5.396) into (5.395), and proceeding a similar way as the perturbation method, we have 𝑢_0𝜉𝜉−𝛼²𝑢₀+8𝛼²𝑢³₀𝑢=0,_1𝜉𝜉−𝛼²𝑢₁+𝛼₁𝑢₀+24𝛼²𝑢²₀𝑢₁+𝛽₁𝑢³₀+𝑎𝑢2𝑏²⁰+1𝑏𝐹𝑢₀=0. (5.397) As an example, we consider the following equation: 𝑢_𝑡+𝑢𝑢_𝑥+𝜂𝑢_𝑥𝑥𝑥=0. (5.398) By the transformation 𝑢(𝑡,𝑥)=𝑢(𝜉), 𝜉=𝑥−𝐷𝑡, we have −𝐷𝑢_𝜉+𝑢𝑢_𝜉+𝜂𝑢_𝜉𝜉𝜉=0, (5.399) or 𝑢^′′−𝐷𝜂1𝑢+𝑢2𝜂²+𝐶=0. (5.400) In order to make the solution process simple, we construct a homotopy equation in the form 𝑢^′′−𝐷𝜂1𝑢+𝑝𝑢2𝜂²+𝐶=0. (5.401) Assume that the solution and the parameter 𝐷/𝜂 can be expanded in the forms 𝑢=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂𝐷+⋅⋅⋅,𝜂=𝜆²₀+𝑝𝜆₁+𝑝²𝜆₂+⋅⋅⋅. (5.402) Substituting (5.402) into (5.401), proceeding as the classical perturbation method does, we have 𝑢₀^′′−𝜆²₀𝑢₀𝑢=0,_1′′−𝜆²₀𝑢₁−𝜆₁𝑢₀+1𝑢2𝜂²⁰𝑢+𝐶=0,_2′′−𝜆²₀𝑢₂−𝜆₂𝑢₀−𝜆₁𝑢₁+1𝜂𝑢₀𝑢₁=0. (5.403) The solution of the first equation of (5.403) is 𝑢₀=𝐴𝑒^𝜆0𝜉+𝐵𝑒^−𝜆0𝜉. (5.404) Substituting (5.404) into the second equation of (5.403) we have 𝑢₁^′′−𝜆²₀𝑢₁=𝜆₁𝐴𝑒^𝜆0𝜉+𝐵𝑒^−𝜆0𝜉−12𝜂𝐴𝑒^𝜆0𝜉+𝐵𝑒^{−𝜆0𝜉2}−𝐶=𝜆₁𝐴𝑒^𝜆0𝜉+𝐵𝑒^−𝜆0𝜉−1𝐴2𝜂²𝑒^2𝜆0𝜉+𝐵²𝑒^{−2𝜆0𝜉}−𝐴𝐵𝜂−𝐶. (5.405) We set 𝜆₁=0,𝐴𝐵=−𝜂𝐶 (5.406) in (5.405) to avoid the secular-like terms 𝜉^𝑘𝑒^𝜆0𝜉, 𝜉^𝑘𝑒^−𝜆0𝜉 or 𝜉^𝑘 (𝑘≥0), and solving the resultant equation, we have the solution for 𝑢₁: 𝑢₁1=−𝐴6𝜂²𝑒^2𝜆0𝜉+𝐵²𝑒^{−2𝜆0𝜉}. (5.407) Substituting 𝑢₀ and 𝑢₁ into the third equation of (5.403) results in 𝑢₂^′′−𝜆²₀𝑢₂=𝜆₂𝐴𝑒^𝜆0𝜉+𝐵𝑒^−𝜆0𝜉+16𝜂²𝐴𝑒^𝜆0𝜉+𝐵𝑒^−𝜆0𝜉𝐴²𝑒^2𝜆0𝜉+𝐵²𝑒^{−2𝜆0𝜉}=𝐴𝜆₂+𝐴𝐵6𝜂²𝑒^𝜆0𝜉+𝐵𝜆₂+𝐴𝐵6𝜂²𝑒^−𝜆0𝜉+16𝜂²𝐴³𝑒^3𝜆0𝜉+𝐵³𝑒^{−3𝜆0𝜉}. (5.408) Similarly, we set 𝜆₂=−𝐴𝐵6𝜂². (5.409) The solution for 𝑢₂ is 𝑢₂=148𝜂²𝐴³𝑒^3𝜆0𝜉+𝐵³𝑒^{−3𝜆0𝜉}. (5.410) If the second order approximate solution is enough, then we have 𝑢=𝑢₀+𝑢₁+𝑢₂=𝐴𝑒^𝜆0𝜉+𝐵𝑒^−𝜆0𝜉−1𝐴6𝜂²𝑒^2𝜆0𝜉+𝐵²𝑒^{−2𝜆0𝜉}+148𝜂²𝐴³𝑒^3𝜆0𝜉+𝐵³𝑒^{−3𝜆0𝜉}. (5.411) Solving 𝜆₀ from the second equation of (5.402), we have 𝜆₀=𝐷𝜂−𝑝𝜆₁+𝑝²𝜆₂+⋅⋅⋅_{𝑝=1,𝜆𝑖=0(𝑖≥3)}=𝐷𝜂+𝐴𝐵6𝜂². (5.412)

Using in the [1,2] exponential Padé approximant [62], we have 𝑢=𝐴𝑒^𝜆0𝜉+𝐵𝑒^−𝜆0𝜉1+𝑎𝑒^𝜆0𝜉+𝑏𝑒^−𝜆0𝜉+𝑐𝑒^2𝜆0𝜉+𝑑𝑒^{−2𝜆0𝜉}, (5.413) where the parameters 𝐴,𝐵,𝑎,𝑏,𝑐, and 𝑑 can be identified by the exponential Padé approximant similar to the Padé approximant.

The [𝑚,𝑛] exponential Padé approximant is defined as follows.

Let 𝐹=𝐹₀+𝜀𝐹₁+𝜀²𝐹₂+⋯,𝜀→0. (5.414)

The [𝑚,𝑛] exponential Padé approximant reads 𝐹_𝑚𝑛(𝜀)=∑^𝑚_𝑖=0𝑐_𝑖exp𝑑_𝑖𝜀∑^𝑛_𝑖=0𝑎_𝑖exp𝑏_𝑖𝜀. (5.415) The coefficients 𝑎_𝑖,𝑏_𝑖,𝑐_𝑖,  and  𝑑_𝑖, are determined from the following condition: the first (2𝑚+2𝑛) components of the expansion of the rational function 𝐹_𝑚𝑛(𝜀) in a Maclaurin series coincide with the first (2𝑚+2𝑛) components of the series 𝐹(𝜀).

5.7.4. Couple Homotopy Perturbation Method with Laplace Transform [73]

Couple of the homotopy perturbation method with the Laplace transform makes the solution procedure much simpler.

Consider the following foam drainage equation: 𝑢_𝑡+2𝑢²𝑢_𝑥−𝑢²_𝑥−12𝑢_𝑥𝑥𝑢=0, (5.416) with initial conditions 𝑢()1𝑥,0=−2+11+𝑒^𝑥. (5.417) Apply Laplace transformation 𝑢()=1𝑥,𝑠𝑠−12+11+𝑒^𝑥−1𝑠𝐿2𝑢²𝑢_𝑥−𝑢²_𝑥−12𝑢_𝑥𝑥𝑢. (5.418) Apply inverse Laplace transformation 𝑢()=−1𝑥,𝑡2+11+𝑒^𝑥−𝐿⁻¹1𝑠𝐿2𝑢²𝑢_𝑥−𝑢²_𝑥−12𝑢_𝑥𝑥𝑢. (5.419) Apply the convex homotopy 𝑢₀+𝑝𝑢₁+𝑝²𝑢₂=−1+⋯2+11+𝑒^𝑥−^𝑡₀𝐿⁻¹1𝑠𝐿2𝑢₀+𝑝𝑢₁+⋯²𝜕𝑢₀𝜕𝑥+𝑝𝜕𝑢₁+𝜕𝑥+⋯𝑑𝑡_𝑡0𝐿⁻¹1𝑠𝐿𝑢₀+𝑝𝑢₁+⋯²_𝑥+12𝜕²𝑢₀𝜕𝑥²𝜕+𝑝²𝑢₁𝜕𝑥²+⋯𝑢₀+𝑝𝑢₁+⋯𝑑𝑡. (5.420) Comparing the coefficient of like powers of 𝑝, 𝑝⁽⁰⁾∶𝑢₀()=−1𝑥,𝑡2+11+𝑒^𝑥,𝑝⁽¹⁾∶𝑢₁(𝑥,𝑡)=4𝑒^𝑥𝑡(1+2𝑒^𝑥)⁴𝑒+8^2𝑥𝑡(1+2𝑒^𝑥)⁴−5𝑒^𝑥𝑡(1+2𝑒^𝑥)³−2𝑒^2𝑥𝑡(1+2𝑒^𝑥)³+3𝑒^𝑥𝑡2(1+2𝑒^𝑥)²−𝑒^𝑥𝑡²16(1+2𝑒^𝑥)³+𝑒^2𝑥𝑡8(1+2𝑒^𝑥)³.⋮ (5.421) The series solution is given by 𝑢()=−1𝑥,𝑡2+11+𝑒^𝑥+4𝑒^𝑥𝑡(1+2𝑒^𝑥)⁴𝑒+8^2𝑥𝑡(1+2𝑒^𝑥)⁴−5𝑒^𝑥𝑡(1+2𝑒^𝑥)³−2𝑒^2𝑥𝑡(1+2𝑒^𝑥)³+3𝑒^𝑥𝑡2(1+2𝑒^𝑥)²−𝑒^𝑥𝑡²16(1+2𝑒^𝑥)³+𝑒^2𝑥𝑡8(1+2𝑒^𝑥)³⋯. (5.422) Applications of the homotopy perturbation method to solitary solutions can be referred to [74–77].

5.8. Parameter-Expansion Method

5.8.1. An Example

Slab detachment or break-off is appreciated as an important geological process. Ananalytical solution was given in [78] for the nonlinear dynamics of high amplitude necking in a free layer of power-law fluid extended in layer-parallel direction due to buoyancy stress.

The power-law flow law for the layer can be written as 𝜀̇=𝐵𝜏^𝑛, (5.423) where 𝜀̇, rate of deformation parallel to the layer of a plane in the layer, 𝜏 is the mean layer-parallel deviatoric stress, and 𝐵 and 𝑛 are a material parameter and the stress exponent, respectively.

Schmalholz obtained the following detachment time [78]: 𝑡𝑡=^𝑐𝑛=1𝑛𝐵((1/2)𝐹)^𝑛, (5.424) where 𝐹 is the layer-parallel force due to buoyancy at the top of the layer.

The power-law fluid assumption, (5.423), is useful because of its simplicity but only approximately describes the behaviour of a real non-Newtonian fluid. We modify (5.423) in the form 𝜀̇=𝐵₁𝜏+𝐵₂𝜏^𝑛, (5.425) where 𝐵₁ and 𝐵₂ are material parameters.

The governing equation for the dynamics of necking becomes −1𝐷𝑑𝐷=𝐵𝑑𝑡¹𝐹2𝐷+𝐵₂𝐹2𝐷^𝑛,𝐷(0)=𝐷₀, (5.426) where 𝐷 is the local thickness of the layer, and 𝐹 is the layer-parallel force.

Rewrite (5.426) in the form 𝑑𝐷𝑑𝑡+𝐴₁+𝐴₂𝐷^1−𝑛=0,𝐷(0)=𝐷₀, (5.427) where 𝐴₁=𝐵₁𝐹/2, 𝐴₂=𝐵₂(𝐹/2)^𝑛.

We use the parameter-expansion method (see Section 5.8.2) to solve (5.427). We expand the solution in the form 𝐷=𝐷+𝑝𝐷+𝑝²𝐷+⋯, (5.428) where 𝐷 is a reference thickness, 𝐷 is a perturbation term, 𝐷 is an even smaller infinitesimal perturbation term, and 𝑝 is a bookkeeping parameter, 𝑝=1 (see Section  5.8.2).

The parameters 𝐴₁ and 𝐴₂ in (5.427) are expended in a similar way: 𝐴₁=𝐴₁+𝑝𝐴₁+𝑝²𝐴₁𝐴+⋯,₂=𝑝𝐴₂+𝑝𝐴₂+𝑝²𝐴₂+⋯. (5.429) Substituting (5.428)-(5.429) into (5.427), collecting terms of the same power of 𝑝, and equating coefficients of like powers of  𝑝 yield a series of linear equations. We write only the equations for 𝐷 and 𝐷, which read 𝑑𝐷+𝑑𝑡𝐴₁=0,𝐷(0)=𝐷₀,𝑑𝐷+𝐴𝑑𝑡¹+𝐴₂𝐷^1−𝑛=0,𝐷(0)=0. (5.430)(5.431) The solution of (5.430) reads 𝐷=𝐷₀−𝐴₁𝑡. (5.432) Substituting the result into (5.431), we have 𝑑𝐷+𝐴𝑑𝑡¹+𝐴₂𝐷₀−𝐴₁𝑡^1−𝑛=0,𝐷(0)=0. (5.433) Its solution is 𝐷=−𝐴₂𝐷₀^2−𝑛(2−𝑛)𝐴₁−𝐴₁𝑡+𝐴₂𝐷₀−𝐴₁𝑡^2−𝑛(2−𝑛)𝐴₁. (5.434) Note that 𝐷 is a perturbation term. When 𝑡 is small, it can be approximated as 𝐷=−𝐴₂𝐷₀^2−𝑛(2−𝑛)𝐴₁−𝐴₁𝑡+𝐴₂(2−𝑛)𝐴₁𝐷₀^2−𝑛−(2−𝑛)𝐷₀^1−𝑛𝐴₁𝑡=−𝐴₁−𝐴₂𝐷₀^1−𝑛𝑡. (5.435) We set 𝐴₁=𝐴₂𝐷₀^1−𝑛. (5.436) If the first-order approximate solution is enough, setting 𝑝=1 in (5.428)-(5.429) and equating higher-order perturbation terms to be zero, we have 𝐴₂=𝐴₂,𝐴₁=𝐴₂𝐷₀^1−𝑛,𝐴₁=𝐴₁−𝐴₂𝐷₀^1−𝑛,𝐷=𝐷+𝐷=𝐷₀−𝐴₁𝐴𝑡+²𝐷₀^2−𝑛(2−𝑛)𝐴₁−𝐴₁𝐴𝑡−²𝐷₀−𝐴₁𝑡^2−𝑛(2−𝑛)𝐴₁. (5.437) For qualitative analysis, we use 𝐷 as an approximate solution 𝐷=𝐷₀−𝐴₁−𝐴₂𝐷₀^1−𝑛𝑡. (5.438) The detachment time is 𝐷𝑡=⁰𝐴₁−𝐴₂𝐷₀^1−𝑛=𝐷₀𝐵₁𝐹/2−𝐵₂(𝐹/2)^𝑛𝐷₀^1−𝑛. (5.439) Equation (5.439) shows that the detachment time depends upon the initial layer thickness and fluid properties and the buoyancy.

The nonlinear equation describing the dynamics of necking is of strong nonlinearity, and in order to solve it analytically, oversimple assumption has to be made. The power-law fluid is simple, but it is not a real non-Newtonian fluid. Recent development of analytical methods increases tantalizing possibility of theoretically seeking the approximate solutions to the present problem.

5.8.2. Parameter-Expansion Method for Solitary Solutions

Parameter-expansion method includes the modified Lindstedt-Poincare method [79] and bookkeeping parameter method [80]. In the review article [7], it was previously called parameter-expanding method. The method does not require to construct a homotopy. To illustrate its solution procedure, we reconsider (5.384), which is rewritten in the form 𝜕𝑢𝜕𝑡+𝑎𝑢𝜕𝑢𝜕𝜕𝑥+𝑏³𝑢𝜕𝑥³+1⋅𝑁(𝑢)=0. (5.440) Supposing that the parameters, 𝑎, 𝑏, and 1, can be expressed in the forms 𝑎=𝑎₀+𝑝𝑎₁+𝑝²𝑎₂+⋯,𝑏=𝑏₀+𝑝𝑏₁+𝑝²𝑏₂+⋯,1=𝑝𝑐₁+𝑝²𝑐₂+⋯, (5.441) where 𝑝 is a bookkeeping parameter, 𝑝=1, substituting (5.441) into (5.440), proceeding the same way as the perturbation method, we can easily obtain the needed solution.

We consider a simple mathematical model in the form 𝑢^′′+𝑢²(0)(1)𝑢=0,𝑢=𝑢=0,_′′+0+1⋅𝑢²=0. (5.442)

We seek an expansion of the form 𝑢=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯, (5.443) where the ellipsis dots stand for terms proportional to powers of 𝑝 greater than 2, 𝑝 is a bookkeeping parameter, 𝑝=1.

The constants, 0 and 1, in the left-hand side of (5.442) can be, respectively, expanded in a similar way 0=2𝑎+𝑎₁𝑝+𝑎₂𝑝²+⋯,1=𝑏₁𝑝+𝑏₂𝑝²+⋯. (5.444) Substituting (5.444) to (5.442), we have 𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯^′′+2𝑎+𝑎₁𝑝+𝑎₂𝑝²+𝑏+⋯₁𝑝+𝑏₂𝑝²+⋯⋅𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯²=0, (5.445) and equating coefficients of like powers of 𝑝 we obtain the same equations as illustrated in previous examples: 𝑢₀^′′+2𝑎=0,𝑢₀(0)=𝑢₀(1)𝑢=0,_1′′+𝑏₁𝑢²₀+𝑎₁=0,𝑢₁(0)=𝑢₁(1)=0. (5.446) Solving 𝑢₀ and 𝑢₁, we have the following first-order approximate solution: 𝑢(𝑡)=𝑢₀(𝑡)+𝑢₁(𝑡)=𝑎𝑡(1−𝑡)+𝑎𝑡²−𝑎²1𝑡30⁶−1𝑡10⁵+1𝑡12⁴−1𝑎−𝑎60²𝑡. (5.447) The discussion on how to determine the constant, 𝑎, is given in [69]. Applications of the parameter-expansion method to various nonlinear problems are available in [81–86].

5.9. Exp-Function Method

5.9.1. Exp-Function Method for Solitary Solutions

Exp-function method [87–89] provides us with a straightforward and concise approach for obtaining generalized solitary solutions and periodic solutions; the solution procedure, by the help of Matlab or Mathematica, is of utter simplicity, see [90–97]. Considering a general nonlinear partial differential equation in the form 𝐹𝑢,𝑢_𝑥,𝑢_𝑦,𝑢_𝑧,𝑢_𝑡,𝑢_𝑥𝑥,𝑢_𝑦𝑦,𝑢_𝑧𝑧,𝑢_𝑡𝑡,𝑢_𝑥𝑦,𝑢_𝑥𝑡,𝑢_𝑦𝑡,…=0. (5.448) The exp-function method assumes that the solitary solution can be expressed in the form 𝑢=∑𝑥,𝑦,𝑧,𝑡^{𝑙𝑛=−𝑘}𝑝_𝑛exp𝑎_𝑛𝑥+𝑏_𝑛𝑦+𝑐_𝑛𝑧+𝑑_𝑛𝑡∑^𝑗_𝑚=−𝑖𝑞_𝑚exp𝑎_𝑚𝑥+𝑏_𝑚𝑦+𝑐_𝑚𝑧+𝑑_𝑚𝑡. (5.449) We can also introduce a transformation 𝜉=𝑎𝑥+𝑏𝑦+𝑐𝑧+𝑑𝑡. (5.450) We can rewrite (5.448) in the following nonlinear ordinary differential equation: 𝐹𝑢,𝑢^′,𝑢^′′,𝑢^′′′,…=0, (5.451) where the prime denotes the derivation with respect to 𝜉.

According to the exp-function method, the traveling wave solutions can be expressed in the form 𝑢(𝜉)=∑^𝑙_𝑛=−𝑘𝑎_𝑛exp(𝑛𝜉)∑^𝑗_𝑚=−𝑖𝑏_𝑚(),exp𝑚𝜉 (5.452) where 𝑖, 𝑗, 𝑘, and 𝑙 are positive integers which could be freely chosen; 𝑎_𝑛 and 𝑏_𝑚 are unknown constants to be determined.

Consider a fractional Benjamin-Bona-Mahony (BBM) equation [97] 𝜕𝑢𝜕𝑡−𝑢𝜕𝑢−𝜕𝜕𝑥𝜕𝜕𝑡²𝑢𝜕𝑥²=0, (5.453) using a transformation 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑘𝑥+𝑤𝑡, (5.454) (5.453) becomes an ordinary differential equation, 𝑤𝑈^′−𝑘𝑈𝑈^′−𝑘²𝑤𝑈^′′′=0. (5.455) By the exp-function method, we assume that the solution of (5.460) can be expressed as 𝑈(𝜉)=𝑎₁exp(𝜉)+𝑎₀+𝑎₋₁exp(−𝜉)𝑏₁exp(𝜉)+𝑏₀+𝑏₋₁(),exp−𝜉 (5.456) where 𝑎_𝑖,𝑏_𝑖(𝑖=1,0,−1) are unknown parameters.

Substituting (5.456) into (5.455), collecting terms of the same term of exp(𝑖𝜉), we have 1𝐴𝐶₇exp(7𝜉)+𝐶₆exp(6𝜉)+𝐶₅exp(5𝜉)+𝐶₄exp(4𝜉)+𝐶₃exp(3𝜉)+𝐶₂exp(2𝜉)+𝐶₁exp(𝜉)=0. (5.457)

Equating coefficients, 𝐶_𝑖(𝑖=1∼7), to zero yields a series of linear equations 𝐶_𝑖=0(𝑖=1∼7). (5.458) Solving the system of algebraic equation, we can identify parameters in (5.456). Finally, we obtain [97] 𝑢()𝑤𝑘𝑥,𝑡=−²−1𝑘+6𝑘²𝑏₀𝑏₁exp[(𝑘𝑥+𝑤𝑡)]+𝑏₀+𝑏²₀/4𝑏₁exp[−(𝑘𝑥+𝑤𝑡)], (5.459) where 𝑏₀ and 𝑏₁ are free parameters.

By 𝑘=𝑖𝐾, 𝑤=𝑖𝑊, the obtained solitary solution can be converted into periodic solution.

Other expressions are listed as follows: 𝑢()=∑𝑥,𝑡^𝑛𝑎_𝑛exp𝑞_𝑛𝑥+𝑝_𝑛𝑡∑_𝑚𝑏_𝑚exp𝛼_𝑚𝑥+𝛽_𝑚𝑡, (5.460) where 𝑎_𝑖,𝑏_𝑖𝛼_𝑖,𝛽_𝑖𝑞_𝑖,  and  𝑝_𝑖 are unknown constants to be further determined.

For a two-wave solution, we can use the double exp-function method [8], which requires 𝑎𝑢=⁻¹exp(−𝜉)+𝑏₁exp(𝜉)+𝑎₀+𝑏₋₁exp−𝜂+𝑏₁exp𝜂𝑐₋₁exp(−𝜉)+𝑐₁exp(𝜉)+𝑏₀+𝑑₋₁exp−𝜂+𝑑₁exp𝜂, (5.461) where 𝜉=𝑎₁𝑥+𝑏₁𝑦+𝑐₁𝑧−𝑑₁𝑡 and 𝜂=𝑎₂𝑥+𝑏₂𝑦+𝑐₂𝑧−𝑑₂𝑡.

5.9.2. The Exp-Function Method for Periodic Solutions

The French mathematician J. Fourier (1768–1830) showed that any periodic motion can be represented by a series of sines and cosines that are harmonically related 𝑢(𝑡)=12𝑎₀+𝑎₁cos𝜔₁𝑡+𝑎₂cos𝜔₂𝑡+⋯+𝑏₁sin𝜔₁𝑡+𝑏₂sin𝜔₂𝑡+⋯, (5.462) where 𝜔_𝑛=𝑛𝜔₁, and 𝑎_𝑛 and 𝑏_𝑛 are, respectively, expressed as 𝑎_𝑛=2𝑇^𝑇/2_−𝑇/2𝑢(𝑡)cos𝜔_𝑛𝑏𝑡𝑑𝑡,_𝑛=2𝑇^𝑇/2_−𝑇/2𝑢(𝑡)sin𝜔_𝑛𝑡𝑑𝑡. (5.463) Using Euler’s formula, we have 1sin𝑡=2𝑒^𝑖𝑡+𝑒^−𝑖𝑡,1cos𝑡=−2𝑖𝑒^𝑖𝑡−𝑒^−𝑖𝑡. (5.464) We, therefore, can rewrite the Fourier series, (5.462), in the form 𝑢(𝑡)=^∞_𝑛=−∞𝑐_𝑛𝑒^{𝑖𝜔𝑛𝑡}, (5.465) where 𝑐₀=(1/2)𝑎₀ and 𝑐_𝑛=(1/2)(𝑎_𝑛−𝑖𝑏_𝑛).

For periodic solution, we can assume that the solution can be expressed in the form 𝑢(𝑡)=∑^𝑑_𝑛=−𝑐𝑝_𝑛exp𝑖𝜔_𝑛𝑡∑^𝑔_𝑚=−𝑓𝑞_𝑚exp𝑖𝜔_𝑚𝑡, (5.466) or 𝑢(𝑡)=^𝑁₀𝑐_𝑛𝑒^{𝑖𝜔𝑛𝑡}. (5.467)

5.10. Subequation Method

There are many subequation methods in the open literature for solitary solution [98–100]. We consider a general nonlinear equation in the form 𝐹𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑦,𝑢_𝑧,𝑢_𝑥𝑥,𝑢_𝑦𝑦,𝑢_𝑧𝑧,…=0. (5.468) The sub-equation method follows the following steps.

Step 1. Let 𝑢=𝑢(𝑥,𝑦,𝑧,𝑡)=𝑢(𝜉), where 𝜉=𝑎𝑥+𝑏𝑦+𝑐𝑧+𝑑𝑡+𝜉₀. Equation (5.468) turns out to be an ordinary differential equation 𝐹𝑢,𝑢_𝜉,𝑢_𝜉𝜉,𝑢_𝜉𝜉𝜉,…=0. (5.469)

Step 2. Assume that the solution can be expressed in the form 𝑢=_𝑛𝑎_𝑛𝑓^𝑛(𝜉), (5.470) or in a more general form, 𝑢=𝑈𝑓(𝜉), (5.471) where 𝑓 is the solution of an auxiliary equation 𝑓^′2=𝑏₀+𝑏₁𝑓+𝑏₂𝑓²+𝑏₃𝑓³+𝑏₄𝑓⁴+⋯. (5.472) The exact solution of (5.472) for fixed 𝑏_𝑖(𝑖=0,1,2,3,…) is known.

Step 3. Use some mathematical software to determine 𝑎_𝑛 in (5.470).

The auxiliary equation can be constructed using the Hamiltonian invariant as discussed in Section 5.3.5. For (5.128), and the following auxiliary equation can be constructed: 𝑢^′=√𝑐𝑢²+2𝑢³. (5.473) For (5.227), the following auxiliary equation should be in the form: 𝑢^′=𝑐𝑢²−12()𝑢𝑛+1)(𝑛+2^𝑛+2. (5.474) In such case, the solution of the auxiliary equation is the special solution of the problem.

5.11. Ancient Chinese Mathematics

5.11.1. Introduction to the Ancient Chinese Mathematics

Can ancient Chinese mathematics still work for nonlinear science? Yes, it can. It is simple but effective.

To illustrate the basic idea of the ancient Chinese method, we consider an algebraic equation: 𝑓(𝑥)=0. (5.475)

Let 𝑥₁ and 𝑥₂ be the approximate solutions of the equation, which lead to the remainders 𝑓(𝑥₁) and 𝑓(𝑥₂), respectively; the ancient Chinese algorithm leads to the result 𝑥𝑥=²𝑓𝑥₁−𝑥₁𝑓𝑥₂𝑓𝑥₁−𝑓𝑥₂. (5.476)

Now we consider a generalized nonlinear oscillator in the form 𝑢^′′+𝑓(𝑢)=0,𝑢(0)=𝐴,𝑢^′(0)=0. (5.477) We use two trial functions 𝑢₁(𝑡)=𝐴cos𝜔₁𝑡 and 𝑢₂=𝐴cos𝜔₂𝑡, which are, respectively, the solutions of the following linear oscillator equations: 𝑢^′′+𝜔²₁𝑢𝑢=0,_′′+𝜔²₂𝑢=0. (5.478) The residuals are 𝑅₁𝜔₁𝑡=−𝜔²₁cos𝜔₁𝑡+𝑓𝐴cos𝜔₁𝑡,𝑅₂𝜔₂𝑡=−𝜔²₂cos𝜔₂𝑡+𝑓𝐴cos𝜔₂𝑡. (5.479) The residuals depend upon 𝑡, and in our previous applications [7], we just use 𝑅₁(0) and 𝑅₂(0) for simplicity when we use the ancient Chinese method, resulting in relatively low accuracy [7] 𝜔²=𝜔²₁𝑅₂(0)−𝜔²₂𝑅₁(0)𝑅₂(0)−𝑅₁(0). (5.480) Equation (5.480) is called He’s frequency formulation or He’s frequency-amplitude formulation in the literature [101–106].

In order to improve its accuracy, we introduce two new residual variables 𝑅₁ and 𝑅₂ defined as [105, 106] 𝑅₁=4𝑇₁^𝑇1₀^/4𝑅₁(𝑡)cos2𝜋𝑇₁𝑡𝑅𝑑𝑡,₂=4𝑇₂^𝑇2₀^/4𝑅₂(𝑡)cos2𝜋𝑇₂𝑡𝑑𝑡. (5.481)(5.482) According to the ancient Chinese method, we can approximately determine 𝜔² in the form 𝜔²=𝜔²₁𝑅₂−𝜔²₂𝑅₁𝑅₂−𝑅₁. (5.483) In this letter, we will use the well-known Duffing equation.

Using trial functions 𝑢₁(𝑡)=𝐴cos𝑡 and 𝑢₂=𝐴cos𝜔₂𝑡, respectively, for (5.481), we obtain the following residuals: 𝑅₁(𝑡)=𝜀𝐴³cos³𝑅𝑡,₂(𝑡)=𝐴1−𝜔²₂cos𝜔₂𝑡+𝜀𝐴³cos³𝜔₂𝑡, (5.484) where 𝜔₂>0 and 𝜔₂≠1.

By simple calculation, we obtain 𝑅₁=4𝑇₁^𝑇1₀^/4𝑅₁(𝑡)cos2𝜋𝑇₁𝑡2𝑑𝑡=𝜋₀^𝜋/2𝜀𝐴³cos⁴3𝑡𝑑𝑡=,𝑅4𝜋₂=4𝑇₂^𝑇2₀^/4𝑅₂(𝑡)cos2𝜋𝑇₂𝑡𝑑𝑡=2𝜔𝜋^𝑇2₀^/4𝐴1−𝜔²₂cos²𝜔₂𝑡+𝜀𝐴³cos⁴𝜔₂𝑡=2𝑑𝑡𝜋₀^𝜋/2𝐴1−𝜔²₂cos²𝑠+𝜀𝐴³cos⁴𝑠𝑑𝑠=𝐴1−𝜔²₂1𝜋+34𝜋𝜀𝐴³. (5.485) Applying (5.483), we have 𝜔²=𝜔²₁𝑅₂−𝜔²₂𝑅₁𝑅₂−𝑅₁=𝐴1−𝜔²₂(1/𝜋)+(3/4𝜋)𝜀𝐴³−(3/4𝜋)𝜀𝐴³𝜔²₂𝐴1−𝜔²₂(1/𝜋)+(3/4𝜋)𝜀𝐴³−(3/4𝜋)𝜀𝐴³=1−𝜔²₂+(3/4)𝜀𝐴³−(3/4)𝜀𝐴³𝜔²₂1−𝜔²₂3=1+4𝜀𝐴². (5.486)

5.11.2. Ancient Chinese Mathematics for Optimization [107]

To elucidate the basic idea of application of the ancient Chinese mathematics to optimal problems, we consider first the following simple example: 𝑓=𝑥𝑦→max,s.t.𝑔𝑥,𝑦=𝑥+𝑦−1=0. (5.487)(5.488)

To make the Lagrange function  𝑓  maximal, it requires 𝑑𝑓=𝑥𝑑𝑦+𝑦𝑑𝑥=0, (5.489) where 𝑑𝑥 and 𝑑𝑦 are not independent, and according to the constraint equation, (5.490), we have 𝑑𝑥+𝑑𝑦=0. (5.490) Considering the relationship between 𝑑𝑥 and 𝑑𝑦, (5.492), (5.491) becomes 𝑑𝑓=𝑥−𝑦𝑑𝑦=0. (5.491) Due to the arbitrary 𝑑𝑦 in (5.491), we have the following Euler equation: 𝑥−𝑦=0. (5.492) Solving (5.488) and (5.492) simultaneously, we find the optimal values 𝑥^∗=𝑦^∗=12. (5.493)

A general solution procedure is given in [8], and the stationary condition (Euler equation) can be simply obtained using He’s brackets [8, 108]: 𝑓,𝑔_𝑥,𝑦=𝑓_𝑥𝑔_𝑦−𝑓_𝑦𝑔_𝑥=0, (5.494) where 𝑓_𝑥=𝜕𝑓/𝜕𝑥=𝑦, 𝑓_𝑦=𝜕𝑓/𝜕𝑦=𝑥, 𝑔_𝑥=𝜕𝑔/𝜕𝑥=1, and 𝑔_𝑦=𝜕𝑔/𝜕𝑦=1.

Equation (5.494) is exactly the same with (5.492).

According to the ancient Chinese mathematics, to solve a problem, we should choose double trial solutions, 𝑥₁ and 𝑥₂. We can choose 𝑥₁ arbitrarily, for example, 𝑥₁=2/7, according to the constraint equation (5.488), and we have 𝑦₁=5/7.

From (5.487), we obtain 𝑓₁=𝑥₁𝑦₁=10.49 (5.495) The condition (5.489) can be approximately expressed as Δ𝑓=𝑥Δ𝑦+𝑦Δ𝑥=0, (5.496) where Δ𝑥=𝑥₂−𝑥₁, Δ𝑦=𝑦₂−𝑦₁, and Δ𝑓=𝑓₂−𝑓₁.

Equation (5.496) implies that 𝑓₂=𝑓₁. (5.497) This means that 𝑓₂=𝑥₂𝑦₂=10.49 (5.498) Considering the constraint equation, (5.488), again, we have 𝑥₂+𝑦₂=1. (5.499) From (5.498) and (5.499), we can obtain immediately 𝑥₂=5/7 and 𝑦₂=2/7.

Substituting 𝑥₁, 𝑥₂, 𝑦₁, and 𝑦₂ into (5.498), we have Δ𝑓=𝑥𝑦₂−𝑦₁+𝑦𝑥₂−𝑥₁2=𝑥7−575+𝑦7−27=0. (5.500) Solving (5.488) and (5.500) simultaneously, we find the optimal values given in (5.493).

Example 5.1. Condiser the following [109]: 𝑓=𝑥²+𝑦²→min,s.t.𝑔𝑥,𝑦=𝑦−2−𝑥²=0. (5.501)(5.502) To make (5.501) minimal, it requires that Δ𝑓=2𝑥Δ𝑥+2𝑦Δ𝑦2𝑥²+𝑦²=𝑥Δ𝑥+𝑦Δ𝑦𝑥²+𝑦²=0. (5.503) Choosing 𝑥₁=1, we have 𝑦₁=3 and 𝑓₁=√10. We set 𝑓₂=√10, which means that 𝑥²₂+𝑦²₂𝑦=10,₂=2+𝑥²₂. (5.504)(5.505) Solving (5.504) and (5.505) simultaneously, we have 𝑥₂=−1 and 𝑦₂=3.
Substituting 𝑥₁, 𝑥₂, 𝑦₁, and 𝑦₂ into (5.503), we obtain 𝑥𝑥Δ𝑓=²−𝑥₁+𝑦𝑦₂−𝑦₁𝑥²+𝑦²=−2𝑥𝑥²+𝑦²=0. (5.506)
We, therefore, obtain the optimal values 𝑥^∗=0 and 𝑦^∗=2.

Example 5.2. Consider the following problem [109]: 𝑓𝑥,𝑦=𝑥²+𝑦²→min,s.t.𝑦²−(𝑥−1)³=0. (5.507)(5.508) The Lagrange multiplier method becomes invalid for this example [109]. Using the basic idea of the ancient Chinese mathematics, we choose 𝑥₁=2, 𝑦₁=1 and 𝑥₂=2, 𝑦₂=−1, and both cases lead to 𝑓₁=𝑓₂=√5.
Making (5.507) stationary requires Δ𝑓=𝑥Δ𝑥+𝑦Δ𝑦𝑥²+𝑦²=𝑥𝑥₂−𝑥₁+𝑦𝑦₂−𝑦₁𝑥²+𝑦²=−2𝑦𝑥²+𝑦²=0, (5.509) which leads to immediately 𝑦^∗=0 and 𝑥^∗=1 if the constraint equation, (5.508), is considered.

5.11.3. Ancient Chinese Mathematics for Nonlinear Wave Equations [110]

Now we consider the nonlinear Schrödinger equation 𝑖𝑢_𝑡+𝑢_𝑥𝑥+2|𝑢|²𝑢=0,𝑢(𝑥,0)=𝑒^𝑖𝑘𝑥. (5.510) We use the trial functions 𝑢₁(𝑥,𝑡)=𝑒^{𝑖(𝑘𝑥+𝑡)},𝑢₂(𝑥,𝑡)=𝑒^{𝑖(𝑘𝑥+𝜔𝑡)}. (5.511) The residuals are 𝑅₁(𝑥,𝑡)=1−𝑘²𝑒^{𝑖(𝑘𝑥+𝑡)},𝑅₂(𝑥,𝑡)=2−𝜔−𝑘²𝑒^{𝑖(𝑘𝑥+𝜔𝑡)}. (5.512) According to frequency-amplitude formulation, (5.480), we have 𝜔²=𝜔²₁𝑅₂(𝑥,0)−𝜔²₂𝑅₁(𝑥,0)𝑅₂(𝑥,0)−𝑅₁(𝑥,0)=−𝜔+𝜔𝑘²−2𝜔+𝑘²−2. (5.513)

We get the equation about 𝜔, and the solution is 𝜔=2−𝑘²,𝑢(𝑥,𝑡)=𝑒^{𝑖{𝑘𝑥+(2−𝑘2)𝜔𝑡}}. (5.514) Now we consider the Schrödinger equation with power law nonlinearity 𝑖𝑢_𝑡+𝑢_𝑥𝑥−2|𝑢|^2𝑟𝑢=0,𝑢(𝑥,0)=2(𝑟+1)sech²(2𝑟𝑥)^1/2𝑟,𝑟≥1. (5.515) Following the analysis presented above, we choose the trial functions 𝑢₁(𝑥,𝑡)=2(𝑟+1)sech²(2𝑟𝑥)^1/2𝑟𝑒^𝑖𝑡,𝑢₂(𝑥,𝑡)=2(𝑟+1)sech²(2𝑟𝑥)^1/2𝑟𝑒^𝑖𝜔𝑡. (5.516) The residuals are 𝑅₁(𝑥,𝑡)=−2(𝑟+1)(sech(2𝑟𝑥))^21/2𝑟𝑒^𝑖𝑡+42(𝑟+1)(sech(2𝑟𝑥))^21/2𝑟(tanh(2𝑟𝑥))²𝑒^𝑖𝑡−42(𝑟+1)(sech(2𝑟𝑥))^21/2𝑟1−(tanh(2𝑟𝑥))²𝑟𝑒^𝑖𝑡+22(𝑟+1)(sech(2𝑟𝑥))^21/2𝑟−1𝑒^𝑖𝑡,𝑅₂(𝑥,𝑡)=−2(𝑟+1)(sech(2𝑟𝑥))^21/2𝑟𝑒^𝑖𝜔𝑡𝜔+42(𝑟+1)(sech(2𝑟𝑥))^21/2𝑟(tanh(2𝑟𝑥))²𝑒^𝑖𝜔𝑡−42(𝑟+1)(sech(2𝑟𝑥))^21/2𝑟1−(tanh(2𝑟𝑥))²𝑟𝑒^𝑖𝜔𝑡+22(𝑟+1)(sech(2𝑟𝑥))^21/2𝑟𝑒^𝑖𝜔𝑡. (5.517) According to the frequency-amplitude formulation, (5.480), setting 𝑥=0 and 𝑡=0,  we have 𝜔²=−4𝑟+2𝜔2^1/2𝑟(𝑟+1)^1/2𝑟2𝑟−𝜔−4𝜔𝑟+2^1/2𝑟(𝑟+1)^1/2𝑟2𝑟. (5.518) Solving 𝜔 from the above equation yields 𝜔=−4𝑟+22^1/2𝑟(𝑟+1)^1/2𝑟2𝑟. (5.519) We, therefore, obtain the solution 𝑢₂(𝑥,𝑡)=2(𝑟+1)sech²(2𝑟𝑥)^1/2𝑟𝑒^𝑖𝜔𝑡. (5.520)

5.11.4. Solitary-Solution Formulation for Nonlinear Equations

Now consider a general nonlinear equation 𝐿(𝑢)+𝑁(𝑢)=0, (5.521) where 𝐿 is a linear operator, and 𝑁 is a nonlinear operator.

We choose two trial equations 𝐿(𝑢)𝐿(𝑢)=0,=0, (5.522) where 𝐿 is a linearized linear operator or a simple nonlinear operator, and its solution 𝐿(𝑢)=0 is known. For example, we consider a nonlinear oscillator: 𝑢^′′+𝑢³=0, and we can choose 𝐿(𝑢)=𝑢^′′+𝑢=0 and 𝐿(𝑢)=𝑢^′′+𝜔²𝑢=0, where 𝜔 is the frequency to be searched for.

Substituting the solutions of (5.522) with same boundary/initial conditions as (5.521) into (5.521), we obtain, respectively, the residuals  𝑅₁ and 𝑅₂.

For traveling wave solutions, we can make a transformation 𝜉=𝑎𝑥+𝑏𝑦+𝑐𝑧+𝜔𝑡. (5.523) The solitary-solution formulation can be expressed as 𝜔²=𝑅₁𝜔²₁−𝑅₂𝜔²₂𝑅₁−𝑅₂, (5.524) or 𝜔^𝑛=𝑅₁𝜔^𝑛₁−𝑅₂𝜔^𝑛₂𝑅₁−𝑅₂. (5.525)

The KdV equation is a nonlinear, dispersive partial differential equation for a function of two real variables, space 𝑥 and time 𝑡: 𝑢_𝑡+6𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=0. (5.526) According to the solitary-solution formulation (5.524), we construct two trial functions in the following forms: 𝑢₁()=𝐴𝑥,𝑡()[−(,𝑢exp𝑘𝑥+𝑡+𝐷+exp𝑘𝑥+𝑡+𝐷)]+𝐵₂()=𝐴𝑥,𝑡()[−(,exp𝑘𝑥+𝜔𝑡+𝐷+exp𝑘𝑥+𝜔𝑡+𝐷)]+𝐵 (5.527) where 𝐴, 𝐵, 𝐷 are arbitrary constants.

Substituting 𝑢₁ and 𝑢₂ into (5.526), respectively, we obtain the following residuals: 𝑅₁()=()[−(𝑥,𝑡−𝐴exp𝑘𝑥+𝑡+𝐷−exp𝑘𝑥+𝑡+𝐷)]exp(𝑘𝑥+𝑡+𝐷)+exp[−(𝑘𝑥+𝑡+𝐷)]+𝐵²−6𝐴²𝑘exp(𝑘𝑥+𝑡+𝐷)−exp[−(𝑘𝑥+𝑡+𝐷)]exp(𝑘𝑥+𝑡+𝐷)+exp[−(𝑘𝑥+𝑡+𝐷)]+𝐵³−𝐴𝑘³exp(𝑘𝑥+𝑡+𝐷)−exp[−(𝑘𝑥+𝑡+𝐷)]exp(𝑘𝑥+𝑡+𝐷)+exp[−(𝑘𝑥+𝑡+𝐷)]+𝐵²−6𝐴𝑘³exp(𝑘𝑥+𝑡+𝐷)−exp[−(𝑘𝑥+𝑡+𝐷)]³exp(𝑘𝑥+𝑡+𝐷)+exp[−(𝑘𝑥+𝑡+𝐷)]+𝐵⁴+6𝐴𝑘³exp(𝑘𝑥+𝑡+𝐷)−exp[−(𝑘𝑥+𝑡+𝐷)]exp(𝑘𝑥+𝑡+𝐷)+exp[−(𝑘𝑥+𝑡+𝐷)]exp(𝑘𝑥+𝑡+𝐷)+exp[−(𝑘𝑥+𝑡+𝐷)]+𝐵³,𝑅₂()=()[−(𝑥,𝑡−𝐴𝜔exp𝑘𝑥+𝜔𝑡+𝐷−exp𝑘𝑥+𝜔𝑡+𝐷)]exp(𝑘𝑥+𝜔𝑡+𝐷)+exp[−(𝑘𝑥+𝜔𝑡+𝐷)]+𝐵²−6𝐴²𝑘exp(𝑘𝑥+𝜔𝑡+𝐷)−exp[−(𝑘𝑥+𝜔𝑡+𝐷)]exp(𝑘𝑥+𝜔𝑡+𝐷)+exp[−(𝑘𝑥+𝜔𝑡+𝐷)]+𝐵³−𝐴𝑘³exp(𝑘𝑥+𝜔𝑡+𝐷)−exp[−(𝑘𝑥+𝜔𝑡+𝐷)]exp(𝑘𝑥+𝜔𝑡+𝐷)+exp[−(𝑘𝑥+𝜔𝑡+𝐷)]+𝐵²−6𝐴𝑘³exp(𝑘𝑥+𝜔𝑡+𝐷)−exp[−(𝑘𝑥+𝜔𝑡+𝐷)]³exp(𝑘𝑥+𝜔𝑡+𝐷)+exp[−(𝑘𝑥+𝜔𝑡+𝐷)]+𝐵⁴+6𝐴𝑘³exp(𝑘𝑥+𝜔𝑡+𝐷)−exp[−(𝑘𝑥+𝜔𝑡+𝐷)]exp(𝑘𝑥+𝜔𝑡+𝐷)+exp[−(𝑘𝑥+𝜔𝑡+𝐷)]exp(𝑘𝑥+𝜔𝑡+𝐷)+exp[−(𝑘𝑥+𝜔𝑡+𝐷)]+𝐵³. (5.528) Using the solitary-solution formulation (5.480), and choosing the location point (0,0), we obtain 𝜔²=𝜔²₁𝑅₂(0,0)−𝜔²₂𝑅₁(0,0)𝑅₂(0,0)−𝑅₁().0,0 (5.529) Simplifying (5.529) results in 𝜔²+𝜔+(𝜔+1)𝑘³+2𝐵𝜔²+2𝜔𝐵+(𝜔+1)6𝑘𝐴−4𝑘³𝐵()+exp−𝐷2𝜔²+𝐵²𝜔²+2𝜔+𝜔𝐵²+(𝜔+1)6𝑘𝐴𝐵−22𝑘³+𝑘³𝐵²()+exp−2𝐷2𝐵𝜔²+2𝜔𝐵+(𝜔+1)6𝑘𝐴−4𝑘³𝐵()+𝜔exp−3𝐷₂+𝜔+(𝜔+1)𝑘³exp(−4𝐷)=0. (5.530) Setting the coefficients of exp(−𝑛𝐷) (𝑛=0,1,2,3,4) to be zero, we obtain 𝜔²+𝜔+(𝜔+1)𝑘³=0,2𝐵𝜔²+2𝜔𝐵+(𝜔+1)6𝑘𝐴−4𝑘³𝐵=0,2𝜔²+𝐵²𝜔²+2𝜔+𝜔𝐵²+(𝜔+1)6𝑘𝐴𝐵−22𝑘³+𝑘³𝐵²=0. (5.531)(5.532)(5.533) Solving (5.531)~(5.533) simultaneously, we obtain the following relationships:(1)the relationship between solitary wavenumber and solitary frequency: 𝜔=−𝑘³, (5.534)(2)the relationship between solitary amplitude and solitary wavenumber: 𝐴=𝑘²𝐵, (5.535)(3)the morphofactor: 𝐵²=4, (5.536) so the solitary wave solution can be readily obtained, which reads 𝑢(𝑥,𝑡)=2𝑘²exp𝑘𝑥−𝑘³𝑡+𝐷+exp−𝑘𝑥−𝑘³,𝑡+𝐷+2 (5.537) or 𝑢(𝑥,𝑡)=−2𝑘²exp𝑘𝑥−𝑘³𝑡+𝐷+exp−𝑘𝑥−𝑘³,𝑡+𝐷−2 (5.538) which are, by chance, exact solutions.

5.11.5. Ancient Chinese Inequality and Application

In an ancient history book, it is write (see [111]):

He Chengtian uses 26/49 as the strong, and 9/17 as the weak. Among the strong and the weak, Chengtian tries to find a more accurate denominator of the fractional day of the Moon. Chengtian obtains 752 as the denominator by using the 15 and 1 as weighting factors, respectively, for the strong and the weak. No other calendar can reach such a high accuracy after Chengtian, who uses heuristically the strong and weak weighting factors.

The statement is rather cryptic, and in modern mathematical term, the statement can be explained as follows.

According to the observation data, He Chengtian (369?~447AD) finds that 2926949days>1Moon>2917days. (5.539) Using the weighting factors (15 and 1), He Chengtian obtains Thefractionalday=26×15+9×1=49×15+17×1399,752 (5.540) so 1Moon=29399752days. (5.541) He Chengtian actually uses the following inequality.

If 𝑎𝑏𝑑<𝑥<𝑐, (5.542) where 𝑎,𝑏,𝑐, and 𝑑 are real numbers, then 𝑎𝑏<𝑚𝑎+𝑛𝑑<𝑑𝑚𝑏+𝑛𝑐𝑐, (5.543) and 𝑥 is approximated by 𝑥=𝑚𝑎+𝑛𝑑,𝑚𝑏+𝑛𝑐 (5.544) where 𝑚 and 𝑛 are weighting factors.

The above inequity was developed by the max-min approach to nonlinear oscillators [112]. Hereby we use the inequity for a chemical problem.

A mass balance on a differential volume element of porous medium for a spherical catalyst pellet gives 𝜕𝑐^′𝜕𝑡=∇⋅𝐷_𝑒∇𝑐^′−𝑅_𝐴, (5.545) where 𝑐^′ is the chemical reactant concentration, −𝑅_𝐴 is the rate of reaction per unit volume, and 𝐷_𝑒  is the effective diffusion coefficient for reactant.

Assuming the coupled process of diffusion and reaction is steady, we reduce (5.545) to the form 𝐷_𝑒𝜕²𝑐^′𝜕𝑟²+2𝑟𝜕𝑐^′𝜕𝑟=𝑅_𝐴. (5.546) The boundary conditions are 𝜕𝑐^′/𝜕𝑟=0,𝑟=0 (center of catalyst), and 𝑐^′=𝑐_𝑠,𝑟=𝑟₀ (surface of catalyst).

Introducing dimensionless variables 𝐶=𝑐^′/𝑐_𝑠 and 𝑅=𝑟/𝑟₀, we have the following nonlinear equation for 𝑛th-order and irreversible reaction at isothermal condition: 𝜕²𝐶𝜕𝑅²+2𝑅𝜕𝐶𝜕𝑅=𝜙²𝐶^𝑛,𝜕𝐶(0)(1)𝜕𝑅=0,𝐶=1, (5.547) where 𝜙 is the Thiele modulus, defined as 𝜙=𝑟²₀𝑘_𝜈𝑐_𝑠^𝑛−1/𝐷_𝑒, where  𝑘_𝜈 is the reaction constant.

Equation (5.547) can also be written in Cartesian geometry, and it is easy to transform Cartesian geometry to cylindrical or spherical geometry.

To investigate the nature of the solution of (5.547), consider the analogous linear equation 𝜕²𝐶𝜕𝑅²+2𝑅𝜕𝐶𝜕𝑅=𝑘,𝜕𝐶(0)(1)𝜕𝑅=0,𝐶=1. (5.548)

Multiplying 𝑅² on both sides of (5.548), we have 𝜕𝑅𝜕𝑅²𝜕𝐶𝜕𝑅=𝑘𝑅²,𝜕𝐶(0)(1)𝜕𝑅=0,𝐶=1. (5.549)

Its solution is 1𝐶=1+6𝑘𝑅²−1. (5.550)

Now we rewrite (5.544) in the form 𝜕²𝐶𝜕𝑅²+2𝑅𝜕𝐶(𝐶),𝜕𝑅=𝑔 (5.551) where 𝑔(𝐶)=𝜙²𝐶^𝑛.

The left side of (5.551) is analogous to 𝑘 in (5.548).

Observe that 0<𝐶<1, so 𝐶 is never less than the solution of the following initial value problem: 𝜕²𝐶𝜕𝑅²+2𝑅𝜕𝐶𝜕𝑅=𝜙²𝐶^𝑛_max=𝜙². (5.552)

The solution of (5.552) reads 1𝐶=1+6𝜙²𝑅²−1. (5.553) So it follows that (see Figure 4) 6+𝜙²𝑅²−16<𝐶<1. (5.554) According to He Chengtian’s interpolation, we have 𝐶=6𝑝+𝑝𝜙²𝑅²−1+𝑞,6𝑝+𝑞 (5.555) or 𝐶=1+𝜉𝜙²𝑅²−1, (5.556) where 𝑝 and 𝑞 are weighting factors, 𝜉=𝑝/(6𝑝+𝑞). The free parameter, 𝜉, in (5.556) can be identified via various methods. Hereby we illustrate a simple approach by the residual method. Substituting (5.556) into (5.544), we obtain the following residual equation: Π()=𝜕𝑅,𝜉²𝐶𝜕𝑅²+2𝑅𝜕𝐶𝜕𝑅−𝜙²𝐶^𝑛=2𝜉𝜙²+4𝜉𝜙²−𝜙²1+𝜉𝜙²𝑅²−1^𝑛. (5.557)

Figure 4 

Qualitative sketch for the diffusion reaction in spherical porous catalyst. The dashed line presents an initial guess 𝐶=1+𝜙2(𝑅2−1)/6, and the continued line presents the real solution.

Chemists and technologists always want to have an accurate halfway through the change. Assume that the above residual vanishes completely at 𝐶=1/2, that is, 1+𝜉𝜙²(𝑅²−1)=1/2, then we have Π(𝑅,𝜉)_𝑅=𝑅∗=6𝜉𝜙²−12^𝑛𝜙², (5.558) where 𝑅=𝑅^∗ is the root of the equation 1+𝜉𝜙²(𝑅²−1)=1/2. From (5.558), the unknown 𝜉 can be identified as 2𝜉=^−𝑛6. (5.559)

Finally, we obtain the following design formulation: 𝐶(𝑅)2=1+^−𝑛6𝜙²𝑅²−1. (5.560)

Comparison with exact solution is illustrated in Figure 5.

Figure 5 

Comparison between exact solution and the approximate solution (𝑛=1, 𝜙=1). The dashed line presents an approximate solution, and the continued line presents the exact solution. The maximal error is about 7.8%.

In order to identify the weighting factors, 𝑝 and 𝑞, in (5.555), we proceed the same manipulation as illustrated above. Substituting (5.555) into (5.544), we obtain a residual equation: Π=𝜕𝑅,𝑝,𝑞²𝐶𝜕𝑅²+2𝑅𝜕𝐶𝜕𝑅−𝜙²𝐶^𝑛=2𝑝𝜙²+6𝑝+𝑞4𝑝𝜙²6𝑝+𝑞−𝜙²6𝑝+𝑝𝜙²𝑅²−1+𝑞6𝑝+𝑞^𝑛=0. (5.561) Locating at 𝑅=0 and 𝑅=1/2, we obtain Π0,𝑝,𝑞=2𝑝𝜙²+6𝑝+𝑞4𝑝𝜙²6𝑝+𝑞−𝜙²6𝑝−𝑝𝜙²+𝑞6𝑝+𝑞^𝑛Π1=0,2,𝑝,𝑞=2𝑝𝜙²+6𝑝+𝑞4𝑝𝜙²6𝑝+𝑞−𝜙²6𝑝−(3/4)𝑝𝜙²+𝑞6𝑝+𝑞^𝑛=0. (5.562)(5.563) Solving (5.562) and (5.563) simultaneously, the values of 𝑝 and 𝑞 can be readily determined.

The accuracy of vanishing residuals depends upon the trial functions, (5.555) and (5.556), which are derived from an analogous linear equation, so the obtained solutions are valid for the whole solution domain, keeping maximal accuracy at the location points.

The preceding analysis has the virtue of utter simplicity, and the illustrating example shows that the suggested method is very effective and convenient in solving nonlinear equations. The suggested method can be readily applied to the search for analytical solutions for various nonlinear equations, which should have the character of monotonic increase or monotonic decrease. Many nonlinear problems arising in chemical engineering can be described by, for example, exponential decay/rise functions, so the ancient Chinese method—the He Chengtian interpolation—might be a powerful mathematical tool in chemical engineering.

6. Solitons in Lattice Systems

6.1. Physical Understanding of Differential-Difference Equations

Governing equations for continuum media are well established by using the Gauss’ divergence theorems: _𝑉∇𝜙𝑑𝑉=_𝜕𝑉𝑛𝜙𝑑𝑆,_𝑉∇⋅𝐴𝑑𝑉=_𝜕𝑉𝑛⋅𝐴𝑑𝑆,_𝑉∇×𝐴𝑑𝑉=_𝜕𝑉𝑛×𝐴𝑑𝑆. (6.1)(6.2)(6.3) Mathematically, the above Gauss’ divergence theorems are invalid for fractal media, such as porous media, weaves. So the governing equations for noncontinuum media cannot be expressed by differential equations in space but can be done by difference equations. Considering the continuous time, such problems can be described by differential-difference equations: 𝜕𝑢_𝑛𝑢𝜕𝑡=𝐹_𝑛−1,𝑢_𝑛,𝑢_𝑛+1,𝜕𝑢_𝑛𝜕𝑡=𝐹𝜕𝑢_𝑛−1,𝜕𝑥𝜕𝑢_𝑛,𝜕𝑥𝜕𝑢_𝑛−1𝜕𝑥,𝑢_𝑛−1,𝑢_𝑛,𝑢_𝑛+1,𝜕𝑢_𝑛𝜕𝑡=𝐹𝜕𝑢_𝑛,𝜕𝑥𝜕𝑢_𝑛,𝜕𝑦𝜕𝑢_𝑛𝜕𝑧,𝑢_𝑛−1,𝑢_𝑛,𝑢_𝑛+1. (6.4)

Unlike difference equations which are fully discretized, DDEs are semidiscretized with some of their spatial variables discretized, while time is usually kept continuous.

Solitons and compactons for difference-differential equations have caught much attention due to the fact that discrete spacetime may be the most radical and logical viewpoint of reality [113].

A better physical understanding of differential-difference equations can be obtained by considering the flow through a lattice where the conservation of mass requires [114] 𝑑𝜌_𝑖𝑑𝑡+𝜌_𝑖+1𝑢_𝑖+1−𝜌_𝑖−1𝑢_𝑖−1=0, (6.5) with  𝜌_𝑖 and 𝑢_𝑖 being, respectively, the gas density and velocity at the 𝑖th lattice point (see Figure 6).

Figure 6 

Conservation of mass.

6.2. Exp-Function Method

Consider a hybrid-lattice system [115] 𝜕𝑢_𝑛=𝜕𝑡1+𝛼𝑢_𝑛+𝛽𝑢²_𝑛𝑢_𝑛−1−𝑢_𝑛+1, (6.6) where 𝛼 and  𝛽 are constants.

Let 𝑢_𝑛=𝑢_𝑛(𝜉),𝜉=𝑑𝑛+𝑐₁𝑥+𝑐₂𝑡+𝜉₀, then (6.6) becomes 𝑐₂𝑢^′_𝑛=1+𝛼𝑢_𝑛+𝛽𝑢²_𝑛𝑢_𝑛−1−𝑢_𝑛+1. (6.7) According to the exp-function method, we assume that 𝑢_𝑛(𝜉)=𝑎₁exp(𝜉)+𝑎₀+𝑎₋₁exp(−𝜉)𝑏₁exp(𝜉)+𝑏₀(),𝑢+exp−𝜉_𝑛−1(𝜉)=𝑎₁exp(𝜉−𝑑)+𝑎₀+𝑎₋₁exp(−𝜉+𝑑)𝑏₁exp(𝜉−𝑑)+𝑏₀(),𝑢+exp−𝜉+𝑑_𝑛+1(𝜉)=𝑎₁exp(𝜉+𝑑)+𝑎₀+𝑎₋₁exp(−𝜉−𝑑)𝑏₁exp(𝜉+𝑑)+𝑏₀().+exp−𝜉−𝑑 (6.8) Substituting (6.8) into (6.7), and by the help of Maple, clearing the denominator, and setting the coefficients of power terms like exp(𝑗𝜉),𝑗=1,2,… to zero yield a system of algebraic equations, and we obtain some generalized exact solutions, one of which is expressed in the form [115] 𝑢_𝑛=2𝛼𝛽𝑎²₀𝑒^2𝑑/𝛼²−4𝛽𝑒^2𝑑−1²exp(𝜉)+𝑎₀−𝛼/2𝛽exp(−𝜉)−4𝛼𝛽²𝑎²₀𝑒^2𝑑/𝛼²−4𝛽𝑒^2𝑑−1²(𝜉)(),exp+exp−𝜉 (6.9) where 𝑐₂=(𝛼²−4𝛽)(𝑒^2𝑑−1)/(4𝛽𝑒^𝑑), 𝜉=𝑑𝑛+𝑐₁𝑥+𝑐₂𝑡+𝜉₀, and 𝑐₁ and 𝑎₀ are free parameters.

The application of the exp-function method to the discrete mKdV lattice and the discrete Schrödinger equation was discussed in [116, 117].

6.3. Variational Iteration Method

Consider the Volterra lattice equation [118] 𝜕𝑢_𝑛𝜕𝑡=𝑢_𝑛𝑢_𝑛+1−𝑢_𝑛−1,𝑢_𝑛(0)=𝑛. (6.10) Using the variational iteration method, an iteration formulation can be constructed as follows: 𝑢_𝑛,𝑚=𝑢_{𝑛,𝑚−1}−^𝑡₀𝑢_{𝑛,𝑚−1}𝑢_{𝑛+1,𝑚−1}−𝑢_{𝑛−1,𝑚−1}𝑑𝑡. (6.11) Begining with (𝑢_𝑛)₀=𝑛, we can obtain the following series solution [118]: 𝑢_𝑛(𝑡)=𝑛1+2𝑡+4𝑡²+8𝑡³+16𝑡⁴+32𝑡⁵+64𝑡⁶+𝑂𝑡⁷, (6.12) which converges to the exact solution 𝑢_𝑛(𝑡)=𝑛/(1−2𝑡).

6.4. Homotopy Perturbation Method

Consider the discretized mKdV lattice equation [119] 𝑑𝑢_𝑛𝑑𝑡=𝑢_𝑛²𝑢_𝑛+1−𝑢_𝑛−1,𝑢_𝑛(0)1=1−𝑛². (6.13) Construct a homotopy 1−𝑝𝑑𝑢_𝑛(𝑡)−𝑑𝑡𝑑𝑢_𝑛,0(𝑡)𝑑𝑡+𝑝𝑑𝑢_𝑛(𝑡)𝑑𝑡−𝑢²_𝑛(𝑡)⋅𝑢_𝑛+1(𝑡)−𝑢_𝑛−1(𝑡)=0. (6.14) Suppose the solution of (6.13) to be in the following form: 𝑢_𝑛(𝑡)=𝑢_𝑛,0(𝑡)+𝑝𝑢_𝑛,1(𝑡)+𝑝²𝑢_𝑛,2+⋯. (6.15) By a simple calculation, beginning with 𝑢_𝑛,0(𝑡)=1−1/𝑛², we have [119] 𝑢_𝑛(𝑡)1=1−𝑛²+4𝑡𝑛³−12𝑡²𝑛⁴+32𝑡³𝑛⁵−80𝑡⁴𝑛⁶+⋯, (6.16) which converges to the exact solution 𝑢_𝑛(𝑡)=1−1/(𝑛+2𝑡)².

6.5. Parameter-Expansion Method

We rewrite (6.13) in the form 𝑑𝑢_𝑛𝑑𝑡=1⋅𝑢²_𝑛𝑢_𝑛+1−𝑢_𝑛−1. (6.17) We expand, respectively, the solution and the coefficient, 1, in the forms 𝑢_𝑛(𝑡)=𝑢_𝑛,0(𝑡)+𝑝𝑢_𝑛,1(𝑡)+𝑝²𝑢_𝑛,2+⋯,𝑝=1,1=𝑝𝑎₁+𝑝²𝑎₂+⋯. (6.18) The solution procedure is given in Section 5.8.

6.6. Parameterized Perturbation Method

Parameterized perturbation method was proposed in 1999 [120]; it is useful to obtain exact and approximate solutions of nonlinear differential equations. The method requires no linearization or discretization; large computational work and round-off errors are avoided. It has been used to solve effectively, easily, and accurately a large class of nonlinear problems with approximations. These approximations converge rapidly to accurate solutions [120, 121].

We consider the following general nonlinear DDEs given in the form [122]: 𝑑𝑢_𝑛(𝑡)𝑢𝑑𝑡=𝑓_𝑛−1,𝑢_𝑛,𝑢_𝑛+1, (6.19) with the initial condition 𝑢_𝑛(0)=𝐴(𝑛), where 𝑎, 𝑏, and 𝑐 are constants, and 𝑓 is a nonlinear function.

According to the PPM, an expanding parameter is introduced by a linear transformation 𝑢_𝑛=𝜀𝜈_𝑛, (6.20) where 𝜀 is the introduced perturbation parameter.

Substituting (6.20) into (6.19) results in 𝑑𝜈_𝑛(𝑡)𝑑𝑡=𝑓𝜀;𝜈_𝑛−1,𝜈_𝑛,𝜈_𝑛+1,𝜈_𝑛(0)=𝐴(𝑛)𝜀, (6.21) supposing that the solution of (6.21) can be expressed in the form 𝜈_𝑛=𝜈_𝑛,0+𝜀𝜈_𝑛,1+𝜀²𝜈_𝑛,2+…, (6.22) and processing in a traditional fashion of perturbation technique.

In the following examples, we will illustrate the usefulness and effectiveness of the proposed technique.

Consider the following Volterra equation: 𝑑𝑢_𝑛𝑑𝑡=𝑢_𝑛𝑢_𝑛+1−𝑢_𝑛−1, (6.23) with the initial condition 𝑢_𝑛(0)=𝑛. (6.24)

Using the transformation (6.20), the original equation (6.23) becomes 𝑑𝜈_𝑛𝑑𝑡=𝜀𝜈_𝑛𝜈_𝑛+1−𝜈_𝑛−1,𝜈_𝑛(0)=𝑛𝜀. (6.25) Substituting (6.22) into (6.25) and equating coefficients of like powers of 𝜀 yield the following equations: 𝑑𝜈_𝑛,0𝑑𝑡=0,𝜈_𝑛,0(0)=𝜈_𝑛(0)=𝑛𝜀,𝑑𝜈_𝑛,𝑖=𝑑𝑡^𝑖−1_𝑘=0𝜈_𝑛,𝑘𝜈_{𝑛+1,𝑖−𝑘−1}−𝜈_{𝑛−1,𝑖−𝑘−1},𝜈_𝑛,𝑖(0)=0,𝑖=1,2,…. (6.26) We can start with 𝜈_𝑛,0(0)=𝑛/𝜀, and we obtain the following successive approximations: 𝜈_𝑛,𝑘(𝑡)=2^𝑘𝑛𝜀^𝑘+1𝑡^𝑘,𝑘=0,1,2,…. (6.27) Hence, the solution series in general gives 𝜈_𝑛=𝑛𝜀1+2𝑡+4𝑡²+8𝑡³+16𝑡⁴+⋯, (6.28) then 𝑢_𝑛=𝜀𝜈_𝑛=𝑛1+2𝑡+4𝑡²+8𝑡³+16𝑡⁴+⋯. (6.29) The closed form of the series is 𝑢_𝑛(𝑡)=𝑛,1−2𝑡 (6.30) which gives the exact solution of the problem.

6.7. Solitary-Solution Formulation [123]

Suppose that the differential-difference equation we discuss in this paper is in the following nonlinear polynomial form: 𝑑𝑢_𝑛(𝑡)𝑢𝑑𝑡=𝑓_𝑛−1,𝑢_𝑛,𝑢_𝑛+1, (6.31) where 𝑢_𝑛=𝑢(𝑛,𝑡) is a dependent variable; 𝑡 is a continuous variable; 𝑛,𝑝_𝑖∈𝑍.

Using the basic idea of the ancient Chinese algorithm, we choose two trial functions in the forms 𝑢_𝑛,1(𝑛,𝑡)=𝑓𝜉_𝑛+𝜔₁𝑡,𝑢_𝑛,2(𝑛,𝑡)=𝑔𝜉_𝑛+𝜔₂𝑡, (6.32) where 𝜉_𝑛=𝑛𝑑+𝜉₀, 𝜉₀ is arbitrary, and 𝑓 and 𝑔 are known functions. If a periodic solution is searched for, 𝑓 and 𝑔 must be periodic functions; if a solitary solution is solved, 𝑓 and 𝑔 must be of solitary structures. In this paper, a bell solitary solution of a differential-difference equation is considered, and trial functions are chosen as follows: 𝑢_𝑛,1()=𝐴𝑛,𝑡𝑒^{𝜉𝑛+𝜔1𝑡}+𝑒^{−(𝜉𝑛+𝜔1𝑡)},𝑢+𝐵_𝑛,2()=𝐴𝑛,𝑡𝑒^{𝜉𝑛+𝜔𝑡}+𝑒^{−(𝜉𝑛+𝜔𝑡)}.+𝐵 (6.33) For 𝑢_𝑛, 𝑢_𝑛−1 and 𝑢_𝑛+1 should be compatible, then 𝑢_𝑛−1,1()=𝐴𝑛,𝑡𝑒^{𝜉𝑛−𝑑+𝑡}+𝑒^{−(𝜉𝑛−𝑑+𝑡)},𝑢+𝐵_𝑛+1,1()=𝐴𝑛,𝑡𝑒^{𝜉𝑛+𝑑+𝑡}+𝑒^{−(𝜉𝑛+𝑑+𝑡)},𝑢+𝐵_𝑛−1,2()=𝐴𝑛,𝑡𝑒^{𝜉𝑛−𝑑+𝜔𝑡}+𝑒^{−(𝜉𝑛−𝑑+𝜔𝑡)},𝑢+𝐵_𝑛+1,2()=𝐴𝑛,𝑡𝑒^{𝜉𝑛+𝑑+𝜔𝑡}+𝑒^{−(𝜉𝑛+𝑑+𝜔𝑡)}.+𝐵 (6.34)(6.35)(6.36)(6.37)

6.8. Solution Procedure

Step 1. Define the residual function 𝑅(𝑡)=𝑑𝑢_𝑛(𝑡)𝑢𝑑𝑡−𝑓_𝑛−1,𝑢_𝑛,𝑢_𝑛+1. (6.38) Substituting (6.40)~(6.43) into (6.37), we can obtain, respectively, the residual functions 𝑅₁ and 𝑅₂.

Step 2. Substitute 𝑅₁ and 𝑅₂ into the following equation: 𝜔²=𝜔²₁𝑅₂(0)−𝜔²₂𝑅₁(0)𝑅₂(0)−𝑅₁(0), (6.39) where 𝜔₂=𝜔.

Step 3. Combining the coefficients of 𝑒^𝜉𝑛 in (6.45), and setting them to be zero, we can solve the algebraic equations to find the values of 𝜔, 𝐴, and 𝐵. Finally, an explicit solution is obtained.
The famous mKdV lattice equation reads 𝑑𝑢_𝑛=𝑑𝑡𝛼−𝑢²_𝑛𝑢_𝑛−1−𝑢_𝑛+1, (6.40) where 𝛼≠0.

Substituting (6.40)~(6.43) into (6.46), we obtain, respectively, the following residual functions 𝑅₁ and 𝑅₂, and using the solitary-solution formulation, (6.45), we have 𝑒^3𝜉𝑛𝜔³+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑−𝜔−𝛼𝑒^−𝑑+𝛼𝑒^𝑑−𝑒^−3𝜉𝑛𝜔³+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑−𝜔−𝛼𝑒^−𝑑+𝛼𝑒^𝑑+𝑒^2𝜉𝑛𝐵𝜔³𝑒^−𝑑+𝜔³𝑒^𝑑−2𝜔²𝛼𝑒^𝑑+2𝜔²𝛼𝑒^−𝑑−𝜔𝑒^𝑑−𝜔𝑒^−𝑑+2𝛼𝑒^𝑑−2𝛼𝑒^−𝑑−𝑒^−2𝜉𝑛𝐵𝜔³𝑒^−𝑑+𝜔³𝑒^𝑑−2𝜔²𝛼𝑒^𝑑+2𝜔²𝛼𝑒^−𝑑−𝜔𝑒^𝑑−𝜔𝑒^−𝑑+2𝛼𝑒^𝑑−2𝛼𝑒^−𝑑+𝑒^𝜉𝑛𝜔³𝐵²−𝜔³+𝜔³𝑒^−2𝑑+𝜔³𝑒^2𝑑+𝜔²𝐴²𝑒^𝑑−𝜔²𝐴²𝑒^−𝑑+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑+𝜔²𝛼𝐵²𝑒^−𝑑−𝜔²𝛼𝐵²𝑒^𝑑−𝜔𝐵²+𝜔−𝜔𝑒^2𝑑−𝜔𝑒^−2𝑑+𝐴²𝑒^−𝑑−𝐴²𝑒^𝑑+𝛼𝐵²𝑒^𝑑−𝛼𝐵²𝑒^−𝑑+𝛼𝑒^𝑑−𝛼𝑒^−𝑑−𝑒^−𝜉𝑛𝜔³𝐵²−𝜔³+𝜔³𝑒^−2𝑑+𝜔³𝑒^2𝑑+𝜔²𝐴²𝑒^𝑑−𝜔²𝐴²𝑒^−𝑑+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑+𝜔²𝛼𝐵²𝑒^−𝑑−𝜔²𝛼𝐵²𝑒^𝑑−𝜔𝐵²+𝜔−𝜔𝑒^2𝑑−𝜔𝑒^−2𝑑+𝐴²𝑒^−𝑑−𝐴²𝑒^𝑑+𝛼𝐵²𝑒^𝑑−𝛼𝐵²𝑒^−𝑑+𝛼𝑒^𝑑−𝛼𝑒^−𝑑=0. (6.41) Setting the coefficients of 𝑒^𝑖𝜉𝑛 to be zero, we have 𝜔³+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑−𝜔−𝛼𝑒^−𝑑+𝛼𝑒^𝑑𝐵𝜔=0,₃𝑒^−𝑑+𝜔³𝑒^𝑑−2𝜔²𝛼𝑒^𝑑+2𝜔²𝛼𝑒^−𝑑−𝜔𝑒^𝑑−𝜔𝑒^−𝑑+2𝛼𝑒^𝑑−2𝛼𝑒^−𝑑𝜔=0,₃𝐵²−𝜔³+𝜔³𝑒^−2𝑑+𝜔³𝑒^2𝑑+𝜔²𝐴²𝑒^𝑑−𝜔²𝐴²𝑒^−𝑑+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑+𝜔²𝛼𝐵²𝑒^−𝑑−𝜔²𝛼𝐵²𝑒^𝑑−𝜔𝐵²+𝜔−𝜔𝑒^2𝑑−𝜔𝑒^−2𝑑+𝐴²𝑒^−𝑑−𝐴²𝑒^𝑑+𝛼𝐵²𝑒^𝑑−𝛼𝐵²𝑒^−𝑑+𝛼𝑒^𝑑−𝛼𝑒^−𝑑=0. (6.42) Solving (6.48)–(6.49) simultaneously, we have (𝑑),√𝜔=2𝛼sinh𝐴=2−𝛼sinh(𝑑),𝐵=0. (6.43)(6.44)(6.45) We, therefore, obtain the following needed solitary solution: 𝑢_𝑛=2√−𝛼sinh(𝑑)𝑒^{𝜉𝑛+2𝛼sinh(𝑑)𝑡}+𝑒^{−(𝜉𝑛+2𝛼sinh(𝑑)𝑡)}=√−𝛼sinh(𝑑)sech2𝛼sinh(𝑑)𝑡+𝑛𝑑+𝜉₀. (6.46) We can also use the following assumptions for solitary solutions: 𝑢_𝑛,1()=𝑑𝑛,𝑡𝑎𝑒^{𝜉𝑛+𝜔1𝑡}+𝑏𝑒^{−(𝜉𝑛+𝜔1𝑡)},𝑢+𝑐_𝑛,2()=∑𝑛,𝑡^{𝑑𝑛=−𝑐}𝑎_𝑛exp𝜉_𝑛+𝜔𝑡∑^𝑔_𝑚=−𝑓𝑏_𝑚exp𝜉_𝑛.+𝜔𝑡 (6.47)(6.48) For compact-like solutions, we can assume that the solution has the form 𝑢_𝑛,1()=𝑑𝑛,𝑡𝑎+cos²𝜉_𝑛+𝜔₁𝑡,𝑢₀(𝑥,𝑡)=𝑎sin²(𝑘𝑥+𝑤𝑡)𝑏+𝑐sin²().𝑘𝑥+𝑤𝑡 (6.49)(6.50)

7. Fractional Differential Equations

7.1. Physical Understanding of Fractional Differential Equations

Fractional differential equations have caught much attention recently due to the exact description of nonlinear phenomena. No analytical method was available before 1998 for such equations even for linear fractional differential equations.

In 1998, the variational iteration method was first proposed to solve fractional differential equations with greatest success, see [124]. Following the above idea, Draganescu, Momani, and Odibat applied the variational iteration method to more complex fractional differential equations, showing effectiveness and accuracy of the used method, see [125, 126]. In 2002, the Adomian method was suggested to solve fractional differential equations [127]. But many researchers found that it is very difficult to calculate the Adomian polynomial, see comments in [128]. Ghorbani and Saberi-Nadjafi suggested a very simple method for calculating the Adomian polynomial using the homotopy perturbation method [129], and He polynomial should be used instead of Adomian polynomial [130].

In 2007, Momani and Odibat [131] applied the homotopy perturbation method to fractional differential equations and revealed that the homotopy perturbation method is an alternative analytical method for fractional differential equations.

Although the fractional calculus was invented by Newton and Leibnitz over three centuries ago, it only became a hot topic recently owing to the development of the computer and its exact description of many real-life problems. To give a physical interpretation of the fractional calculus, we begin with a simple function 𝑦=𝑥². (7.1) When 𝑥=𝑥₁ and 𝑥=𝑥₂, we have 𝑦₁=𝑥²₁,𝑦₂=𝑥²₂. (7.2) We therefore have the difference Δ𝑦=𝑦₁−𝑦₂=𝑥²₁−𝑥²₂=𝑥₁−𝑥₂𝑥₁+𝑥₂=𝑥₁+𝑥₂Δ𝑥. (7.3) In case 𝑥₁→𝑥₂ or Δ𝑥→0, we have the differential 𝑑𝑦=2𝑥𝑑𝑥. (7.4) The above derivation, however, is only valid for continuous functions. To show its applications, we consider the action in String theory [132] 𝑆=𝑚𝑐𝑑𝑠, (7.5) where 𝑚 is the mass of a particle, 𝑐 is the speed of light, and 𝑑𝑠 is the relativistic metric that can be expressed in the form 𝑑𝑠=𝑐²𝑑𝑡²−𝑑𝑥²−𝑑𝑦²−𝑑𝑧²=𝑐𝑢1−²𝑐²𝑑𝑡. (7.6) Substitution of (7.6) into (7.5) leads to 𝑆=𝑚𝑐²𝑢1−²𝑐²𝑑𝑡, (7.7) from which the equation of free motion is obtained in the usual way 𝑑𝑑𝑡𝑚𝑢1−𝑢²/𝑐²=0. (7.8)

The above derivation assumes that space and time are both continuous. The distance between two points, for example, cannot be expressed in the form of (7.6) in a discontinuous world (e.g., in the Julia set).

Now consider a plane with fractal structure (see Figure 7). The shortest path between two points is not a line, and we have 𝑑𝑠_𝐸=𝑘𝑑𝑠^𝐷, (7.9) where 𝑑𝑠_𝐸 is the actual distance between two points (discontinuous line in Figure 7), 𝑑𝑠 is the line distance between two points (continuous line in Figure 7), 𝐷 is the fractal dimension, and 𝑘 is a constant.

Figure 7 

The distance between two points in a discontinuous spacetime.

The action in a discontinuous spacetime can therefore be written in the following form using fractional calculus: 𝑆=𝑚𝑐𝑑𝑠_𝐸=𝑚𝑐^1+𝐷𝑘𝑢1−²𝑐^2𝐷/2𝑑𝑡^𝐷. (7.10) Fractional calculus is therefore valid for discontinuous problems. Now we consider a well-known predator-prey model (the Lotka-Volterra equation) [133] 𝑑𝑥,𝑑𝑡=𝑥𝑎−𝑏𝑦𝑑𝑦,𝑑𝑡=−𝑦𝑐−𝑑𝑦 (7.11) where 𝑦 is the number of predators (e.g., wolves), 𝑥 is the number of its prey (e.g., rabbits), and 𝑎,𝑏,𝑐, and 𝑑 are parameters representing the interaction of the two species.

In general, the growth of the two populations is discontinuous, and a simple modification of the predator-prey model is to replace 𝑑𝑦/𝑑𝑡  and  𝑑𝑥/𝑑𝑡  by fractional derivatives 𝐷^𝛼𝑥𝐷𝑡^𝛼,𝐷=𝑥𝑎−𝑏𝑦_𝛽𝑦𝐷𝑡^𝛽=−𝑦𝑐−𝑑𝑦, (7.12) where the populations of the predator and prey may be greatly affected by the fractional orders, 𝛼 and 𝛽.

7.2. Variational Iteration Method

Consider the following general fractional differential equation:𝐷^𝛼𝑢𝐷𝑡^𝛼+𝑓=0. (7.13a) In the case 0<𝛼<1, we rewrite (7.13a) in the form 𝑑𝑢+𝐷𝑑𝑡^𝛼𝑢𝐷𝑡^𝛼−𝑑𝑢𝑑𝑡+𝑓=0, (7.13b)and the variational iteration algorithms are given as follows: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)−^𝑡₀𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢₀(𝑡)−^𝑡₀𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑𝑢_𝑛𝑑𝑡+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢₀(𝑡)−^𝑡₀𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑𝑢_𝑛𝑑𝑡+𝑓_𝑛−𝐷^𝛼𝑢_𝑛−1𝐷𝑡^𝛼−𝑑𝑢_𝑛−1𝑑𝑡+𝑓_𝑛−1𝑑𝑠. (7.14) In the case 1<𝛼<2, the above iteration formulas are also valid. We can rewrite (7.13a) in the form 𝑑²𝑢𝑑𝑡²+𝐷^𝛼𝑢𝐷𝑡^𝛼−𝑑²𝑢𝑑𝑡²+𝑓=0, (7.15) and the following iteration formulae are suggested: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡₀()𝐷𝑠−𝑡^𝛼𝑢_𝑛𝐷𝑡^𝛼+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢₀(𝑡)+^𝑡₀()𝐷𝑠−𝑡^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑²𝑢_𝑛𝑑𝑡²+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡₀(𝐷𝑠−𝑡)^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑²𝑢_𝑛𝑑𝑡²+𝑓_𝑛−𝐷^𝛼𝑢_𝑛−1𝐷𝑡^𝛼−𝑑²𝑢_𝑛−1𝑑𝑡²+𝑓_𝑛−1𝑑𝑠. (7.16) When 𝛼 is close to 1, (7.14) is better, while (7.16) is recommended for 𝛼 approaching 2.

For the case 𝑁<𝛼<𝑁+1, where 𝑁 is a natural number, the iteration formulas 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+(−1)^𝑁𝑡₀1()!()𝑁−1𝑠−𝑡^𝑁−1𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢₀(𝑡)+(−1)^𝑁𝑡₀1()!()𝑁−1𝑠−𝑡^𝑁−1𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑^𝑁𝑢_𝑛𝑑𝑡^𝑁+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+(−1)^𝑁×^𝑡₀1()!()𝑁−1𝑠−𝑡^𝑁−1×𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑^𝑁𝑢_𝑛𝑑𝑡^𝑁+𝑓_𝑛−𝐷^𝛼𝑢_𝑛−1𝐷𝑡^𝛼−𝑑^𝑁𝑢_𝑛−1𝑑𝑡^𝑁+𝑓_𝑛−1𝑑𝑠, (7.17) or 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+(−1)^𝑁+1𝑡₀1()𝑁!𝑠−𝑡^𝑁𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢₀(𝑡)+(−1)^𝑁+1𝑡₀1()𝑁!𝑠−𝑡^𝑁𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑^𝑁+1𝑢_𝑛𝑑𝑡^𝑁+1+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+(−1)^𝑁+1×^𝑡₀1()𝑁!𝑠−𝑡^𝑁𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑^𝑁+1𝑢_𝑛𝑑𝑡^𝑁+1+𝑓_𝑛−𝐷^𝛼𝑢_𝑛−1𝐷𝑡^𝛼−𝑑^𝑁+1𝑢_𝑛−1𝑑𝑡^𝑁+1+𝑓_𝑛−1𝑑𝑠 (7.18) can be used. When 𝛼 is close to 𝑁, (7.17) is recommended, and (7.18) works more effectively for 𝛼 approaching 𝑁+1.

We consider the fractional calculus version of the standard vibration equation in one dimension as [133] 𝜕²𝑢𝜕𝑟²+1𝑟𝜕𝑢=1𝜕𝑟𝑐²𝜕^𝛽𝑢𝜕𝑡^𝛽,𝑟≥0,𝑡≥0,1<𝛽≤2, (7.19) with initial conditions 𝑢(𝑟,0)=𝑟²,𝜕𝑢()𝜕𝑡𝑟,0=𝑐𝑟. (7.20) Equation (7.19) can be written as 𝜕²𝑢𝜕𝑡²=𝑐²𝜕^2−𝛽𝜕𝑡^2−𝛽𝜕²𝑢𝜕𝑟²+1𝑟𝜕𝑢.𝜕𝑟 (7.21) According to the variational iteration method, we consider the correction functional in 𝑡-direction in the following form: 𝑢_𝑛+1(𝑟,𝑡)=𝑢_𝑛(𝑟,𝑡)+^𝑡₀(𝜕𝜉−𝑡)²𝑢_𝑛(𝑟,𝜉)𝜕𝜉²−𝑐²𝜕^2−𝛽𝜕𝑡^2−𝛽𝜕²𝑢_𝑛(𝑟,𝜉)𝜕𝑟²+1𝑟𝜕𝑢_𝑛(𝑟,𝜉)𝜕𝑟𝑑𝜉. (7.22) Beginning with an initial approximation 𝑢₀(𝑟,𝑡)=𝑢(𝑟,0)+𝑡𝑢_𝑡(𝑟,0)=𝑟²+𝑐𝑡𝑟, (7.23) we obtain [133] 𝑢(𝑟,𝑡)=lim_𝑛→∞𝑢_𝑛(𝑟,𝑡)=𝑟²+𝑐𝑡𝑟+4𝑐²𝑡^𝛽Γ+𝑐𝛽+1³𝑡^𝛽+1+𝑐𝑟Γ𝛽+2⁵𝑡^2𝛽+1𝑟³Γ+2𝛽+29𝑐⁷𝑡^3𝛽+1𝑟⁵Γ3𝛽+2+⋯=𝑟²+4𝑐²𝑡^𝛽Γ𝛽+1+𝑐𝑡𝑟𝐸_𝛽,2𝑐²𝑟²𝑘𝑡^𝛽, (7.24) where 𝑘^𝑛=[1×3×5×…×(2𝑛−3)]² and 𝐸_𝛽,𝑏(𝑡)=∑^∞_𝑛=0𝑡^𝑛/Γ(𝑛𝛽+𝑏) are the generalized Mittag-Leffler function.

Consider another nonlinear fractional differential equation [134] 𝐷^𝛼_∗𝑢(𝑥)+𝑢^′(𝑥)+𝑢³(𝑥)=0,𝑥>0, (7.25) where 1<𝛼≤2, subject to the initial condition 𝑢(0)=1,𝑢^′(0)=0. (7.26) We can construct a correction functional in the form 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+^𝑥₀𝜆(𝜉)𝐷⁽²⁾𝑢_𝑛(𝜉)+𝑀𝑢_𝑛(𝜉)−𝐷⁽²⁾𝑢_𝑛(𝜉)+𝐷^𝛼𝑢_𝑛(𝜉)𝑑𝜉. (7.27) After identification of the Lagrange multiplier, we obtain the following iteration formulation: 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+^𝑥₀(𝜉−𝑥)𝐷^𝛼_∗𝑢_𝑛(𝜉)+𝑢^′_𝑛(𝜉)+𝑢³(𝜉)𝑑𝜉. (7.28) Consequently, beginning with 𝑢₀=1, we find the following approximations: 𝑢₀(𝑥)𝑢=1,₁(𝑥)𝑥=1−²2,𝑢₂(𝑥)=1−𝑥²+𝑥³6+𝑥⁴8−𝑥⁶+𝑥40⁸−Γ()𝑥4484−𝛼^4−𝛼Γ()Γ()+𝑥3−𝛼5−𝛼Γ(^4−𝛼).4−𝛼 (7.29)

7.3. Homotopy Perturbation Method

The homotopy perturbation method becomes an effective tool to fractional differential equations [134–137]. Consider the nonlinear time-fractional RLW equation [134] 𝜕^𝛼𝑢𝜕𝑡^𝛼−𝑢_𝑥𝑥𝑡+𝑢²2_𝑥=0,𝑡>0, (7.30) where 0<𝛼≤1, subject to the initial condition 𝑢(𝑥,0)=𝑥. (7.31) We can construct the following homotopy: 𝜕𝑢𝜕𝑡=𝑝𝜕𝑢𝜕𝑡+𝑢_𝑥𝑥𝑡−𝑢²2_𝑥−𝐷^𝛼_𝑡𝑢. (7.32) Processing the same solution procedure as the homotopy perturbation method, we obtain 𝑢₀𝑢=𝑥,₁𝑢=−𝑥𝑡,₂=𝑥−𝑡+𝑡²+𝑡^2−𝛼Γ(),𝑢3−𝛼₃=𝑥−𝑡+2𝑡²−𝑡³+2𝑡^2−𝛼Γ()−3−𝛼4𝑡^3−𝛼Γ()−𝑡4−𝛼Γ(^3−2𝛼),4−2𝛼 (7.33) and the solution can be approximately expressed as 𝑢=𝑢₀+𝑢₁+𝑢₂+𝑢₃+…. (7.34)

7.4. Fractional Complex Transform

The fractional complex transform [138, 139] was originally suggested to convert a fractional differential equation with Jumarie’s modification of Riemann-Liouville derivative into its classical differential partner.

Consider the following general fractional differential equation: 𝑓𝑢,𝑢_𝑡^(𝛼),𝑢_𝑥^(𝛽),𝑢_𝑦^(𝛾),𝑢_𝑧^(𝜆),𝑢_𝑡^(2𝛼),𝑢_𝑥^(2𝛽),𝑢_𝑦^(2𝛾),𝑢_𝑧^(2𝜆),…=0,0<𝛼≤1,0<𝛽≤1,0<𝛾≤1,0<𝜆≤1, (7.35) where 𝑢_𝑡^(𝛼)=𝐷^𝛼_𝑡𝑢=𝐷^𝛼𝑢/𝐷𝑡^𝛼 denotes Jumarie’s fractional derivation, which is a modified Riemann-Liouville derivative defined as [140] 𝐷^𝛼_𝑡𝑢=1𝑡,𝑥,𝑦,𝑧Γ()𝑑1−𝛼𝑑𝑡^𝑡₀(𝑡−𝜉)^−𝛼𝑢𝜉,𝑥,𝑦,𝑧−𝑢0,𝑥,𝑦,𝑧𝑑𝜉, (7.36) where 𝑢 is a continuous (but not necessarily differentiable) function 𝜕^𝛼𝑐𝜕𝑡^𝛼𝜕=0,_𝛼𝜕𝑡^𝛼[]𝜕𝑐𝑢=𝑐^𝛼𝑢𝜕𝑡^𝛼,𝜕^𝛼𝑡^𝛽𝜕𝑡^𝛼=Γ1+𝛽Γ𝑡1+𝛽−𝛼^𝛽−𝛼,𝛽≥𝛼>0. (7.37)(7.38)(7.39) In our previous publications [138], there were some flaws. The solution procedure should be followed as follows.

Using the following transforms: 𝑠=𝑡^𝛼,𝑋=𝑥^𝛽,𝑌=𝑦^𝛾,𝑍=𝑧^𝜆, (7.40) we have 𝜕^𝛼𝑢𝜕𝑡^𝛼=𝜕𝑢⋅𝜕𝜕𝑠^𝛽𝑠𝜕𝑥^𝛽=𝜎_𝑠𝜕𝑢,𝜕𝜕𝑠_𝛽𝑢𝜕𝑥^𝛽=𝜕𝑢⋅𝜕𝜕𝑋^𝛽𝑋𝜕𝑥^𝛽=𝜎_𝑋𝜕𝑢,𝜕𝜕𝑋_𝛾𝑢𝜕𝑦^𝛾=𝜕𝑢⋅𝜕𝜕𝑌^𝛾𝑌𝜕𝑦^𝛾=𝜎_𝑌𝜕𝑢,𝜕𝜕𝑌_𝜆𝑢𝜕𝑧^𝜆=𝜕𝑢⋅𝜕𝜕𝑍^𝜆𝑍𝜕𝑧^𝜆=𝜎_𝑍𝜕𝑢,𝜕𝑍 (7.41) where 𝜎_𝑠, 𝜎_𝑋, 𝜎_𝑌, and 𝜎_𝑍 are fractal indexes [139]. We can, therefore, easily convert fractional differential equations into partial differential equations, so that everyone familiar with advanced calculus can deal with fractional calculus without any difficulty.

To determine  𝜎_𝑠, we consider a special case 𝑠=𝑡^𝛼 and  𝑢=𝑠^𝑚, and we have 𝜕^𝛼𝑢𝜕𝑡^𝛼=Γ(1+𝑚𝛼)⋅𝑡^{𝑚𝛼−𝛼}Γ(1+𝑚𝛼−𝛼)=𝜎𝜕𝑢𝜕𝑠=𝜎𝑚𝑡^{𝑚𝛼−𝛼}. (7.42) We, therefore, can determine 𝜎_𝑠 as follows: 𝜎_𝑠=Γ(1+𝑚𝛼)().𝑚Γ1+𝑚𝛼−𝛼 (7.43) Other fractal indexes (𝜎_𝑋, 𝜎_𝑌, 𝜎_𝑍) can be determined in a similar way.

As an example, consider the fractional differential equation 𝜕^𝛼𝑢𝜕𝑡^𝛼+𝐵𝑢=0,0<𝛼<1,𝑢(0)=1. (7.44) Assume that the solution can be expressed in a series in the form 𝑢=^∞_𝑚=0𝑎_𝑚𝑠^𝑚, (7.45) where 𝑎_𝑚(𝑚=0,1,2,3,…) are constants to be further determined.

Using the transform 𝑠=𝑡^𝛼, and submitting (7.45) into (7.44), we have 𝜕𝜕𝑠^∞_𝑚=0𝜎_𝑠𝑚𝑎_𝑚𝑠^𝑚+𝐵^∞_𝑚=0𝑎_𝑚𝑠^𝑚=0, (7.46) or ^∞_𝑚=0𝑚𝜎_𝑠𝑚𝑎_𝑚𝑠^𝑚−1+𝐵^∞_𝑚=0𝑎_𝑚𝑠^𝑚=0. (7.47) According to (7.43), the fractal index 𝜎_𝑠𝑚 can be determined as follows: 𝜎_𝑠𝑚=Γ(1+𝑚𝛼)().𝑚Γ1+𝑚𝛼−𝛼 (7.48) From (7.47) and (7.48), we obtain Γ(1+𝑚𝛼)Γ()𝑎1+𝑚𝛼−𝛼^𝑚+𝐵𝑎_𝑚−1=0. (7.49) Generally, we begin with 𝑢₀=𝑢(0)=1. After a simple calculation, we have 𝑎_𝑚=(−𝐵)^𝑚Γ().1+𝑚𝛼 (7.50) We, therefore, obtain the following solution: 𝑢(𝑠)=^∞_𝑚=0(−𝐵)^𝑚Γ()𝑠1+𝑚𝛼^𝑚, (7.51) or 𝑢(𝑡)=^∞_𝑚=0(−𝐵)^𝑚Γ()𝑡1+𝑚𝛼^𝑚𝛼=𝐸_𝛼−𝐵𝑡^𝛼, (7.52) where 𝐸_𝛼 is a Mittag-Leffler function defined as 𝐸_𝛼(𝑡)=^∞_𝑚=0𝑡^𝑚Γ().1+𝑚𝛼 (7.53) Equation (7.53) is the exact solution of the example.

7.5. Yang-Laplace Transform [141]

The local fractional functional analysis was first proposed by Yang [141]. The Yang-Laplace transform of 𝑓(𝑥) is defined as𝐿_𝛼𝑓(𝑥)=𝑓_𝑠^𝐿,𝛼(𝑠)1∶=Γ(1+𝛼)^∞₀𝐸_𝛼−𝑠^𝛼𝑥^𝛼𝑓(𝑥)(𝑑𝑥)^𝛼,0<𝛼≤1, (7.54) and the inverse formula is 𝐿_𝛼⁻¹𝑓_𝑠^𝐿,𝛼(𝑠)(𝑡)1=𝑓=∶(2𝜋)^{𝛼𝛽+𝑖∞}_{𝛽−𝑖∞}𝐸_𝛼𝑠^𝛼𝑥^𝛼𝑓_𝑠^𝐿,𝛼(𝑠)(𝑑𝑠)^𝛼, (7.55) where 𝑠^𝛼=𝛽^𝛼+𝑖^𝛼∞^𝛼 and Re(𝑠^𝛼)=𝛽^𝛼>0^𝛼.

Table of the Yang-Laplace Transform of Elementary Functions (1)𝐿_𝛼𝛿_𝛼(𝑥(2)𝐿)=1._𝛼𝛿_𝛼(𝑥−𝑎)=𝐸_𝛼−𝑎^𝛼𝑠^𝛼.(3)𝐿_𝛼𝐻_𝛼(𝑥=1)𝑠^𝛼.(4)𝐿_𝛼𝐻_𝛼(𝑥)𝑥^𝛼=1𝑠^2𝛼.(5)𝐿_𝛼𝛿_𝛼^(𝑛𝛼)(𝑥)=𝑠^𝑛𝛼.(6)𝐿_𝛼{1}=1𝑠^𝛼.(7)𝐿_𝛼𝐸_𝛼𝑎^𝛼𝑥^𝛼=1𝑠^𝛼−𝑎^𝛼.(8)𝐿_𝛼𝑥^𝛼𝐸_𝛼𝑎^𝛼𝑥^𝛼=1(𝑠−𝑎)^2𝛼.(9)𝐿_𝛼sin_𝛼(𝑎𝑥)^𝛼=𝑎^𝛼𝑠^2𝛼−𝑎^2𝛼.(10)𝐿_𝛼cos_𝛼(𝑎𝑥)^𝛼=𝑠^𝛼𝑠^2𝛼+𝑎^2𝛼.(11)𝐿_𝛼𝐸_𝛼−𝑏^𝛼𝑥^𝛼sin_𝛼(𝑎𝑥)^𝛼=𝑎^𝛼(𝑠+𝑏)^2𝛼+𝑎^2𝛼.(12)𝐿_𝛼𝐸_𝛼−𝑏^𝛼𝑥^𝛼cos_𝛼(𝑎𝑥)^𝛼=𝑎^𝛼+𝑏^𝛼(𝑠+𝑏)^2𝛼+𝑎^2𝛼.(13)𝐿_𝛼𝑥^𝑘𝛼=Γ(1+𝑘𝛼)𝑠^(𝑘+1)𝛼.(14)𝐿_𝛼𝑥^𝑘𝛼𝐸_𝛼𝑎^𝛼𝑥^𝛼=Γ(1+𝑘𝛼)(𝑠−𝑎)^(𝑘+1)𝛼.(15)𝐿_𝛼𝐸_𝛼𝑎^𝛼𝑥^𝛼−𝐸_𝛼𝑏^𝛼𝑥^𝛼=𝑎^𝛼−𝑏^𝛼(𝑠−𝑎)^𝛼(𝑠−𝑏)^𝛼. (7.56) The Yang-Laplace transform is an effective mathematical tool for solving local fractional differential equations. Consider the following local fractional differential equation: 𝑦^(𝛼)(𝑡)+2𝑦(𝑡)=𝐸_𝛼−𝑡^𝛼,0<𝛼≤1, (7.57) with initial condition 𝑦(𝑡)|_𝑡=0=0.

By the Yang-Laplace transform, (7.55) becomes 𝑠^𝛼𝑦_𝑠^𝐿,𝛼(𝑠)+2𝑦_𝑠^𝐿,𝛼(𝑠)=1.1+𝑠 (7.58) Solving 𝑦_𝑠^𝐿,𝛼 from (7.58), we have 𝑦_𝑠^𝐿,𝛼(𝜔)=11+𝑠^𝛼−12+𝑠^𝛼. (7.59) The inverse Yang-Laplace transform of (7.59) gives 𝑦(𝑡)=𝐸_𝛼−𝑡^𝛼−𝐸_𝛼−2𝑡^𝛼. (7.60) Consider another local fractional differential equation given by 𝜕^𝛼𝑢𝜕𝑦^𝛼+𝑘^𝛼𝑢=0,𝑦>0,0<𝛼≤1,𝑢(0)=1. (7.61) Taking the Yang-Laplace transform, we have 𝑠^𝛼𝑢(𝑠)+𝑘^𝛼𝑢(𝑠)−𝑢(0)=0. (7.62) It is obvious that 𝑢(𝑠)=1𝑠^𝛼+𝑘^𝛼. (7.63) The inverse Yang-Laplace transform results in the following solution: 𝑢𝑦=𝐸_𝛼𝑘^𝛼𝑦^𝛼. (7.64)

7.6. Yang-Fourier Transform

Yang-Fourier transform is also an effective method for local fractional differential equations. The Yang-Fourier transform of 𝑓(𝑥) is given by 𝐹_𝛼𝑓(𝑥)=𝑓_𝜔^𝐹,𝛼(𝜔)1∶=Γ(1+𝛼)^∞_−∞𝐸_𝛼−𝑖^𝛼𝜔^𝛼𝑥^𝛼𝑓(𝑥)(𝑑𝑥)^𝛼, (7.65) and the inverse formula of the Yang-Fourier transform is 𝑓(𝑥)=𝐹_𝛼⁻¹𝑓_𝜔^𝐹,𝛼(𝜔)1∶=(2𝜋)^𝛼∞_−∞𝐸_𝛼𝑖^𝛼𝜔^𝛼𝑥^𝛼𝑓_𝜔^𝐹,𝛼(𝜔)(𝑑𝜔)^𝛼. (7.66) The basic properties of the Yang-Fourier transform are as follows: 𝐹_𝛼𝑎𝑓(𝑥)+𝑏𝑔(𝑥)=𝑎𝐹_𝛼𝑓(𝑥)+𝑏𝐹_𝛼𝑔(𝑥,𝐹_𝛼)𝑓^(𝛼)(𝑥)=𝑖^𝛼𝜔^𝛼𝐹_𝛼𝑓(𝑥). (7.67)

Table of Local Fractional Fourier Transforms (1)𝐹_𝛼𝛿_𝛼(𝑥(2)𝐹)=1._𝛼{1}=(2𝜋)^𝛼Γ()𝛿1+𝛼^𝛼(𝜔).(3)𝐹_𝛼𝐸_𝛼𝑖^𝛼𝜔^𝛼₀𝑥^𝛼=(2𝜋)^𝛼Γ()𝛿1+𝛼^𝛼𝜔−𝜔₀.(4)𝐹_𝛼𝛿_𝛼𝑥−𝑥₀𝑓(𝑥)=𝐸_𝛼−𝑖^𝛼𝜔^𝛼𝑥^𝛼₀𝑓𝑥₀.(5)𝐹_𝛼𝛿_𝛼𝑥−𝑥₀=𝐸_𝛼−𝑖^𝛼𝜔^𝛼𝑥^𝛼₀.(6)𝐹_𝛼cos_𝛼(𝑎𝑥)^𝛼=𝜋^𝛼𝛿_𝛼(𝜔+𝑎)+𝛿_𝛼(.(7)𝐹𝜔−𝑎)_𝛼sin_𝛼(𝑎𝑥)^𝛼=𝜋^𝛼𝑖^𝛼𝛿_𝛼(𝜔+𝑎)−𝛿_𝛼(.(8)𝐹𝜔−𝑎)_𝛼𝐻_𝛼(𝑥)𝐸_𝛼𝑖^𝛼𝜔^𝛼₀𝑥^𝛼=1𝑖^𝛼(𝜔−𝑎)^𝛼+(2𝜋)^𝛼()𝛿2Γ1+𝛼^𝛼().(9)𝐹𝜔−𝑎_𝛼|𝑥|^𝛼2=−𝜔^2𝛼.(10)𝐹_𝛼𝛿_𝛼^(𝑛)(𝑥)=𝑗^𝑛𝛼𝜔^𝑛𝛼.(11)𝐹_𝛼𝑥^𝑛𝛼=(2𝜋)^𝛼Γ()𝛿1+𝛼^{𝛼(𝑛𝛼)}(𝜔).(12)𝐹_𝛼𝐻_𝛼(𝑥)𝑥^𝑛𝛼=Γ(1+𝑛𝛼)𝑖^𝑛𝛼𝜔^𝑛𝛼+(2𝜋)^𝛼()𝑖2Γ1+𝛼^𝑛𝛼𝛿_𝛼^(𝑛𝛼)(𝜔). (7.68) Consider a local fractional differential equation 𝑦^(𝛼)(𝑡)+2𝑦(𝑡)=𝐸_𝛼−𝑡^𝛼,0<𝛼≤1, (7.69) with initial condition 𝑦(0)=0.

Taking Yang-Fourier transform for (7.69), 𝑖^𝛼𝜔^𝛼𝑦_𝜔^𝐹,𝛼(𝜔)+2𝑦_𝜔^𝐹,𝛼(𝜔)=11+𝑖^𝛼𝜔^𝛼, (7.70)

we have 𝑦_𝜔^𝐹,𝛼(𝜔)=11+𝑖^𝛼𝜔^𝛼−12+𝑖^𝛼𝜔^𝛼. (7.71)

The inverse Yang-Fourier transform gives 𝑓(𝑡)=𝐸_𝛼−𝑡^𝛼−𝐸_𝛼−2𝑡^𝛼. (7.72)

Consider another local fractional differential equation 𝜕^𝛼𝑢𝜕𝑦^𝛼+𝑘^𝛼𝑢=0,𝑦∈ℝ,0<𝛼≤1,𝑢(0)=1. (7.73)

Taking the Yang-Fourier transform yields 𝑖^𝛼𝜔^𝛼𝑢(𝜔)+𝑘^𝛼𝑢(𝜔)−1=0, (7.74) or 𝑢(𝜔)=1𝑖^𝛼𝜔^𝛼+𝑘^𝛼. (7.75) The inverse Yang-Fourier transform gives the solution 𝑢𝑦=𝐸_𝛼𝑘^𝛼𝑦^𝛼. (7.76)

7.7. Fractal Derivative and q-Derivative

The fractal derivative is defined by transforming the standard integer-dimensional space time (𝑥,𝑡) into a fractal space time [142] 𝑑𝑢(𝑡)𝑑𝑡^𝐷=lim_𝑠→𝑡𝑢(𝑡)−𝑢(𝑠)𝑡^𝐷−𝑠^𝐷, (7.77) where 𝐷 is the order of the fractal derivative. This definition is much simpler but lacks physical understanding.

As an example, consider the fractal derivative relaxation equation [142] 𝑑𝑢𝑑𝑡^𝐷+𝐵𝑢=0,0<𝐷<1,𝑢(0)=1, (7.78) whose analytical solution is [142] 𝑢(𝑡)=exp−𝐵𝑡^𝐷. (7.79) We write down a general fractal differential equation of the form 𝑑𝑢𝑑𝑡^𝐷+𝑓=0, (7.80) and use the transformation 𝑡^𝐷=𝑥 to convert (7.80) into an ordinary differential equation 𝑑𝑢𝑑𝑥+𝑓=0, (7.81) so that the variational iteration algorithms above can be directly applied: 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)−^𝑥₀𝑑𝑢_𝑛𝑑𝑥+𝑓_𝑛𝑢𝑑𝑥,_𝑛+1(𝑥)=𝑢₀(𝑥)−^𝑥₀𝑓_𝑛𝑢𝑑𝑥,_𝑛+1(𝑥)=𝑢₀(𝑥)−^𝑥₀𝑓_𝑛−𝑓_𝑛−1𝑑𝑥, (7.82) or 𝑢_𝑛+1𝑡^𝐷=𝑢_𝑛𝑡^𝐷−^𝑡𝐷₀𝑑𝑢_𝑛𝑑𝑡^𝐷+𝑓_𝑛𝐷𝑡^𝐷−1𝑢𝑑𝑡,_𝑛+1𝑡^𝐷=𝑢₀𝑡^𝐷−^𝑡𝐷₀𝑓_𝑛𝐷𝑡^𝐷−1𝑢𝑑𝑡,_𝑛+1𝑡^𝐷=𝑢₀𝑡^𝐷−^𝑡𝐷₀𝑓_𝑛−𝑓_𝑛−1𝐷𝑡^𝐷−1𝑑𝑥. (7.83) As another example, consider the 𝑁th-order fractal differential equation 𝑑^𝑁𝑢𝑑𝑡^𝐷+𝑓=0, (7.84) which can be converted by the transformation 𝑡^𝐷=𝑥^𝑁 (7.85) into the ordinary differential equation 𝑑^𝑁𝑢𝑑𝑥^𝑁+𝑓=0, (7.86) and the variational iteration algorithms are 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+(−1)^𝑁𝑥₀1()!()𝑁−1𝑠−𝑥^𝑁−1𝑢_𝑛^(𝑁)+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑥)=𝑢₀(𝑥)+(−1)^𝑁𝑡₀1()!()𝑁−1𝑠−𝑥^𝑁−1𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+(−1)^𝑁𝑡₀1()!()𝑁−1𝑠−𝑡^𝑁−1𝑓_𝑛−𝑓_𝑛−1𝑑𝑠. (7.87) The fractal derivative is simpler than its fractional counterpart in many applications, and it is also valid for discontinuous cases. We rewrite (7.77) in the form 𝑑𝑢(𝑥)𝑑𝑥^𝐷=1𝑘lim_𝐴→𝐵𝑢(𝐴)−𝑢(𝐵)𝑥_𝐴−𝑥_𝐵, (7.88) where 𝑘 is a constant, and 𝐴 and 𝐵 are arbitrary points in discontinuous space or spacetime (as shown in Figure 8). (𝑥_𝐴, 𝑥_𝐵) are called the fractal coordinates and are defined by 𝑥_𝐴=𝑘𝑥_𝐴−0^𝐷=𝑘𝑥_𝐴^𝐷,𝑥_𝐵=𝑘𝑥_𝐵−0^𝐷=𝑘𝑥_𝐵^𝐷, (7.89)(7.90) where (𝑥_𝐴,𝑥_𝐵) are the coordinates, and 𝐷 is the fractional dimension in 𝑥-direction. Substituting (7.88) and (7.89) into (7.88), we obtain 𝑑𝑢(𝑥)𝑑𝑥^𝐷=lim_𝐴→𝐵𝑢(𝐴)−𝑢(𝐵)𝑥_𝐴^𝐷−𝑥_𝐵^𝐷. (7.91) The fractal differential model is particularly suitable for describing discontinuous matter and is the preferred model for describing flow or heat conduction through porous media. For instance, the principle of mass conservation can be written in the form 𝜕𝜌𝜕𝑡+𝑘₁𝜕𝜌𝑢𝜕𝑥^𝐷1+𝑘₂𝜕𝜌𝑣𝜕𝑦^𝐷2+𝑘₃𝜕𝜌𝑤𝜕𝑧^𝐷3=0, (7.92) where 𝐷₁, 𝐷₂, and 𝐷₃ are the fractal dimensions of porosity in the 𝑥, 𝑦, and 𝑧 directions, respectively, and 𝑘_𝑖(𝑖=1,2,3) are constants that are related to the fractal dimensions. In particular, we have 𝑘_𝑖=1, when 𝐷_𝑖=1. Similarly, the momentum equation for one-dimensional porous flow can be written in the form 𝜕𝑢𝜕𝑡+𝑘𝑢𝜕𝑢𝜕𝑥^𝐷𝑘=−𝜌𝜕𝑃𝜕𝑥^𝐷𝜕+𝑘𝜕𝑥^𝐷𝜇𝑘𝜕𝑢𝜕𝑥^𝐷, (7.93) when 𝐷=1 and 𝑘=1,  and (7.93) turns out to be the classical one.

Figure 8 

A schematic diagram of distance between 𝑂 and 𝐴 (i.e., the fractal coordinate of 𝐴) in a fractal space, and 𝐷=ln2/ln3, while the fractional dimensions for the plane are ln8/ln3.

The one-dimensional heat conduction equation in porous media can be expressed as 𝜕𝑇𝜕𝜕𝑡+𝑘𝜕𝑥^𝐷𝜇𝑘𝜕𝑇𝜕𝑥^𝐷=0, (7.94) where 𝜇 is the conduction coefficient and 𝑘=1 when 𝐷=1.

An oscillator swinging in a porous medium can also be described by fractal differential equations. For example, the Duffing equation with fractal damp can be expressed as 𝑑²𝑢𝑑𝑡²+𝑢+𝑘𝜇𝑑𝑢𝑑𝑡^𝐷+𝜀𝑢³=0. (7.95) It is easy to establish fractal differential equations for discontinuous media by replacing 𝜕/𝜕𝑥 in the classical approach by 𝑘𝜕/𝜕𝑥^𝐷.

Now we consider a fractal media illustrated in Figure 9, and assume that the smallest measure is 𝐿₀, any discontinuity less than 𝐿₀ is ignored, then the distance between two points of 𝐴 and 𝐵 in Figure 1 can be expressed using fractal geometry. Hereby we introduce a new fractal derivative for engineering application: 𝐷𝑢(𝑡)𝐷𝑥^𝛼=lim_{Δ𝑥→𝐿0}𝑢(𝐴)−𝑢(𝐵)=Thedistancebetweentwopoints𝑑𝑢𝑑𝑠=lim_{Δ𝑥→𝐿0}𝑢(𝐴)−𝑢(𝐵)𝑘𝐿^𝛼₀, (7.96) where 𝑘 is a constant, 𝛼 is the fractal dimension, and the distance between two points in a discontinuous space can be expressed as 𝑑𝑠=𝑘𝐿^𝛼₀.

Figure 9 

The distance between two points in a discontinuous spacetime.

Please note in the above definition that Δ𝑥 does tend to zero, but to the smallest measure size, 𝐿₀.

Example 7.1. As a simple application, we consider the Fourier’s law heat conduction, which reads 𝜕𝑞_ℎ𝜕𝑡=−𝑐𝑑𝑇,𝑑𝑛 (7.97) where 𝑇 is the thermal potential, and 𝑞_ℎ is heat flow.
In the discontinuous media, the Fourier’s law can be simply modified as 𝜕𝑞_ℎ𝜕𝑡=−𝑐𝐷𝑇𝐷𝑛^𝛼, (7.98) where 𝐷𝑇/𝐷𝑛^𝛼 is a fractal derivative defined in (7.96).

The suggested fractal derivative is easy to be used for any discontinuous problems, and equations with fractal derivative can be easily solved using classical calculus.

Let 𝑓(𝑥;𝑦;…) be a multivariable real continuous function. The 𝑞-derivative and the partial 𝑞-derivative are defined by [143]: 𝐷^𝑥_𝑞𝑓(𝑥)=𝑓𝑞𝑥−𝑓(𝑥)𝑥,𝜕𝑞−1_𝑥𝑞𝑓=𝑓𝑥;𝑦;…𝑞𝑥;𝑦;…−𝑓𝑥;𝑦;…𝑥,𝑞−1 (7.99) and 𝜕^𝑥_𝑞𝑓(𝑥;𝑦;…)|_𝑥=0=lim_𝑛→∞(𝑓(𝑥𝑞^𝑛;𝑦;…)−𝑓(0;𝑦;…))/𝑥𝑞^𝑛, 𝐷^𝑡_𝑞𝑢+𝑓(𝑢)=0,0<𝑞<1. (7.100) In the case 0<𝛼<1, we rewrite (7.13a) in the form 𝑑𝑢+𝐷𝑑𝑡^𝑡𝑞𝑢+𝑓(𝑢)−𝑑𝑢𝑑𝑡=0, (7.101) and the variational iteration algorithms are given as follows: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)−^𝑡₀𝐷^𝑡_𝑞𝑢_𝑛+𝑓𝑢_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢₀(𝑡)−^𝑡₀𝐷^𝑡_𝑞𝑢_𝑛−𝑑𝑢_𝑛𝑢𝑑𝑡+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢₀(𝑡)−^𝑡₀𝐷^𝑡_𝑞𝑢_𝑛−𝑑𝑢_𝑛𝑢𝑑𝑡+𝑓_𝑛−𝐷^𝛼𝑢_𝑛−1𝐷𝑡^𝛼−𝑑𝑢_𝑛−1𝑢𝑑𝑡+𝑓_𝑛−1𝑑𝑠. (7.102) In the case 1<𝛼<2, the above iteration formulas are also valid. We can rewrite (7.100) in the form 𝑑²𝑢𝑑𝑡²+𝐷^𝛼𝑢𝐷𝑡^𝛼−𝑑²𝑢𝑑𝑡²+𝑓=0 (7.103) and the following iteration formulae are suggested 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡₀()𝐷𝑠−𝑡^𝛼𝑢_𝑛𝐷𝑡^𝛼+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢₀(𝑡)+^𝑡₀()𝐷𝑠−𝑡^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑²𝑢_𝑛𝑑𝑡²+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡₀(𝐷𝑠−𝑡)^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑²𝑢_𝑛𝑑𝑡²+𝑓_𝑛−𝐷^𝛼𝑢_𝑛−1𝐷𝑡^𝛼−𝑑²𝑢_𝑛−1𝑑𝑡²+𝑓_𝑛−1𝑑𝑠, (7.104) when 𝛼 is close to 1, (7.102) is better, while (7.104) is recommended for 𝛼 approaching 2.

For the case 𝑁<𝛼<𝑁+1, where 𝑁 is a natural number, the iteration formulas are as follows 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+(−1)^𝑁𝑡₀1()!()𝑁−1𝑠−𝑡^𝑁−1𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼+𝑓_𝑛𝑢𝑑𝑠_𝑛+1(𝑡)=𝑢₀(𝑡)+(−1)^𝑁𝑡₀1()!()𝑁−1𝑠−𝑡^𝑁−1𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑^𝑁𝑢_𝑛𝑑𝑡^𝑁+𝑓_𝑛𝑢𝑑𝑠_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+(−1)^𝑁+^𝑡₀1()!()𝑁−1𝑠−𝑡^𝑁−1×𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑^𝑁𝑢_𝑛𝑑𝑡^𝑁+𝑓_𝑛−𝐷^𝛼𝑢_𝑛−1𝐷𝑡^𝛼−𝑑^𝑁𝑢_𝑛−1𝑑𝑡^𝑁+𝑓_𝑛−1𝑑𝑠, (7.105)

or 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+(−1)^𝑁+1𝑡₀1()𝑁!𝑠−𝑡^𝑁𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢₀(𝑡)+(−1)^𝑁+1𝑡₀1()𝑁!𝑠−𝑡^𝑁𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑^𝑁+1𝑢_𝑛𝑑𝑡^𝑁+1+𝑓_𝑛𝑢𝑑𝑠,_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+(−1)^𝑁+1×^𝑡₀1()𝑁!𝑠−𝑡^𝑁𝐷^𝛼𝑢_𝑛𝐷𝑡^𝛼−𝑑^𝑁+1𝑢_𝑛𝑑𝑡^𝑁+1+𝑓_𝑛−𝐷^𝛼𝑢_𝑛−1𝐷𝑡^𝛼−𝑑^𝑁+1𝑢_𝑛−1𝑑𝑡^𝑁+1+𝑓_𝑛−1𝑑𝑠. (7.106)

8. Nonlinear Wave Transform for Nonlinear Wave Equations

The linear wave transform is widely used in nonlinear community to search for solitary solutions. Take the well-known KdV equation as an example 𝜕𝑢𝜕𝑡−6𝑢𝜕𝑢+𝜕𝜕𝑥³𝑢𝜕𝑥³=0. (8.1) The general solution approach is to introduce a wave transform 𝜉=𝑥−𝑉𝑡 (8.2) to convert (8.1) into an ordinary differential equation. In the linear wave transform, 𝑉 has a definite physical understanding, and it is the wave speed.

The nonlinear wave transform introduces a complex variable 𝜉 defined as 𝜉=𝑎(𝑡)𝑥+𝑏(𝑡)𝑦+𝑐(𝑡)𝑧+𝑑(𝑡), (8.3) where 𝑎,𝑏,𝑐, and 𝑑 are unknown functions of time.

In order to elucidate the solution procedure, we consider the following equation: 𝑢_𝑡+𝑡²𝑢_𝑥+𝑠𝑡𝑢𝑢_𝑥+𝑟𝑡𝑢_𝑥𝑥𝑥=0, (8.4) where 𝑟 and 𝑠 are constants.

By a nonlinear wave transform, 𝜉=𝑥−𝑎₁𝑡−𝑎₂𝑡²−𝑎₃𝑡³, (8.5) where 𝑎₁, 𝑎₂, and 𝑎₃ are constants, we can convert (8.4) into an ordinary differential equation, −𝑎₁−2𝑎₂𝑡−3𝑎₃𝑡²𝑢^′+𝑡²𝑢^′+𝑠𝑡𝑢𝑢^′+𝑟𝑡𝑢^′′′=0, (8.6) where prime denotes the derivation with respect to 𝜉.

In order to eliminate the time-dependent coefficients in (8.6), we set 𝑎₁=0,𝑎₃=13. (8.7) Equation (8.6) becomes −2𝑎₂𝑢^′+𝑠𝑢𝑢^′+𝑟𝑢^′′′=0. (8.8)

We rewrite (8.8) in the form 𝑢^′−𝑠2𝑎₂𝑢𝑢^′−𝑟2𝑎₂𝑢^′′′=0. (8.9) Equation (8.9) can be effectively solved by the exp-function method (see Section 5.9). Hereby we consider a special case 𝑠2𝑎₂𝑟=6,2𝑎₂=−1. (8.10) Equation (8.9) becomes the standard KdV equation, which admits the following solitary solution: 𝐴𝑢=−2sech²𝐴4𝐴𝜉=−2sech²𝐴4𝑥−𝑎₂𝑡²−13𝑡³. (8.11)

Acknowledgments

This work is supported by PAPD (a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions), National Natural Science Foundation of China under Grants nos. 11061028 and 50806011, and Natural Science Foundation of Yunnan Province under Grant no. 2010CD086.