The Bäcklund Transformations, Exact Solutions, and Conservation Laws for the Compound Modified Korteweg-de Vries-Sine-Gordon Equations which Describe Pseudospherical Surfaces

Abstract

I show that the compound modified Korteweg-de Vries-Sine-Gordon equations describe pseudospherical surfaces, that is, these equations are the integrability conditions for the structural equations of such surfaces. I obtain the self-Bäcklund transformations for these equations by a geometrical method and apply the Bäcklund transformations to these solutions and generate new traveling wave solutions. Conservation laws for the latter ones are obtained using a geometrical property of these pseudospherical surfaces.

1. Introduction

A differential equation (DE) for a real-valued function 𝑢(𝑥,𝑡) is said to describe pseudospherical surfaces (pss) if it is the necessary and sufficient condition for the existence of smooth functions 𝑓_𝑖𝑗,  1≤𝑖≤3,  1≤𝑗≤2, depending on 𝑢 and its derivatives, such that the 1-forms 𝜔_𝑖=𝑓_𝑖1𝑑𝑥+𝑓_𝑖2𝑑𝑡,  1≤𝑖≤3 satisfy the structure equations of a surface of a constant Gaussian curvature equal to −1, that is, 𝑑𝜔₁=𝜔₃∧𝜔₂,𝑑𝜔₂=𝜔₁∧𝜔₃,𝑑𝜔₃=𝜔₂∧𝜔₁. (1) It is equivalent to say that the DE for 𝑢(𝑥,𝑡) is necessary and sufficient for the integrability of the linear system [1–6] 𝜙𝑑𝜙=Ω𝜙,𝜙=¹𝜙₂, (2) where 𝑑 denotes exterior differentiation, 𝜙 is a column vector, and the 2×2 matrix Ω (Ω_𝑖𝑗, 𝑖,𝑗=1,2) is traceless 1Ω=2𝜔₂𝜔₁−𝜔₃𝜔₁+𝜔₃−𝜔₂. (3) Take 𝜂Ω=2𝜂𝑑𝑥+𝐴𝑑𝑡𝑞𝑑𝑥+𝐵𝑑𝑡𝑟𝑑𝑥+𝐶𝑑𝑡−2𝑑𝑥−𝐴𝑑𝑡=𝑃𝑑𝑥+𝑄𝑑𝑡, (4) from (2) and (4), we obtain 𝜙_𝑥=𝑃𝜙,𝜙_𝑡=𝑄𝜙, (5) where 𝑃 and 𝑄 are two 2×2 null-trace matrices 𝜂𝑃=2𝑞𝜂𝑟−2,𝑄=𝐴𝐵𝐶−𝐴. (6) Here, 𝜂 is a parameter, independent of 𝑥 and 𝑡, while 𝑞 and 𝑟 are functions of 𝑥 and 𝑡. Now, 0=𝑑²𝜙=𝑑Ω𝜙−Ω∧𝑑𝜙=(𝑑Ω−Ω∧Ω)𝜙, (7) which requires the vanishing of the two forms: Θ≡𝑑Ω−Ω∧Ω=0, (8) or in the component form: 𝐴_𝑥𝑞=𝑞𝐶−𝑟𝐵,_𝑡−2𝐴𝑞−𝐵_𝑥𝐶+𝜂𝐵=0,_𝑥=𝑟_𝑡+2𝐴𝑟−𝜂𝐶. (9) Many partial differential equations (PDEs) which are of interest to study and investigate due to the role they play in various areas of mathematics and physics are included in this category [7–25].

The formulation of classical theory of surfaces in a form is familiar to the soliton theory, which makes possible an application of the analytical methods of this theory to integrable cases [26].

The results of [27] were obtained by inverse spectral method. The results of [28] were obtained using algorithm for constructing certain exact solutions, such as solutions describing the interaction of two traveling waves.

The Fokas transform method for solving boundary value problems for linear and integrable nonlinear PDEs can be viewed as an extension of the Fourier transform method, and, indeed, how in simple cases it reduces to the Fourier transform. The unifying character of the steps involved in the Fokas method makes it attractive from the theoretical and formal point of view. For nonlinear integrable problems, this approach is, at present, the only existing method yielding results in a general context [29–32].

The Fokas method has a much broader domain of applicability than it is possible to present in [30–33]. For example, elliptic problems can also be treated by this general approach. In this case, the analysis of the global relation, which is the crucial step in the methodology, may involve the solution of additional Reimann-Hilbert problems. As regards numerical approximations, preliminary results indicate that, using this approach, the Dirichlet-to-Neumann map for linear elliptic problems can also be evaluated, at least for some important examples, with exponential accuracy [34, 35].

The current paper directions include the implementation of the geometrical properties and the Bäcklund transformations (BTs) to generate a new soliton solution and conservation laws for the compound modified Korteweg-de Vries-sine-Gordon (cmKdV-SG) equations.

The paper is organized as follows. In Section 2, I show that the cmKdV-SG equations describe pss. In Section 3, we find the BTs for the cmKdV-SG equations. Exact soliton solution class from a known constant solution is obtained for the cmKdV-SG equations. On the other hand, a new exact traveling wave solutions for the cmKdV-SG equations are obtained by using the BTs to generate a new soliton solution class in Section 4. In Section 5, I obtain an infinite number of conserved densities for the cmKdV-SG equations which describe pss using a theorem of Khater et al. [18] and Sayed et al. [23–25]. Finally, I give some conclusions in Section 6.

2. The cmKdV-SG Equations which Describe pss

The notion of a DE describing pss was first introduced by Chern and Tenenblat [2], who observed that most of the nonlinear evolution equations (NLEEs) solvable by the method of inverse scattering [3–5], such as the KdV and mKdV equations, have the property of describing pss. They also showed that if 𝑓₂₁=𝜂 and the functions 𝑓₁₁ and 𝑓₃₁ do not depend on 𝜂, then the linear system (2) reduces to the inverse scattering problem (ISP) considered by Ablowitz et al. in [3], with 𝜂 corresponding to the spectral parameter. Let 𝑀² be a two-dimensional differentiable manifold parameterized by coordinates 𝑥,  𝑡. We consider a metric on 𝑀² defined by 𝜔₁,  𝜔₂. The first two equations in (1) are the structure equations which determine the connection from 𝜔₃, and the last equation in (1), the Gauss equation, determines that the Gaussian curvature of 𝑀² is −1,  that is,  𝑀² is a pss. Moreover, the one-forms 𝜔₁=𝑓₁₁𝑑𝑥+𝑓₁₂𝑑𝑡,𝜔₂=𝑓₂₁𝑑𝑥+𝑓₂₂𝜔𝑑𝑡,₃=𝑓₃₁𝑑𝑥+𝑓₃₂𝑑𝑡 (10) satisfy the structure equations (1) of a pss. It has been known, for a long time, that the SG equation describes a pss. In this paper, we extend the same analysis to include the cmKdV-SG equations: 𝛽𝑢_𝑟𝜃+1ℎ16⁴𝑢_𝑟²𝑢_𝑟𝑟+1ℎ24⁴𝑢_{𝑟𝑟𝑟𝑟}−𝛼sin𝑢=0, (11) where 𝛼,  𝛽,  ℎ are constants. This equation can be thought of as a generalization of the mKdV and SG equations. As particular cases:(i) when 𝛼=0, (11) becomes the mKdV equation in 𝑢_𝑟 which is retrieved,(ii) while the neglect of the terms in ℎ⁴ leads to the SG equation. Moreover, the introduction of the variables 𝑥=(24)^1/4𝑟ℎ,𝑡=(24)^−1/4ℎ𝜃𝛽, (12)reduces (11) to the form 𝑢_𝑥𝑡+𝑢_{𝑥𝑥𝑥𝑥}+32𝑢²_𝑥𝑢_𝑥𝑥−𝛼sin𝑢=0. (13) Let 𝑀² be a differentiable surface, parameterized by coordinates 𝑥,  𝑡. Consider that 𝜔₁=𝜂𝑢_𝑥𝑥+𝛼𝜂𝜔sin𝑢𝑑𝑡,₂𝛼=𝜂𝑑𝑥+𝜂cos𝑢−𝜂³−𝜂2𝑢²_𝑥𝜔𝑑𝑡,₃=𝑢_𝑥𝑑𝑥+−𝑢_𝑥𝑥𝑥−𝜂²𝑢_𝑥−12𝑢³_𝑥𝑑𝑡, (14) then 𝑀² is a pss if and only if 𝑢 satisfies the cmKdV-SG equations (13).

3. The Self-Bäcklund Transformation for the cmKdV-SG Equations

In this section, we show how the geometrical properties of a pss may be applied to obtain analytical results for the cmKdV-SG equations which describe pss.

The classical Bäcklund theorem originated in the study of pss, relating solutions of the SG equation. Other transformations have been found relating solutions of specific equations in [6–9]. Such transformations are called BTs after the classical one. A BT which relates solutions of the same equation is called a self-Bäcklund transformation (sBT). An interesting fact which has been observed is that DEs which have sBT also admit a superposition formula. The importance of such formulas is due to the following: if 𝑢₀ is a solution of the NLEE and 𝑢₁,  𝑢₂ are solutions of the same equation obtained by the sBT, then the superposition formula provides a new solution 𝑢^′ algebraically. By this procedure, one obtains the soliton solutions of an NLEE. In what follows, we show that geometrical properties of pss provide a systematic method to obtain the BTs for some NLEEs which describe pss.

Proposition 1. Given a coframe {𝜔₁,𝜔₂} and corresponding connection one-form 𝜔₃ on a smooth Riemannian surfaces 𝑀², there exists a new coframe {𝜔^′₁,𝜔^′₂} and new connection one-form 𝜔^′₃ satisfying the equations 𝑑𝜔^′₁=0,𝑑𝜔^′₂=𝜔^′₂∧𝜔^′₁,𝜔^′₃+𝜔^′₂=0, (15) if and only if the surface 𝑀² is pss. For the sake of clarity, one gives a revised proof of [10].

Proof. Assume that the orthonormal dual to the coframes {𝜔₁,𝜔₂} and {𝜔^′₁,𝜔^′₂} possess the same orientation. The one-forms 𝜔_𝑖 and 𝜔^′_𝑖  (𝑖=1,2,3) are connected by means of 𝜔^′₁=𝜔₁cos𝜓−𝜔₂sin𝜓,𝜔^′₂=𝜔₁sin𝜓+𝜔₂cos𝜓,𝜔^′₃=𝜔₃−𝑑𝜓. (16) It follows that 𝜔^′₁,  𝜔^′₂,  𝜔^′₃  satisfying (15) exist if and only if the Pfaffian system 𝜔₃−𝑑𝜓+𝜔₁sin𝜓+𝜔₂cos𝜓=0, (17) on the space of coordinates (𝑥,𝑡,𝜓) is completely integrable for 𝜓(𝑥,𝑡), and this happens if and only if 𝑀² is pss.
Geometrically, (15) and (17) determine geodesic coordinates on 𝑀². Now, if 𝑢_𝑡=𝐹(𝑢,𝑢_𝑥,…,𝑢_𝑥𝑘)(𝑢_𝑥𝑘=𝜕^𝑘𝑢/𝜕𝑥^𝑘) describes pss with associated one-forms 𝜔_𝑖=𝑓_𝑖1𝑑𝑥+𝑓_𝑖2𝑑𝑡, (15) and (17) imply that the Pfaffian system, 𝜔₃−𝑑𝜓+𝜔₁sin𝜓+𝜔₂cos𝜓=0, (18) is completely integrable for 𝜓(𝑥,𝑡) whenever 𝑢(𝑥,𝑡) is a local solution of 𝑢_𝑡=𝐹(𝑢,𝑢_𝑥,…,𝑢_𝑥𝑘) [2, 11].

Proposition 2. Let 𝑢_𝑡=𝐹(𝑢,𝑢_𝑥,…,𝑢_𝑥𝑘) be an NLEE which describes a pss with associated one-forms (10). Then, for each solution 𝑢(𝑥,𝑡) of 𝑢_𝑡=𝐹(𝑢,𝑢_𝑥,…,𝑢_𝑥𝑘), the system of equations for 𝜓(𝑥,𝑡), 𝜓_𝑥−𝑓₃₁+𝑓₁₁𝜓sin𝜓+𝜂cos𝜓=0,_𝑡−𝑓₃₂+𝑓₁₂sin𝜓+𝑓₂₂cos𝜓=0, (19) is completely integrable. Moreover, for each solution of 𝑢(𝑥,𝑡) of 𝑢_𝑡=𝐹(𝑢,𝑢_𝑥,…,𝑢_𝑥𝑘) and corresponding solution 𝜓, 𝑓₁₁cos𝜓−𝜂sin𝜓𝑑𝑥+𝑓₁₂cos𝜓−𝑓₂₂sin𝜓𝑑𝑡, (20) is a closed one-form [2].

Eliminating 𝜓(𝑥,𝑡) from (19), by using the substitution cos𝜓=2Γ1+Γ², (21) where 𝜙Γ=¹𝜙₂, (22) then (19) is reduced to the Riccati equations: 𝜕Γ1𝜕𝑥=𝜂Γ+2𝑓₁₁1−Γ²−12𝑓₃₁1+Γ²,𝜕Γ𝜕𝑡=𝑓₂₂1Γ+2𝑓₁₂1−Γ²−12𝑓₃₂1+Γ². (23)(24) The procedure in the following is that one constructs a transformation Γ^′ satisfying the same equation as (24) with a potential 𝑢^′(𝑥), where 𝑢^′(𝑥)=𝑢(𝑥)+𝑓Γ,𝜂. (25) Thus eliminating Γ in (23), (24), and (25), we have a BT to a desired NLEE. We consider the following example (BT for the cmKdV-SG equations).

For (13), we consider the functions defined by 𝑓₁₁=0,𝑓₁₂=𝜂𝑢_𝑥𝑥+𝛼𝜂𝑓sin𝑢,₂₁=𝜂,𝑓₂₂=𝛼𝜂cos𝑢−𝜂³−𝜂2𝑢²_𝑥,𝑓₃₁=𝑢_𝑥,𝑓₃₂=−𝑢_3𝑥−𝜂²𝑢_𝑥−12𝑢³_𝑥, (26) for any solution 𝑢(𝑥,𝑡) of (13), the above functions satisfy (8). Then, (23) becomes 𝜕Γ𝑢𝜕𝑥=𝜂Γ−^𝑥21+Γ². (27) If we choose Γ^′ and 𝑢^′ as Γ^′=1Γ,𝑢^′=𝑢+4tan⁻¹Γ, (28) then Γ^′ and 𝑢^′ satisfy (27). If we eliminate Γ in (27) and (24) with (28), we get the BT 𝑢^′+𝑢_𝑥1=−2𝜂sin2𝑢−𝑢^′,𝑢−𝑢^′_𝑡=2𝑓₃₂−2𝑓₁₂1cos2𝑢−𝑢^′+2𝑓₂₂1sin2𝑢−𝑢^′. (29) Equation (29) is the BT for the cmKdV-SG equations (13) with 𝑓₁₂,  𝑓₂₂, and 𝑓₃₂ given in (26).

4. A New Traveling Wave Solutions for the cmKdV-SG Equations

For any solution 𝑢(𝑥,𝑡) of the cmKdV-SG equations (13), the matrices 𝑃 and 𝑄 are𝜂𝑃=2−𝑢_𝑥2𝑢_𝑥2−𝜂2,1𝑄=2−𝜂³−𝜂𝑢²_𝑥2+𝛼𝜂1cos𝑢2𝜂𝑢_𝑥𝑥+𝑢_𝑥𝑥𝑥+𝑢³_𝑥2+𝜂²𝑢_𝑥+𝛼𝜂1sin𝑢2𝜂𝑢_𝑥𝑥−𝑢_𝑥𝑥𝑥−𝑢³_𝑥2−𝜂²𝑢_𝑥+𝛼𝜂−1sin𝑢2−𝜂³−𝜂𝑢²_𝑥2+𝛼𝜂.cos𝑢 (30)

Substitute 𝑢=𝑛Π,  𝑛=0,±1,±2,±3,… into the matrices 𝑃 and 𝑄 in (30), then by (5) we have 𝑑𝜙=𝜙_𝑥𝑑𝑥+𝜙_𝑡𝑑𝑡=𝑃𝜙𝑑𝜌_𝑛, (31) where 𝜂𝑃=20𝜂0−2,𝜌_𝑛=𝑥−𝑘𝑡,𝑘=𝜂²−𝛼𝜂²(−1)^𝑛. (32) The solution of (31) is 𝜙_𝑛=𝑒^𝜌𝑛𝑃𝜙₀=𝐼+𝜌_𝑛𝜌𝑃+^2𝑛𝑃²+𝜌2!^3𝑛𝑃³𝜙3!+⋯₀, (33) where 𝜙₀ is a constant column vector. The solution of (33) is 𝜙_𝑛=𝜂cosh2𝜌_𝑛𝜂+sinh2𝜌_𝑛0𝜂0cosh2𝜌_𝑛𝜂−sinh2𝜌_𝑛𝜙₀. (34) Now, we choose 𝜙₀=(1,1)^𝑇 in (34), then we have 𝜙_𝑛=𝑒^{𝜂𝜌𝑛/2}𝑒^{−𝜂𝜌𝑛/2}. (35) Substitute (35) into (22), then, by (28), we obtain the new solutions of the cmKdV-SG equations (13): 𝑢^′(𝑥,𝑡)=𝑛Π+4tan⁻¹𝑒^𝜂𝜌𝑛,𝑛=0,±1,±2,±3,…. (36) Consequently, the solution of (11) is 𝑢^′(𝑟,𝜃)=𝑛Π+4tan⁻¹𝑒^𝜂𝜌𝑛,𝜌_𝑛=(24)^1/4𝑟ℎ−(24)^−1/4ℎ𝜃𝛽𝑘,𝑘=𝜂²−𝛼𝜂²(−1)^𝑛𝑛=0,±1,±2,±3,…. (37) By means of the same procedures above, we obtain the solution of mKdV equation:(i) when 𝛼=0 in (13), we obtain the mKdV equation in 𝑢_𝑥𝑢_𝑥𝑡+𝑢_{𝑥𝑥𝑥𝑥}+32𝑢_𝑥²𝑢_𝑥𝑥=0, (38)    and its solutions is 𝑢^′()𝜕𝑥,𝑡=4𝜕𝑥tan⁻¹𝑒^𝜂𝜌,𝜌=𝑥−𝜂²𝑡, (39)(ii) when 𝛼=0 in (11), we obtain the mKdV equation in 𝑢_𝑟𝛽𝑢_𝑟𝜃+1ℎ16⁴𝑢_𝑟²𝑢_𝑟𝑟+1ℎ24⁴𝑢_{𝑟𝑟𝑟𝑟}=0, (40)    and its solutions is 𝑢^′()𝜕𝑟,𝜃=4𝜕𝑟tan⁻¹𝑒^𝜂𝜌,𝜌=(24)^1/4𝑟ℎ−(24)^−1/4ℎ𝜃𝛽𝜂². (41)

Now, we use a known traveling wave solutions for the cmKdV-SG equations to generate a new solution for the cmKdV-SG equations by means of the BTs.

We will find a new traveling wave solutions 𝑢^′(𝑥,𝑡) of the cmKdV-SG equations (13) and substitute these solution into the corresponding matrices 𝑃 and 𝑄. Next we solve (22) for 𝜙₁ and 𝜙₂. Then by (22) and the corresponding BTs (28) we will obtain the new solution classes. We take 𝑢=4tan⁻¹𝑒^𝜂𝜌,𝜌=𝑥−𝑘𝑡,𝑘=𝜂²−𝛼𝜂², (42) as a traveling wave solution class of the cmKdV-SG equations (13). The traveling wave known solution of the cmKdV-SG equations takes the form 𝑢=𝑢𝜌,𝜌=𝑥−𝑘𝑡. (43) In this case the AKNS system (5) and (6) has a general solution. Let us consider the more general case. Suppose that the components 𝑞 and 𝑟 of the matrix 𝑃 are function of 𝜌 [8, 12]: 𝑞=𝑞𝜌,𝑟=𝑟𝜌; (44) then the components 𝐴, 𝐵 and 𝐶 of the matrix 𝑄 as determined by (6) are also functions of 𝜌: 𝐴=𝐴𝜌,𝐵=𝐵𝜌,𝐶=𝐶𝜌. (45) Under these assumptions, the following result holds, which is crucial in the subsequent exact solution. The quantity 𝛽₁=𝜂𝐴+𝑘2²+𝐵+𝑘𝑞(𝐶+𝑘𝑟), (46) is constant with respect to 𝜌 (or 𝑥 and 𝑡). Using the result of [13] and the constant 𝛽₁ defined by (46) is greater than zero and therefore the corresponding solution of the AKNS system (5) and (6) is: 𝜙₁𝜙₂=𝑐₁(𝐶+𝑘𝑟)^−1/2𝜂𝐴+𝑘2sinh𝜔𝜉+𝑐₂+𝜔cosh𝜔𝜉+𝑐₂𝑐₁(𝐶+𝑘𝑟)^1/2sinh𝜔𝜉+𝑐₂,when𝛽₁>0,𝜔²=𝛽₁, (47)where 𝑐₁ and 𝑐₂ are constants and 𝜉=𝑡+𝑟𝑑𝜌.𝐶+𝑘𝑟 (48)

Now applying the results obtained here and the known traveling wave solutions for the cmKdV-SG equations respectively to construct new solution class of the corresponding cmKdV-SG equations by means of the BTs. The constant 𝛽₁ and 𝜉 defined by (46), (48) can be determined by using (42) 𝜂𝜉=𝑡−²2𝜂⁴−+2𝛼8𝜂¹⁰9𝜂¹²+3𝜂⁸𝛼−5𝛼²𝜂⁴+𝛼³𝜌−𝜂12𝜂⁴𝑒−4𝛼^−2𝜂𝜌−4𝜂⁹9𝜂¹²+3𝜂⁸𝛼−5𝛼²𝜂⁴+𝛼³×ln𝑒^2𝜂𝜌+𝛼−3𝜂⁴𝛼+𝜂⁴. (49) Consequently, we obtain Γ from (47) for 𝛽₁>0Γ=(𝐶+𝑘𝑟)⁻¹𝜂𝐴+𝑘2+𝜔coth𝜔𝜉+𝑐₂, (50) then substituting this Γ into the BTs (28) and using (42), we arrive at the new solution 𝑢^′ of the cmKdV-SG equations (13) corresponding to the known traveling wave solution class (42), then 𝑢^′(𝑥,𝑡)=4tan⁻¹𝑒^𝜂𝜌+tan⁻¹Γ. (51)

Consequently, the solution of (11) is 𝑢^′(𝑟,𝜃)=4tan⁻¹𝑒^𝜂𝜌+tan⁻¹Γ,𝜌=(24)^1/4𝑟ℎ−(24)^−1/4ℎ𝜃𝛽𝑘,𝑘=𝜂²−𝛼𝜂². (52)

By means of the same procedures above,(i) we obtain the solution of mKdV equation (38), 𝑢^′()𝜕𝑥,𝑡=4𝜕𝑥tan⁻¹𝑒^𝜂𝜌+tan⁻¹Γ,𝜌=𝑥−𝜂²7𝑡,𝜉=𝑡+9𝜂²1𝜌−12𝜂³𝑒^−2𝜂𝜌−49𝜂³ln𝑒^2𝜂𝜌−3, (53)(ii) we obtain the solution of mKdV equation (40), 𝑢^′()𝜕𝑟,𝜃=4𝜕𝑥tan⁻¹𝑒^𝜂𝜌+tan⁻¹Γ,𝜌=(24)^1/4𝑟ℎ−(24)^−1/4ℎ𝜃𝛽𝜂²,𝜉=(24)^−1/4ℎ𝜃𝛽+79𝜂²1𝜌−12𝜂³𝑒^−2𝜂𝜌−49𝜂³ln𝑒^2𝜂𝜌−3. (54)

5. Conservation Laws for the cmKdV-SG Equations

One of the most widely accepted definitions of integrability of PDEs requires the existence of soliton solutions, that is, of a special kind of traveling wave solutions that interact “elastically,” without changing their shapes. The analytical construction of soliton solutions is based on the general ISM. In the formulation of Zakharov and Shabat [14], all known integrable systems supporting solitons can be realized as the integrability condition of a linear problem of the form (5). Thus, an equation (5) is kinematically integrable if it is equivalent to the curvature condition 𝑃_𝑥−𝑄_𝑡+[𝑃,𝑄]=0. (55) As mentioned in the previous sections, Sasaki [15], Chern and Tenenblat [2], and Cavalcante and Tenenblat [16] have given a geometrical method for constructing conservation laws of equations describing pss. The formal content of this method is contained in the following theorem, which may be seen as generalizing the classical discussion on conservation laws appearing in Wadati et al. [17].

Theorem 3. Suppose that 𝑢_𝑡=𝐹(𝑢,𝑢_𝑥,…,𝑢_𝑥𝑘) or more generally 𝐹(𝑥,𝑡,𝑢,𝑢_𝑥,…,𝑢_{𝑥𝑛𝑡𝑚})=0 is an NLEE describing pss. The systems 𝐷_𝑥𝜙₁𝐷=𝑞𝑟+^𝑥𝑞𝑞−𝜂𝜙₁−𝜙²₁,𝐷_𝑡𝜂2+𝜙₁=𝐷_𝑥𝐵𝐴+𝑞𝜙₁,𝐷_𝑥𝜙₂𝐷=−𝑞𝑟+^𝑥𝑟𝑟+𝜂𝜙₂+𝜙²₂,𝐷_𝑡𝜂2+𝜙₂=𝐷_𝑥𝐶𝐴+𝑟𝜙₂, (56)(57)(58)(59) in which 𝐷_𝑥 and 𝐷_𝑡 are the total derivative operators defined by 𝐷_𝑥=𝜕+𝜕𝑥^∞_𝑘=0𝑢_𝑘+1𝜕𝜕𝑢_𝑘,𝐷_𝑡=𝜕+𝜕𝑡^∞_𝑘=0𝐷^𝑘_𝑥𝑓𝜕𝜕𝑢_𝑘, (60) are integrable on solutions of the equation 𝑢_𝑡=𝐹(𝑢,𝑢_𝑥,…,𝑢_𝑥𝑘) or generally 𝐹(𝑥,𝑡,𝑢,𝑢_𝑥,…,𝑢_{𝑥𝑛𝑡𝑚})=0 [18].

This theorem provides one with at least one 𝜂-dependent conservation law of the NLEE 𝑢_𝑡=𝐹(𝑢,𝑢_𝑥,…,𝑢_𝑥𝑘) or 𝐹(𝑥,𝑡,𝑢,𝑢_𝑥,…,𝑢_{𝑥𝑛𝑡𝑚})=0, to wit, (56) and (57) or ((58) and (59)). One obtains a sequence of 𝜂-independent conservation laws by expanding 𝜙₁ or 𝜙₂ in inverse powers of 𝜂 [19, 22]. Moreover, 𝜙₂=^∞_𝑛=1𝜙₂^(𝑛)𝜂^−𝑛, (61) and the consideration of (58) yields the recursion relation 𝜙₂⁽¹⁾𝜙=−𝑞𝑟,_2(𝑛+1)=𝐷_𝑥𝑟𝑟𝜙₂^(𝑛)+𝐷_𝑥𝜙₂^(𝑛)+^𝑛−1_𝑖=1𝜙₂^(𝑖)𝜙₂^{(𝑛−𝑖)},𝑛≥1, (62) which in turn, by replacing into (59), yields the sequence of conservation laws of equations integrable by AKNS inverse scattering found by Wadati et al. [17]. This section ends with the example.

For (13), we consider the functions of 𝑢(𝑥,𝑡) defined by 𝑢𝑟=^𝑥2𝑢,𝑞=−^𝑥2,1𝐴=2−𝜂³−𝜂𝑢²_𝑥2+𝛼𝜂,1cos𝑢𝐵=2𝜂𝑢_𝑥𝑥+𝑢_𝑥𝑥𝑥+𝑢³_𝑥2+𝜂²𝑢_𝑥+𝛼𝜂,1sin𝑢𝐶=2𝜂𝑢_𝑥𝑥−𝑢_𝑥𝑥𝑥−𝑢³_𝑥2−𝜂²𝑢_𝑥+𝛼𝜂sin𝑢. (63)(64)(65) Equation (58) becomes 𝐷_𝑥𝜙₂=14𝑢²_𝑥+𝑢_𝑥𝑥𝑢_𝑥+𝜂𝜙₂+𝜙²₂. (66) Assume that 𝜙₂ can be expanded in a series of the form (61).

Equation (64) implies that 𝜙₂ is determined by the recursion relation 𝜙₂⁽¹⁾=14𝑢²_𝑥,𝜙₂^(𝑛+1)=𝑢_𝑥𝑥𝑢_𝑥𝜙₂^(𝑛)+𝐷_𝑥𝜙₂^(𝑛)+^𝑛−1_𝑖=1𝜙₂^(𝑖)𝜙₂^{(𝑛−𝑖)},𝑛≥1, (67) whenever 𝑢(𝑥,𝑡) is a solution of the the cmKdV-SG equations. This recursion relation yields a sequence of conserved densities given by the coefficients of the series in 𝜂𝜂2+^∞_𝑛=1𝜙₂^(𝑛)𝜂^−𝑛, (68) which one obtains from (59).

6. Conclusions

In this paper, I show how the geometrical properties of a pss may be applied to obtain analytical results for the cmKdV-SG equations which describe pss. It has been shown that the implementation of certain BTs for a class of NLEE requires the solution of the underlying linear differential equation whose coefficients depend on the known solution 𝑢(𝑥,𝑡) of the NLEE. I obtain a new traveling wave solutions for the cmKdV-SG equations by using BTs. Next, an infinite number of conservation laws is derived for the cmKdV-SG equations just mentioned using a theorem by Khater et al. [18] and Sayed et al. [23–25].

Acknowledgment

The author would like to thank the anonymous referees for these helpful comments. The author thinks that revising the paper according to its suggestions will be quite straightforward.