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

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.


Introduction
A differential equation (DE) for a real-valued function (, ) is said to describe pseudospherical surfaces (pss) if it is the necessary and sufficient condition for the existence of smooth functions   , 1 ≤  ≤ 3, 1 ≤  ≤ 2, depending on  and its derivatives, such that the 1-forms   =  1  +  2 , 1 ≤  ≤ 3 satisfy the structure equations of a surface of a constant Gaussian curvature equal to −1, that is, It is equivalent to say that the DE for (, ) is necessary and sufficient for the integrability of the linear system [1][2][3][4][5][6] where  denotes exterior differentiation,  is a column vector, and the 2 × 2 matrix Ω (Ω  , ,  = 1, 2) is traceless Take from ( 2) and ( 4), we obtain where  and  are two 2 × 2 null-trace matrices ) ,  = (    − ) .
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][30][31][32].
The Fokas method has a much broader domain of applicability than it is possible to present in [30][31][32][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 Vriessine-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][24][25].Finally, I give some conclusions in Section 6.

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][4][5] 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: where , , ℎ are constants.This equation can be thought of as a generalization of the mKdV and SG equations.As particular cases: (i) when  = 0, (11) becomes the mKdV equation in   which is retrieved, (ii) while the neglect of the terms in ℎ 4 leads to the SG equation.Moreover, the introduction of the variables reduces (11) to the form Let  2 be a differentiable surface, parameterized by coordinates , .Consider that then  2 is a pss if and only if  satisfies the cmKdV-SG equations ( 13).

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][7][8][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  0 is a solution of the NLEE and  1 ,  2 are solutions of the same equation obtained by the sBT, then the superposition formula provides a new solution   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.
if and only if the surface  2 is pss.For the sake of clarity, one gives a revised proof of [10].
Eliminating (, ) from (19), by using the substitution where then ( 19) is reduced to the Riccati equations: The procedure in the following is that one constructs a transformation Γ  satisfying the same equation as (24) with a potential   (), where 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 for any solution (, ) of (13), the above functions satisfy (8).Then, (23) becomes If we choose Γ  and   as then Γ  and   satisfy (27).If we eliminate Γ in ( 27) and ( 24) with ( 28), we get the BT Equation ( 29) is the BT for the cmKdV-SG equations ( 13) with  12 ,  22 , and  32 given in (26).
Consequently, the solution of ( 11) is By means of the same procedures above, we obtain the solution of mKdV equation: (i) when  = 0 in (13), we obtain the mKdV equation in and its solutions is (ii) when  = 0 in (11), we obtain the mKdV equation in and its solutions is (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   (, ) of the cmKdV-SG equations (13) and substitute these solution into the corresponding matrices  and .Next we solve (22) for  1 and  2 .Then by (22) and the corresponding BTs (28) we will obtain the new solution classes.We take 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 In this case the AKNS system ( 5) and ( 6) has a general solution.Let us consider the more general case.Suppose that the components  and  of the matrix  are function of  [8,12]: then the components ,  and  of the matrix  as determined by ( 6) are also functions of : Under these assumptions, the following result holds, which is crucial in the subsequent exact solution.The quantity is constant with respect to  (or  and ).Using the result of [13] and the constant  1 defined by ( 46) is greater than zero and therefore the corresponding solution of the AKNS system ( 5) and ( 6) is: where  1 and  2 are constants and 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  1 and  defined by ( 46), (48) can be determined by using (42) Consequently, we obtain Γ from (47) for  1 > 0 then substituting this Γ into the BTs (28) and using (42), we arrive at the new solution   of the cmKdV-SG equations (13) corresponding to the known traveling wave solution class (42), then Consequently, the solution of ( 11) is By means of the same procedures above, (i) we obtain the solution of mKdV equation (38), (ii) we obtain the solution of mKdV equation (40), (54)

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 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].

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 (, ) 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][24][25].