Exact solutions for the KdV–mKdV equation with time-dependent coefficients using the modified functional variable method

In this article, the functional variable method (fvm for short) is introduced to establish new exact travelling solutions of the combined KdV–mKdV equation. The technique of the homogeneous balance method is used in second stage to handle the appropriated solutions. We show that, the method is straightforward and concise for several kinds of nonlinear problems. Many new exact travelling wave solutions are successfully obtained. Subjects: Applied Mathematics; Computer Mathematics; Dynamical Systems; Mathematical Physics; Mathematics & Statistics; Science


Introduction
There are several forms of nonlinear partial differential equations that have been presented in the past decades to investigate new exact solutions. Several methods (Benjamin, Bona, & Mahony, 1972;Dye & Parker, 2000;Freeman & Johnson, 1970;Khelil, Bensalah, Saidi, & Zerarka, 2006;Korteweg & de Vries, 1895;Sirendaoreji, 2004Sirendaoreji, , 2007Wang & Li, 2008;Wazwaz, 2002Wazwaz, , 2007aZerarka, 1996, ABOUT THE AUTHORS W. Djoudi obtained her master's degree from the University of Science and Technology in 2011. She is currently pursuing her PhD degree at the Laboratory of Mathematics. Her research interest includes the nonlinear structures of resonant interaction and dispersion of waves. A. Zerarka is Professor of Physics and Applied Mathematics at the University Med Khider at Biskra, Algeria. He obtained his PhD from the University of Bordeaux, France. His domain of interest is theoretical physics and applied mathematics. He has been teaching and conducting research since 1981, has widely published in international journals and he is the author and co-author of numerous refereed papers. He is the reviewer of numerous papers (Elsevier, Taylor and Francis, Wiley, Springer, Inderscience, Academicjournals and more). He is an active member of the Academy of Science, New York.

PUBLIC INTEREST STATEMENT
We presente an important direct formulation namely the functional variable method for constructing exact analytical solutions of nonlinear partial differential equations. This method is based on the linear combination of the unknown function assumed as a variable of the system, and the technique of the homogeneous balance is used to handle the appropriated solutions which include new exact travelling waves and solitons. We note that almost all of the solutions obtained in the literature are based on trial functions with some free parameters or on the supposition of a known ansatz. In general, these techniques offer solutions that are not unique. The main advantage of this method is the flexibility to give exact solutions to nonlinear PDEs without any need for perturbation techniques, and does not require linearizing or discrediting. Another possible merit is that, this method only uses the techniques of direct integration for integrable models.
2005) have been proposed to handle a wide variety of linear and nonlinear wave equations. As is well known, the description of these nonlinear model equations appeared to supply different structures to the solutions. Among these are the auto-Backlund transformation, inverse scattering method, Hirota method, Miura's transformation.
The availability of symbolic computation packages can facilitate many direct approaches to establish solutions to nonlinear wave equations (Hirota, 1971;Iskandar, 1989;Jinquing & Wei-Guang, 1992;Xu & Zhang, 2007;Zhou, Wang, & Wang, 2003). Various extension forms of the sine-cosine and tanh methods proposed by Malfliet and Wazwaz have been applied to solve a large class of nonlinear equations (Fan & Zhang, 2002;Malfliet, 1992). More importantly, another mathematical treatment is established and used in the analysis of these nonlinear problems, such as Jacobian elliptic function expansion method, the variational iteration method, pseudo spectral method and many other powerful methods (Liu & Yang, 2004;Yomba, 2005).
One of the major goals of the present article is to provide an efficient approach based on the functional variable method to examine new developments in a direct manner without requiring any additional condition on the investigation of exact solutions for the combined KdV-mKdV equation. Abundant exact solutions are obtained together with the aid of symbol calculation software, such as Mathematica.

Description of the method fvm
To clarify the basic idea of fvm proposed in our paper (Zerarka, Ouamane, & Attaf, 2011), we present the governing equation written in several independent variables as where the subscripts denote differentiation, while u(t, x, y, z, …) is an unknown function to be determined. Equation (1) is a nonlinear partial differential equation that is not integrable, in general. Sometime it is difficult to find a complete set of solutions. If the solutions exist, there are many methods which can be used to handle these nonlinear equations.
The following transformation is used for the new wave variable as where i are distinct variables, and when p = 1, = 0 0 + 1 1 + , and if the quantities 0 , 1 are constants, then, they are called the wave pulsation and the wave number k respectively, if 0 , 1 are the variables t and x, respectively. We give the travelling wave reduction transformation for Equation (1) as and the chain rule Upon using (3) and (4), the nonlinear problem (1) with suitably chosen variables becomes an ordinary differential equation (ODE) like If we consider a function as a variable, it is sometimes easy to transform a nonlinear equation into a linear equation. Thus, if the unknown function U is treated as a functional variable in the form then, the solution can be found by the relation here a 0 is a constant of integration which is set equal to zero for convenience. Some successive differentiations of U in terms of F are given as where " ′ " stands for d dU . The ordinary differential equation (5) can be reduced in terms of U, F and its derivatives upon using the expressions of (8) into (5) gives The key idea of this particular form (9) is of special interest because it admits analytical solutions for a large class of nonlinear wave type equations. After integration, the Equation (9) provides the expression of F, and this in turn together with (6) give the relevant solutions to the original problem.

KdV-mKdV solutions
Let us consider the KdV-mKdV equation (Li & Wang, 2007;Triki, Thiab, & Wazwaz, 2010;Wazwaz, 2007b) is written as This equation arises in many physical problems including the motions of waves in nonlinear optics, plasma or fluids, water waves, ion-acoustic waves in a collisionless plasma, where h i , i = 0, 1, 2, 3, 4, are arbitrary smooth model functions that symbolize the coefficients of the time variable t and the subscripts denote the partial differentiations with respect to the corresponding variable. The first element u t designates the evolution term which governs how the wave evolves with respect to time, while the second one shows the term of dispersion. We first combine the independent variables, into a wave variable using as where and are time-dependent functions, and we write the travelling wave solutions of Equation (10) with (11) as u(x, t) = U(t, ). Using the chain rule (4), the differential equation (10) can be transformed as:

Partial structures
In order to prepare the complete solution for Equation (10), we begin with the special case when h 4 = 0 and , and h i , i = 0, 1, 2, 3 are constant functions. Thus, Equation (10) is then integrated as long as all terms contain derivatives where integration constants are considered zeros, and it looks like following: from Equation (13) and the method fvm, we obtain where From (14), we obtain the desired solution as (Sirendaoreji, 2007;Yomba, 2004)

Full structures
Now we reveal the main features of solutions by working directly from (14) (16), and , and are free parameters and M is an undetermined integer and q k are coefficients to be determined later. One of the most useful techniques for obtaining the parameter M in (17) is the homogeneous balance method. Substituting from (17) into Equation (12) and by making balance between the linear term (cubic dispersion) U ′′′ and the nonlinear term (cubic nonlinearity) U 2 U ′ to determine the value of M, and by simple calculation we have got that M + 1 = 3M − 1, this in turn gives M = 1, and the solution (17) takes the form Now, we substitute (19) into (12) along with (18) and set each coefficient of Φ k Φ � l and xΦ � (k = 0, 1, 2 and l = 0, 1) to zero to obtain a set of algebraic equations for q 0 , q 1 , , and as, Solving the system of algebraic equations, we would end up with the explicit pulse parameters for q 0 , q 1 , and , we obtain where q 00 , q 10 , 0 and 0 are the integration constants and are identified from initial data of the pulse. Notice that (20e) and (20f) serve as constraint relations between the coefficient functions and the pulse parameters.
We note that, Equation (18) admits several other types of solutions, it is easy to see that we can include more solutions as listed in Sirendaoreji (sire1), we omit these results here. The propagation velocity of the soliton pulse is related to parameters describing the process and is expressed by the relation v(t) = d A qualitative plot of the solution (24a), u 1 (x, t) = U 1 (t, ) is presented in Figures 1 and 2 shows the physical wave (24d), u 4 (x, t) = U 4 (t, ). It is apparent that the amplitude contributes to the formation of solitons mainly through the model function h 4 (t) (for full structures). Consequently, in the absence of the coefficient function h 4 (t), the partial and full structures become substantially equivalent versions.

Conclusion
In this work, we applied a new analytical technique namely, the functional variable method (fvm) to establish some new exact analytic wave structures to the KdV-mKdV equation with time-dependent coefficients. On one side, for the first case study, we obtained the limited solutions using the partial structures for the function h 4 = 0. On the other side, for the second case study, we introduced the results obtained for the partial structures to solve the KdV-mKdV nonlinear differential equation for the full structures in the presence of the function h 4 ≠ 0. Four variants of complete travelling wave solutions are obtained. The present method provides a reliable technique that requires less work if compared with the difficulties arising from computational aspect. The main advantage of this method is the flexibility to give exact solutions to nonlinear PDEs without any need for perturbation techniques, and does not require linearizing or discrediting. All calculations are performed using Mathematica.
We may conclude that, this method can be easily extended to find the solution of some highdimensional nonlinear problems. These points will be investigated in a future research.