Travelling Wave Solutions of the Schrödinger-Boussinesq System

and Applied Analysis 3 2. Description of the G′/G -Expansion Method The objective of this section is to outline the use of the G′/G -expansion method for solving certain nonlinear partial differential equations PDEs . Suppose that a nonlinear equation, say in two independent variables x and t, is given by P u, ux, ut, uxx, ux,t, utt, . . . 0, 2.1 where u x, t is an unknown function, P is a polynomial in u x, t and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved. The main steps of the G′/G -expansion method are the following: Step 1. Combining the independent variables x and t into one variable ξ kx ωt, we suppose that u x, t U ξ , ξ kx ωt. 2.2 The travelling wave variable 2.2 permits us to reduce 2.1 to an ODE for u x, t U ξ , namely, P ( U, kU′, ωU′, k2U′′, kωU′′, ω2U′′, . . . ) 0, 2.3 where prime denotes derivative with respect to ξ. Step 2. We assume that the solution of 2.3 can be expressed by a polynomial in G′/G as follows:


Introduction
It is well known that the nonlinear Schr ödinger NLS equation models a wide range of physical phenomena including self-focusing of optical beams in nonlinear media, the modulation of monochromatic waves, propagation of Langmuir waves in plasmas, and so forth.The nonlinear Schr ödinger equations play an important role in many areas of applied physics, such as nonrelativistic quantum mechanics, laser beam propagation, Bose-Einstein condensates, and so on see 1 .Some properties of solutions for the nonlinear Schr ödinger equations on R n have been extensively studied in the last two decades e.g., see 2 .
The Boussinesq-type equations are essentially a class of models appearing in physics and fluid mechanics.The so-called Boussinesq equation was originally derived by Boussinesq 3 to describe two-dimensional irrotational flows of an inviscid liquid in a uniform rectangular channel.It also arises in a large range of physical phenomena including the propagation of ion-sound waves in a plasma and nonlinear lattice waves.The study on the soliton solutions for various generalizations of the Boussinesq equation has recently attracted considerable attention from many mathematicians and physicists see 4 .We should remark that it was the first equation proposed in the literature to describe this kind of physical phenomena.This equation was also used by Zakharov 5 as a model of nonlinear string and by Falk et al. 6 in their study of shape-memory alloys.
In the laser and plasma physics, the Schr ödinger-Boussinesq system hereafter referred to as the SB-system has been raised.Consider the SB-system where t > 0, x ∈ 0, L , for some L > 0, and a, b are real constants.Here u and v are, respectively, a complex-valued and a real-valued function defined in space-time 0, L × R. The SB-system is considered as a model of interactions between short and intermediate long waves, which is derived in describing the dynamics of Langmuir soliton formation and interaction in a plasma 7 and diatomic lattice system 8 .The short wave term u x, t : 0, L × R → C is described by a Schr ödinger type equation with a potential v x, t : 0, L × R → R satisfying some sort of Boussinesq equation and representing the intermediate long wave.The SB-system also appears in the study of interaction of solitons in optics.The solitary wave solutions and integrability of nonlinear SB-system has been considered by several authors see 7, 8 and the references therein.
In the literature, there is a wide variety of approaches to nonlinear problems for constructing travelling wave solutions, such as the inverse scattering method 9 , Bcklund transformation 10 , Hirota bilinear method 11 , Painlevé expansion methods 12 , and the Wronskian determinant technique 13 .
With the help of the computer software, most of mentioned methods are improved and many other algebraic method, proposed, such as the tanh/coth method 14 , the Exp-function method 15 , and first integral method 16 .But, most of the methods may sometimes fail or can only lead to a kind of special solution and the solution procedures become very complex as the degree of nonlinearity increases.
Recently, the G /G -expansion method, firstly introduced by Wang et al. 17 , has become widely used to search for various exact solutions of NLEEs 17-19 .
The main idea of this method is that the traveling wave solutions of nonlinear equations can be expressed by a polynomial in G /G , where G G ξ satisfies the second order linear ordinary differential equation G ξ λG ξ μG ξ 0, where ξ kx ωt and k, ω are arbitrary constants.The degree of this polynomial can be determined by considering the homogeneous balance between the highest order derivatives and the non-linear terms appearing in the given non-linear equations.
Our first interest in the present work is in implementing the G /G -expansion method to stress its power in handling nonlinear equations, so that one can apply it to models of various types of nonlinearity.The next interest is in the determination of exact traveling wave solutions for the SB-system 1.1 .

Description of the G /G -Expansion Method
The objective of this section is to outline the use of the G /G -expansion method for solving certain nonlinear partial differential equations PDEs .Suppose that a nonlinear equation, say in two independent variables x and t, is given by where u x, t is an unknown function, P is a polynomial in u x, t and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved.The main steps of the G /G -expansion method are the following: Step 1. Combining the independent variables x and t into one variable ξ kx ωt, we suppose that The travelling wave variable 2.2 permits us to reduce 2.1 to an ODE for u x, t U ξ , namely, where prime denotes derivative with respect to ξ.
Step 2. We assume that the solution of 2.3 can be expressed by a polynomial in G /G as follows: where m is called the balance number, α 0 , and α i , are constants to be determined later, G ξ satisfies a second order linear ordinary differential equation LODE : where λ and μ are arbitrary constants.The positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in ODE 2.3 .
Step 3. By substituting 2.4 into 2.3 and using the second order linear ODE 2.5 , collecting all terms with the same order of G /G together, the left-hand side of 2.3 is converted into another polynomial in G /G .Equating each coefficient of this polynomial to zero yields a set of algebraic equations for k, ω, λ, μ, α 0 , α 1 , . . ., α m .
Step 4. Assuming that the constants k, ω, λ, μ, α 0 , α 1 , ..., α m can be obtained by solving the algebraic equations in Step 3, since the general solutions of the second order linear ODE 2.5 is well known for us, then substituting k, ω, λ, μ, α 0 , ..., α m and the general solutions of 2.5 into 2.4 we have more travelling wave solutions of the nonlinear evolution 2.1 .
In the subsequent section, we will illustrate the validity and reliability of this method in detail with complex model of Schr ödinger-Boussinesq System 1.1 .

Application
To look for the traveling wave solution of the Schr ödinger-Boussinesq System 1.1 , we use the gauge transformation: where ξ kx ωt, η px qt, and p, q, ω are constants and i √ −1.We substitute 3.1 into 1.1 to obtain nonlinear ordinary differential equation In order to simplify, integrating 3.3 twice and taking integration constant to zero, the system 3.2 -3.3 reduces to the following system:

3.4
Suppose that the solution of the nonlinear ordinary differential system 3.4 can be expressed by a polynomial in G /G as follows: where m, n are called the balance number, α i , i 0, 1, . . ., m and β j , j 0, 1, . . ., n are constants to be determined later, G ξ satisfies a second order linear ordinary differential equation 2.5 .The integers m, n can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in nonlinear ordinary differential system 3.4 as follow: 3.6 so that m n 2. We then suppose that 3.4 has the following formal solutions:

3.7
Substituting 3.7 along with 2.5 into 3.4 and collecting all the terms with the same power of G /G together, equating each coefficient to zero, yields a set of simultaneous algebraic equations for k, ω, λ, μ, α j , and β j , j 0, 1, 2 , as follows:

3.8
Then, explicit and exact wave solutions can be constructed through our ansatz 3.7 via the associated solutions of 2.5 .
In the process of constructing exact solutions to the Schr ödinger-Boussinesq system 1.1 , 2.5 is often viewed as a key auxiliary equation, and the types of its solutions determine the solutions for the original system 1.1 indirectly.In order to seek more new solutions to 1.1 , we here combine the solutions to 2.5 which were listed in 18, 19 .Our computation results show that this combination is an efficient way to obtain more diverse families of explicit exact solutions.
Solving 3.8 by use of Maple, we get the following reliable results: 3.9 where k and ω are free constant parameters and k / ± ω.Therefore, substitute the above case in 3.7 , we get

3.10
Substituting the general solutions of ordinary differential equation 2.5 into 3.10 , we obtain two types of traveling wave solutions of 1.1 in view of the positive and negative of λ 2 − 4μ.When D λ 2 −4μ ω 2 −k 2 /k 4 > 0, using the general solutions of ordinary differential equation 2.5 and relationships 3.10 , we obtain hyperbolic function solutions u H x, t and v H x, t of the Schr ödinger-Boussinesq system 1.1 as follows: and k, ω, C 1 , and C 2 are arbitrary constants and k / ± ω.

Conclusions
This study shows that the G /G -expansion method is quite efficient and practically well suited for use in finding exact solutions for the Schr ödinger-Boussinesq system.With the aid of Maple, we have assured the correctness of the obtained solutions by putting them back into the original equation.To illustrate the obtained solutions, the hyperbolic type of obtained solutions, 3.12a and 3.12b , are attached as Figures 1, 2, and 3. We hope that they will be useful for further studies in applied sciences.