Exact solutions for some non-linear differential equations using complex hyperbolic function methods

Abstract

Based on computerized symbolic computation, the coth–csch complex hyperbolic-function method is proposed for the general non-linear equations of mathematical physics in a unified way. In this article, we assume that exact solutions for a given general non-linear equations be the superposition of different powers of the coth-function, csch-function and/or their combinations. After finishing some direct calculations, we can finally obtain the exact solutions expressed by the complex hyperbolic function. The characteristic feature of this method is that we can derive exact solutions to the general non-linear equations directly without transformation. Some illustrative equations, such as the coupled non-linear Schrödinger equation, the generalized non-linear Schrödinger-like equation, coupled non-linear Schrödinger KdV system, Davey Stewartson equation and the (2 + 1)-dimensional generalization of coupled non-linear Schrödinger KdV system equation are investigated by this method and new exact solutions are found.

Keywords:

• general non-linear equations
• symbolic computation
• complex hyperbolic-function method
• exact solutions

AMS Subject Classifications:

• 35K99
• 35P05
• 35P99