Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method

Abstract: In this paper, we will use the functional variable method to construct exact solutions of some nonlinear systems of partial differential equations, including, the (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation, the WhithamBroer-Kaup-Like systems and the Kaup-Boussinesq system. This approach can also be applied to other nonlinear systems of partial differential equations which can be converted to a second-order ordinary differential equation through the travelling wave transformation.


Introduction
Finding the exact solutions nonlinear systems of partial differential equations plays an important role in the study of many physical phenomena in various fields such as fluid mechanics, solid state physics, plasma physics, chemical physics and optical.Thus, it is important to investigate the exact explicit solutions of nonlinear systems of partial differential equations.In recent years, various powerful methods have been presented for finding exact solutions of the nonlinear systems of partial differential equations in mathematical physics, such as modified simple equation method [1], Algebraic method [2], sine-cosine method [3], F-expansion method [4], generalized hyperbolic function [5] and functional variable method [6].Among these methods, the functional variable method is a powerful mathematical tool to solve nonlinear systems of partial differential equations.This method were first proposed by Zerarka et al [7] to find the exact solutions for a wide class of linear and nonlinear wave equations.The functional variable method was further developed by many authors [8,9].The advantage of this method is that one treats nonlinear problems by essentially linear methods, based on which it is easy to construct in full the exact solutions such as soliton-like waves, compacton and noncompacton solutions, trigonometric function solutions, pattern soliton solutions, black solitons or kink solutions, and so on.The aim of this paper is to construct exact solutions of the the (2+1)-dimensional Bogoyavlenskii's breaking soliton equation, the Whitham-Broer-Kaup-Like systems and the Kaup-Boussinesq system by using the functional variable method.Also, we presented two useful theorems of the functional variable method for finding traveling wave solutions of nonlinear partial differential equations.
The rest of this paper is organized as follows.In Section 2, brief description of the functional variable method for finding traveling wave solutions of nonlinear system of partial differential equations is given.In Section 3, the method is employed for obtaining the exact solutions of the (2+1)-dimensional Bogoyavlenskii's breaking soliton equation, the Whitham-Broer-Kaup-Like systems and the Kaup-Boussinesq system.Finally, some conclusions are given in Section 4.

The functional variable method
Now, we describe the main steps of the functional variable method for finding exact solutions of nonlinear system of partial differential equations.
Consider the following nonlinear system of partial differential equations with independent variables x and t and dependent variables u and v Applying the travelling wave transformations u(x, t) = U (ξ) and v(x, t) = V (ξ) where ξ = x − wt, converts Eq.(2.1) into a system of ordinary differential like Using some mathematical operations, the system (2.2) is converted into a secondorder ordinary differential equation as Then we make a transformation in which the unknown function U is considered as a functional variable in the form and where " ′ " stands for d dU .Substituting (2.5) into Eq.(2.3) and after the mathematical manipulations, we reduce the ordinary differential equation (2.3) in terms of U , F as K(U, F ) = 0. (2.6) The key idea of this particular form Eq.(2.6) is of special interest because it admits analytical solutions for a large class of nonlinear wave type equations.After integration, Eq.(2.6) provides the expression of F , and this, together with Eq.(2.4), give appropriate solutions to the original problem.
Theorem 2.1.Consider the following second-order ordinary differential equation where k 1 and k 2 = 0 are constants and U is a functional variable in the form (2.4).
Then using (2.5) transformation, the exact solutions of the Eq.(2.7) are obtained as Type I.When k 1 > 0, the soliton solutions of Eq.(2.7) are Type II.When k 1 < 0, the periodic wave solutions of Eq.(2.7) are (2.11) Proof: According to Eq.(2.5), we get from (2.7) an expression for the function where the prime denotes differentiation with respect to ξ. Integrating Eq.(2.12) with respect to U and after the mathematical manipulations, we have or After changing the variables with differentiation from Eq.(2.16) (2.17) Now, using (2.17), (2.4) and (2.14), we have with integrating from Eq.(2.18) and with setting the constant of integration as zero In this case we have thus, according to (2.21), we have thus, according to (2.23), we have therefore, according to (2.25), we have thus, according to (2.27), we have (2.28) Here, using the relations (2.16), (2.22), (2.24), (2.26) and (2.28), the solutions of Eq.(2.7) are in the following forms When k 1 > 0, the soliton solutions of Eq.(2.7) are When k 1 < 0, the periodic wave solutions of Eq.(2.7) are Consider the following second-order ordinary differential equation where k 1 = 0, and k 2 are constants and U is a functional variable in the form (2.4).Then using (2.5) transformation, the exact solutions of the Eq.(2.29) are obtained as Type I.When k 1 > 0, the soliton solutions of Eq.(2.29) are (2.31) Type II.When k 1 < 0, the periodic wave solutions of Eq.(2.29) are (2.33) Proof: According to Eq.(2.5), we get from (2.29) an expression for the function where the prime denotes differentiation with respect to ξ. Integrating Eq.(2.34) with respect to U and after the mathematical manipulations, we have (2.36) Now, using (2.4) and (2.36), we have with integrating from Eq.(2.37) and with setting the constant of integration as zero In this case we have , two cases will be considered separately.Case 1. suppose that k 1 > 0, then thus, according to (2.40), we have (2.41) Now, suppose that k 1 < 0, then thus, according to (2.42), we have therefore, according to (2.44), we have Now, assume that k 1 < 0, then thus, according to (2.46), we have (2.47) Here, using the relations (2.41), (2.43), (2.45) and (2.47), the solutions of Eq.(2.29) are in the following forms When k 1 > 0, the soliton solutions of Eq.(2.29) are When k 1 < 0, the periodic wave solutions of Eq.(2.29) are ξ) .

Applications
Here, we will apply the functional variable method to obtain the exact solutions for the following three nonlinear systems.

The (2+1)-dimensional Bogoyavlenskii's breaking soliton equation
Let us consider the (2+1)-dimensional Bogoyavlenskii's breaking soliton equation which reads its equivalent form [10] which describes the (2+1)-dimensional interaction of a Riemann wave propagating along the y axis with a long wave long the x axis.The u = u(x, y, t) represents the physical field and v = v(x, y, t) some potential.This equation is typical of the so-called "breaking soliton" equation and was studied by Bogoyavenskii, where overlapping solutions were generated [11].Now, to look for travelling wave solutions of eq.(3.2), we first make the transformations 2), we obtain ordinary differential equations Travelling wave solutions of nonlinear systems of PDEs by using the . . .221 By integrating the Eq.(3.5) with respect to ξ, and neglecting the constant of integration, we have Substituting Eq.(3.6) into Eq.(3.4), after integrating with respect to ξ choosing constant of integration to zero, we obtain or Then we use the transformation substituting Eq.(2.5) into Eq.(3.8) we obtain where the prime denotes differentiation with respect to ξ. Integrating Eq.(3.10) with respect to U and after the mathematical manipulations, we have Using the relations (3.9), (2.8), (2.9), (2.10) and (2.11), when w > 0, the solutions of Eq.(3.7) are in the following forms ) and, when w < 0, the solutions of Eq.(3.7) are in the following forms For w > 0, using the travelling wave transformations (3.3), we obtain the following soliton solutions of the (2+1)-dimensional Bogoyavlenskii's breaking soliton equation (3.17) For w < 0, we obtain the periodic wave solutions These solutions are all new exact solutions.Figure 1 shown that the periodic wave solution u 4 (x, y, t) and v 4 (x, y, t) of the Eq.( 3

The Whitham-Broer-Kaup-Like systems
Let us consider the the Whitham-Broer-Kaup-Like systems [12,13], in the form where u = u(x, t) is the field of horizontal velocity, v = v(x, t) is the height that deviates from the equilibrium position of the liquid and α, β, γ are constants.It is necessary to point out that when the parameters are taken as different values, the following celebrated nonlinear systems can be derived from Eq.(3.20).
(i) When γ = 1, we have the Whitham-Broer-Kaup equations [14,15] (ii) When α = 0, γ = 1, we get the approximate equations for long wave equations (iii) When α = γ = 1, β = 0, we obtain the variant Boussinesq equations [16] (iv) When α = 1 3 , γ = 1, β = 0, we get the dispersive long wave equations [17] It is clear to see that Eq.(3.20) is very important in the field of mathematical physics.Therefore, it is a significant task to search for explicit solutions of the Eq.(3.20).Now, we apply the functional variable method to find the solitary wave solutions for Whitham-Broer-Kaup-Like systems.Firstly, we let Then (3.20) is converted to ordinary differential equations By integrating the Eq.(3.26) with respect to ξ, and neglecting the constant of integration, we have or Integrating Eq.(3.27) with respect to ξ choosing constant of integration to zero, we obtain Substituting Eq.(3.29) into Eq.(3.30) yields or Then we use the transformation and Eq.(2.5) to convert Eq.(3.32) to where the prime denotes differentiation with respect to ξ. Integrating Eq.(3.34) with respect to U and after the mathematical manipulations, we have and, when 1 αγ+β 2 < 0, the solutions of Eq.(3.32) are in the following forms For 1 αγ+β 2 > 0, we can easily obtain following solito solutions For 1 αγ+β 2 < 0, we obtain the periodic wave solutions

The Kaup-Boussinesq system
Consider the Kaup-Boussinesq system [18] where = u(x, t) denotes the height of the water surface above a horizontal bottom and v = v(x, t) is related to the horizontal velocity field.At this time, by means of the functional variable method, we will find some solitary wave solutions of the Kaup-Boussinesq system.By considering the wave transformations We change Eq.(3.49) into a system of ordinary differential equations given by Now, integrating Eq.(3.53) with respect to ξ and choosing constant of integration to zero, we obtain Then we use the transformation and Eq.(2.5) to convert Eq.(3.54) to where the prime denotes differentiation with respect to ξ. Integrating Eq.(3.56) with respect to V and after the mathematical manipulations, we have

Conclusion
In this paper, the functional variable method was successfully applied to obtain travelling wave solutions of some important nonlinear systems, including, the (2+1)-dimensional Bogoyavlenskii's breaking soliton equation, the Whitham-Broer-Kaup-Like systems and the Kaup-Boussinesq systems, which were not discussed elsewhere using that method.The travelling wave solutions were expressed by the hyperbolic functions and the trigonometric functions.Also, we conclude that the proposed method is powerful for nonlinear partial differential systems which can be converted to a second-order ordinary differential equations through the travelling wave transformation.