New $\phi^{6}$ ϕ 6 -model expansion method and its applications to the resonant nonlinear Schrödinger equation with parabolic law nonlinearity

Abstract

With the aid of symbolic computation, the new $\phi^{6}$ -model expansion method is applied, in this article, for the first time to the resonant nonlinear Schrodinger equation with parabolic law nonlinearity to find families of Jacobi elliptic function solutions. In particular, when the modulus of the Jacobi elliptic functions tends to one or to zero, we can get the hyperbolic and trigonometric function solutions, respectively. This new method presents a wider applicability for handling the nonlinear partial differential equations. Comparison of our new results with the well-known results are given. At the end of this paper, we use the solutions of the Lienard equation to find more different solutions for the proposed resonant nonlinear Schrodinger equation mentioned above.