Elementary functions for constructing exact solutions to nonlinear partial differential equations with applications to nonlinear Schrödinger type and MHD equations

In this paper we present an improvement to those methods based on the using of elementary functions like exponential, trigonometric and hyperbolic functions in obtaining exact solutions to nonlinear partial differential equations (NPDEs). The improved method is applied to stable nonlinear Schrödinger (NLS), unstable NLS, generalized NLS, High-order NLS and derivative NLS equations. New solutions for these equations are obtained. The obtained solutions are more general than a wide class of previous solutions. Solutions of a derivative NLS equation that describes the large-amplitude solitons propagating in an arbitrary direction in a high-β hall plasma are also obtained. Moreover, the method is applied to magnetohydrodynamics (MHD) equations describing an ideal incompressible flow in the steady state. One of the most important advantages of the solution method presented here is it deals with several types of nonlinearities associated with PDEs without making a transformation of the original equation to another one.