Travelling Wave Solutions of the Schrödinger-Boussinesq System

Abstract

We establish exact solutions for the Schrödinger-Boussinesq System $i{u}_t+u_{xx}-a\mathrm{uv}=0$, $v_{tt}-v_{xx}+v_{xxxx}-b{({\left|u\right|}^{\mathrm{2}})}_{xx}=0$, where $a$ and $b$ are real constants. The ($G^{\prime }/G$)-expansion method is used to construct exact periodic and soliton solutions of this equation. Our work is motivated by the fact that the ($G^{\prime }/G$)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. As a result, hyperbolic function solutions and trigonometric function solutions with parameters are obtained. These solutions may be important and of significance for the explanation of some practical physical problems.