The Investigation of Solutions to the Coupled Schrödinger-Boussinesq Equations

and Applied Analysis 3 we obtain the algebraic system


Introduction
In laser and plasma physics, the important problems under interactions between a nonlinear complex Schrödinger field and a real Boussinesq field have been raised.In particular, the study of the coupled Schrödinger-Boussinesq equations has attracted much attention of mathematicians and physicists (see [1][2][3]).The existence of the global solution of the initialboundary problem for the equations was investigated in [1].The existence of a periodic solution for the equations was considered in [2].Kılıcman and Abazari [3] used the (  /)expansion method to construct periodic and soliton solutions for the Schrödinger-Boussinesq equations   +  −V = 0, V  − V  + V  − (|| 2 )  = 0, where  and  are real constants.The investigation of nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena (see [4][5][6][7]).
In this paper, we consider the following coupled Schrödinger-Boussinesq equations: where (, ) is a complex unknown function, (, ) is a real unknown function, and  1 and  2 are real positive constants.System (1) is known to describe various physical processes in laser and plasma physics, such as formation, Langmuir field amplitude, intense electromagnetic waves, and modulational instabilities (see [8]).The approximate solutions and conservation law for the coupled system (1) have been studied in [9].In [10], Chen and Xu used the -expansion method to obtain a number of periodic wave solutions expressed by various Jacobi elliptic functions for (1).Cai et al. [11] studied same equations by the modified expansion method.
In the present paper, we use the (  /)-expansion method and the symbolic computation system Mathematica to investigate the coupled Schrödinger-Boussinesq system (1).Here, we state that the previous works do not obtain the solutions presented in this paper.
The layout of this paper is as follows.In Section 2, we give the description of the generalized (  /)-expansion method.In Section 3, we apply this method to solve (1).A conclusion will be obtained in Section 4.
Using travelling transformation (, ) = (),  =  − .Equation ( 3) can be integrated as long as all terms contain derivatives where integration constants are considered to be zeros.
Step 2. Suppose that the solution of (3) can be expressed as a polynomial in (  /) where  = () satisfies the second-order ODE with respect to .Namely, where  1 , . . .,   ̸ = 0, , and  are constants to be determined later.The positive integer  can be determined by balancing the highest-order derivatives with highest-order nonlinear terms appearing in (3).It is easy to check that (5) admits three types of solutions in which  =  2 − 4.
Step 3. By substituting (4) into (3) and using ( 5), collecting all terms with the same order of (  /) together, the left-hand side of (3) can be written as a polynomial in (  /).Letting each coefficient of this polynomial be zero yields a system of algebraic equations for  1 , . . .,   , , , and .

Solutions of the Coupled Schrödinger-Boussinesq Equations
Following the procedure described in Section 2, we adopt the ansatz solution of (1) in the form where (, ) is a real function, ,  are constants to be determined, and  0 is an arbitrary constant.Substituting ( 7) into (1) yields We take where  1 is an arbitrary constant.Substituting ( 11) into ( 9), one gets Suppose that It follows from ( 9), ( 10), (11), and (13) that Balancing   with V in (14) and V  with  2 in (15) leads to  = 1,  = 2. Thus we can search for the solutions of ( 14) and (15) in the following forms: Substituting ( 16) and ( 17) into ( 14) and (15), using (5), and setting the coefficients of (  /)  ( = 0, . . ., 4) to be zero, we obtain the algebraic system Solving this system with the Mathematica, we find or where  and  are arbitrary constants.

Conclusion
The (  /)-expansion method is effectively employed to deal with the coupled Schrödinger-Boussinesq equations.The hyperbolic function solutions, the trigonometric function solutions, and the rational function solutions to the equations in the case of  > 0,  < 0, and  = 0 are obtained.
In particular, the well-known soliton solutions are only the special case of the hyperbolic-type solutions.We find several properties of solutions when  → ∞.