Exact Solutions for the Generalized BBM Equation with Variable Coefficients

Abstract

The variational iteration algorithm combined with the exp-function method is suggested to solve the generalized Benjamin-Bona-Mahony equation (BBM) with variable coefficients. Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered.

1. Introduction

The BBM equation

𝑢_𝑡+𝑢𝑢_𝑥+𝑢_𝑥−𝜇𝑢_𝑥𝑥𝑡=0,(1.1) which describes approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems, has been proposed by Benjamin et al. in 1972 [1] as a more satisfactory model than the KdV equation [2]

𝑢_𝑡+𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=0.(1.2) It is easy to see that (1.1) can be derived from the equal width EW-equation [3]:

𝑢_𝑡+𝑢𝑢_𝑥−𝜇𝑢_𝑥𝑥𝑡=0,(1.3) by means of the change of variable 𝑢=𝑢+1, that is, by replacing 𝑢 with 𝑢+1. This last equation is considered as an equally valid and accurate model for the same wave phenomena simulated by (1.1) and (1.2). On the other hand, some researches analyzed the generalized KdV equation with variable coefficients

𝑢_𝑡+𝜎(𝑡)𝑢^𝑝𝑢_𝑥+𝜇(𝑡)𝑢_𝑥𝑥𝑥=0,(1.4) because this model has important applications in several fields of science [4–7].

Motivated by these facts, we will consider here the generalized EW-equation with variable coefficients

𝑢_𝑡+𝜎(𝑡)𝑢^𝑝𝑢_𝑥−𝜇(𝑡)𝑢_𝑥𝑥𝑡=0.(1.5) Using the solutions of (1.5) we obtain exact solutions to the generalized BBM equation

𝑢_𝑡+𝜎(𝑡)(𝑢+1)^𝑝𝑢_𝑥−𝜇(𝑡)𝑢_𝑥𝑥𝑡=0,(1.6) of order 𝑝>0.

2. Exact Solutions to Generalized BBM Equation

2.1. The Variational Iteration Method

Consider the following nonlinear equation:

𝐿𝑢(𝑥,𝑡)+𝑁𝑢(𝑥,𝑡)=𝑔(𝑥,𝑡),(2.1) where 𝐿 and 𝑁 are linear and nonlinear operators, respectively, and 𝑔(𝑥,𝑡) is an inhomogeneous term. According to the variational iteration method (VIM) [8–14], a functional correction to (2.1) is given by

𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡0𝜃(𝜏)𝐿𝑢_𝑛(𝑥,𝜏)+𝑁𝑢_𝑛̃(𝑥,𝜏)−𝑔(𝑥,𝜏)𝑑𝜏,(2.2) where 𝜃(𝜏) is a general Lagrange's multiplier, which can be identified via the variational theory; the subscript 𝑛≥0 denotes the 𝑛th order approximation and 𝑢̃ is a restricted variation which means 𝛿𝑢̃=0. In this method, we first determine the Lagrange multiplier 𝜃(𝜏) that will be identified optimally via integration by parts. The successive approximation 𝑢_𝑛+1 of the solution 𝑢 will be readily obtained upon using the determined Lagrangian multiplier and any selective function 𝑢₀. One of the advantages of the VIM, is the free choice of the initial solution 𝑢₀(𝑥,𝑡). If we consider a special form to 𝑢₀ with arbitrary parameters, using the relations

𝑢_𝑛(𝑥,𝑡)=𝑢_𝑛+1(𝑥,𝑡),𝜕^𝑘𝜕𝑡^𝑘𝑢_𝑛(𝑥,𝑡)=𝜕^𝑘𝜕𝑡^𝑘𝑢_𝑛+1(𝑥,𝑡),(2.3) we can obtain a set of algebraic equations in the unknowns given by the parameters that appear in 𝑢₀. Solving this system, we have exact solutions to (2.1). To solve (1.5), we construct the following functional equation

𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡0𝜃(𝜏)𝐿𝑢_𝑛(𝑥,𝜏)+𝑁𝑢_𝑛̃(𝑥,𝜏)𝑑𝜏,(2.4) where

𝑁𝑢_𝑛̃𝐿𝑢_𝑛(𝑥,𝜏)=𝑢_𝑛_𝜏(𝑥,𝜏),(𝑥,𝜏)=𝜎(𝜏)(𝑢̃+1)^𝑝𝑢_𝑥̃(𝑥,𝜏)−𝜇(𝜏)𝑢̃_𝑥𝑥𝜏(𝑥,𝜏).(2.5) Taking in (2.4) variation with respect to the independent variable 𝑢_𝑛, and noticing that 𝛿𝑁𝑢_𝑛̃=0 we have

𝛿𝑢_𝑛+1(𝑥,𝑡)==𝛿𝛿𝑢𝑢_𝑛𝑛(𝑥,𝑡)+𝛿^𝑡0𝜃(𝜏)𝐿𝑢_𝑛(𝑥,𝜏)+𝑁𝑢_𝑛̃(𝑥,𝜏)𝑑𝜏(𝑥,𝑡)+𝜃(𝑡)𝛿𝑢_𝑛(𝑥,𝑡)−^𝑡0𝜃^(𝜏)𝛿𝑢_𝑛(𝑥,𝜏)𝑑𝜏=0.(2.6) This yields the stationary conditions

1𝜃+𝜃(𝑡)=0,_(𝑡)=0.(2.7) Therefore,

𝜃(𝑡)=−1.(2.8) Substituting this value into (2.4) we obtain the formula

𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)−^𝑡0𝐿𝑢_𝑛(𝑥,𝜏)+𝑁𝑢_𝑛̃(𝑥,𝜏)𝑑𝜏.(2.9)

Using the wave transformation

𝜉=𝑥+𝜆𝑡+𝜉₀,(2.10) setting

𝜕𝜕𝑡𝑢¹(𝜉)=𝜕𝜕𝑡𝑢⁰(𝜉),(2.11) and performing one integration, (2.9) reduces to

𝜆𝑢₀(𝜉)+𝜎(𝑡)𝑝+1𝑢^0𝑝+1(𝜉)−𝜆𝜇(𝑡)𝑢^0(𝜉)=0,(2.12) where for sake of simplicity we set the constant of integration equal to zero. With the change of variable

𝑢₀(𝜉)=𝑣^2/𝑝(𝜉),(2.13) equation (2.12) converts to

𝜆𝑣²(𝜉)−2𝜇(𝑡)(2−𝑝)𝑝²𝜆𝑣^²−2𝜇(𝑡)𝑝𝜆𝑣(𝜉)𝑣^(𝜉)+𝜎(𝑡)𝑝+1𝑣(𝜉)⁴=0.(2.14) Observe that if 𝑣(𝜉) is a solution to (2.14), then −𝑣(𝜉) is also a solution to this equation.

2.2. The Exp-Function Method

Recently, He and Wu [15] have introduced the Exp-function method to solve nonlinear differential equations. In particular, the Exp-function method is an effective method for solving nonlinear equations with high nonlinearity. The method has been used in a satisfactory way by other authors to solve a great variety of nonlinear wave equations [15–21]. The Exp-function method is very simple and straightforward, and can be briefly revised as follows: Given the nonlinear partial differential equation

𝐹𝑢,𝑢_𝑥,𝑢_𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑡,𝑢_𝑡𝑡,…=0,(2.15) it is transformed to ordinary differential equation

𝐹𝑢,𝑢^,𝑢^,𝑢^,𝑢_𝑥𝑡,…=0,(2.16) by mean of wave transformation 𝜉=𝑥+𝜆𝑡+𝜉₀. Solutions to (2.16) can then be found using the expression

𝑢(𝜉)=∑^{𝑑𝑛=−𝑐}𝑎_𝑛exp(𝑛𝜉)∑^{𝑞𝑛=−𝑝}𝑏_𝑛exp(𝑛𝜉),(2.17) where 𝑐,𝑑,𝑝, and 𝑞 are positive integers which are unknown to be determined later, 𝑎_𝑛 and 𝑏_𝑛 are unknown constants.

After balancing, we substitute (2.17) into (2.16) to obtain an algebraic systems in the variable 𝜁=exp(𝑛𝜉). Solving the algebraic system we can obtain exact solutions to (2.16) and reversing, solutions to (2.15) in the original variables.

3. Solutions to (2.14) by the Exp-Function Method

Using the Exp-function method, we suppose that solutions to (2.14) can be expressed in the form

𝑣(𝜉)=∑^1𝑛=−1𝑎_𝑛exp(𝑛𝑟𝜉)∑^1𝑚=−1𝑏_𝑚exp(𝑚𝑟𝜉)=𝑎⁻¹exp(−𝑟𝜉)+𝑎₀+𝑎₁exp(𝑟𝜉)𝑏₋₁exp(−𝑟𝜉)+𝑏₀+𝑏₁exp(𝑟𝜉).(3.1) We obtain following solutions to (2.14): 𝑣₁=±2𝜆𝑘(𝑝+1)(𝑝+2)𝑣2(𝑝+1)(𝑝+2)𝜆exp𝑝/2√𝜇(𝑡)𝜉−𝑘²𝜎(𝑡)exp−𝑝/2√𝜇(𝑡)𝜉,𝜆=𝜆(𝑡),₂=±2𝜆𝑘(𝑝+1)(𝑝+2)𝜎(𝑡)𝑘²exp𝑝/2√𝜇(𝑡)𝜉−2𝜆(𝑝+1)(𝑝+2)exp−𝑝/2√𝜇(𝑡)𝜉,𝜆=𝜆(𝑡).(3.2) Some special solutions are obtained if

𝜆=𝜆(𝑡)=±𝑘²2𝑝²+3𝑝+2𝜎(𝑡).(3.3) This choice gives solutions

𝑣₃=±𝑘𝑣2csch2√𝑝𝜇(𝑡)𝜉,𝜆=2𝑝²+𝑘3²𝑝+2𝜎(𝑡),(3.4)₄=𝑘𝑣2sech2√𝑝𝜇(𝑡)𝜉,𝜆=−2𝑝²+𝑘3²𝑝+2𝜎(𝑡),(3.5)₅=±𝑘2csc2√𝑝𝜇(𝑡)𝜉,𝜆=−2𝑝²+𝑘3²𝑝+2𝜎(𝑡).(3.6) Solution (3.6) follows from (3.4) with the identifications 𝜇(𝑡)→−𝜇(𝑡) and 𝑘→−𝑘√−1.

𝑣₆=−𝑘2sec2√𝑝𝜇(𝑡)𝜉,𝜆=−2𝑝²+𝑘3²𝑝+2𝜎(𝑡).(3.7) Solution (3.7) follows from (3.4) with the identifications 𝜇(𝑡)→−𝜇(𝑡) and 𝑘→−𝑘.

4. Particular Cases

4.1. Case 1: Solutions to (2.14) When 𝑝=2

Equation (2.14) takes the form

𝜆𝑣²(𝜉)−𝜆𝜇(𝑡)𝑣(𝜉)𝑣^(𝜉)+13𝜎(𝑡)𝑣(𝜉)⁴=0.(4.1) From (3.2) with 𝑝=2:

𝑣₇=±24𝜆𝑘𝑣24𝜆exp1/√𝜇(𝑡)𝜉−𝑘²𝜎(𝑡)exp−1/√𝜇(𝑡)₈𝜉,=±24𝜆𝑘𝑘²𝜎(𝑡)exp1/√𝜇(𝑡)𝜉−24𝜆exp−1/√𝜇(𝑡)𝜉.(4.2) From (3.3)–(3.7) with 𝑝=2:

𝑣₉=±𝑘𝑣2csch√𝜇1(𝑡)𝜉,𝜆=2𝑘4²𝜎(𝑡),₁₀=±𝑘𝑣2sech√𝜇1(𝑡)𝜉,𝜆=−2𝑘4²𝜎(𝑡),₁₁=±𝑘𝑣2csc√−1𝜇(𝑡)𝜉,𝜆=−2𝑘4²𝜎(𝑡),₁₂=±𝑘2sec√𝜇1(𝑡)𝜉,𝜆=−2𝑘4²𝜎(𝑡).(4.3) Other exact solutions are:

𝑣₁₃=±3𝑎²exp2√−2/𝜇(𝑡)𝜉+2√55𝑎exp√−2/𝜇(𝑡)𝜉−22𝑘3𝑎²exp2√−2/𝜇(𝑡)𝜉+22𝑎exp√−2/𝜇(𝑡)𝜉+22,𝜆=−31𝑘²𝑣𝜎(𝑡),₁₄=±3𝑎²±2√55𝑎exp√−2/𝜇(𝑡)𝜉−22exp2√−2/𝜇(𝑡)𝜉𝑘3𝑎²+22𝑎exp√−2/𝜇(𝑡)𝜉+22exp2√−2/𝜇(𝑡)𝜉,𝜆=−13𝑘²𝑣𝜎(𝑡),₁₅=±𝑘⎛⎜⎜⎝1−448+√553𝑎11+√55exp√−2/𝜇(𝑡)𝜉+228+√55⎞⎟⎟⎠,𝜆=−31𝑘²𝑣𝜎(𝑡),₁₆=±𝑘𝑎±sinh√−2/𝜇(𝑡)𝜉√𝑎²+1±cosh√−2/𝜇(𝑡)𝜉,𝜆=−13𝑘²𝑣𝜎(𝑡),₁₇=±𝑘𝑎±cosh√−2/𝜇(𝑡)𝜉√𝑎²+1±sinh√−2/𝜇(𝑡)𝜉,𝜆=−13𝑘²𝑣𝜎(𝑡),₁₈=±𝑘cos√2/𝜇(𝑡)𝜉1±sin√2/𝜇(𝑡)𝜉,𝜆=𝑘3²𝜎(𝑡).(4.4)

4.2. Case 2: Solutions to (2.14) When 𝑝=4

Equation (2.14) takes the form

𝜆𝑣²(𝜉)+𝜇(𝑡)𝜆𝑣^²−𝜆2𝜇(𝑡)𝑣(𝜉)𝑣^(𝜉)+15𝜎(𝑡)𝑣(𝜉)⁴=0.(4.5) From (3.2) with 𝑝=4:

𝑣₁₉=±60𝜆𝑘𝑣60𝜆exp2/√𝜇(𝑡)𝜉−𝑘²𝜎(𝑡)exp−2/√𝜇(𝑡)𝜉,𝜆=𝜆(𝑡),₂₀=±60𝜆𝑘𝜎(𝑡)𝑘²exp2/√𝜇(𝑡)𝜉−60𝜆exp−2/√𝜇(𝑡)𝜉,𝜆=𝜆(𝑡).(4.6) From (3.3)–(3.7) with 𝑝=4:

𝑣₂₁=±𝑘𝑣2csch√𝜇2(𝑡)𝜉,𝜆=6𝑘0²𝜎(𝑡),₂₂=±𝑘𝑣2sech√𝜇2(𝑡)𝜉,𝜆=−6𝑘0²𝜎(𝑡),₂₃=±𝑘𝑣2csc√−2𝜇(𝑡)𝜉,𝜆=−6𝑘0²𝜎(𝑡),₂₄=±𝑘2sec√−2𝜇(𝑡)𝜉,𝜆=−6𝑘0²𝜎(𝑡).(4.7) Other exact solutions are:

𝑣₂₅=±𝑘𝑎exp2/√−𝜇(𝑡)𝜉−4²𝑎²exp4/√−𝜇(𝑡)𝜉+16𝑎exp2/√−𝜇(𝑡)𝜉+16,𝜆=−51𝑘²𝑣𝜎(𝑡),₂₆=±𝑘4exp2/√−𝜇(𝑡)𝜉−𝑎²𝑎²+16𝑎exp2/√−𝜇(𝑡)𝜉+16exp4/√−𝜇(𝑡)𝜉,𝜆=−15𝑘²𝑣𝜎(𝑡),₂₇=±2𝑘⎛⎜⎜⎝1−32±cos2/√𝜇(𝑡)𝜉⎞⎟⎟⎠,𝜆=−45𝑘²𝑣𝜎(𝑡),₂₈=±2𝑘⎛⎜⎜⎝1−32±sin2/√𝜇(𝑡)𝜉⎞⎟⎟⎠,𝜆=−45𝑘²𝜎(𝑡).(4.8) It is clear that using (2.13) we obtain solutions to (1.5). Finally, observe that if 𝑢₀(𝑥,𝑡) is a solution of (1.5), then the solutions 𝑢(𝑥,𝑡) to the generalized BBM equation (1.6) are obtained as follows:

𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)−1.(4.9)

5. Conclusions

We have considered the generalized EW-equation with variable coefficients and the generalized BBM-equation with variable coefficients. We obtained analytic solutions by using the variational iteration method combined with the exp-function method. With the aid of Mathematica we have derived a lot of different types of solutions for these two models. Combined formal soliton-like solutions as well as kink solutions have been formally derived. The results obtained show that the technique used here can be considered as a powerful method to analyze other types of nonlinear wave equations.

According to [22], there are alternative iteration alorithms, which might be useful for future work. Furthermore, various modifications of the exp-function method have been appeared in open literature, for example, the double exp-function method [23, 24].

Other methods for solving nonlinear differential equations may be found in [25–35].

We think that the results presented in this paper are new in the literature.