New extended auxiliary equation method and its applications to nonlinear Schrödinger-type equations

Abstract

A new extended auxiliary equation method has been applied to construct many new types of Jacobi elliptic function solutions of nonlinear partial differential equations (PDEs) in mathematical physics. The effectiveness of this method is demonstrated by applications to three nonlinear PDEs, namely, the (2+1)-dimensional nonlinear cubic–quintic Ginzburg–Landau equation, the (1+1)-dimensional resonant nonlinear Schrödinger's equation with parabolic law nonlinearity and the generalized Zakharov system of equations. The solitary wave solutions or trigonometric functions solutions are obtained from the Jacobi elliptic function solutions when the modulus of the Jacobi elliptic functions approaches to one or zero respectively. Comparing our new results and the well-known results are given.