A sixth-order nonlinear Schrödinger equation as a reduction of the nonlinear Klein–Gordon equation for slowly modulated wave trains

Abstract

We use the method of multiple scales to derive a sixth-order nonlinear Schrodinger equation governing the evolution of slowly modulated plane-wave solutions to the nonlinear Klein–Gordon equation with polynomial nonlinearity. The coefficients of this sixth-order equation are expressed explicitly in terms of the velocity parameter as well as linear, quadratic, cubic, quadruple, and quintic nonlinear coefficients of the original nonlinear Klein–Gordon equation.