A high-order nonlinear Schrödinger equation as a variational problem for the averaged Lagrangian of the nonlinear Klein–Gordon equation

Abstract

We use Whitham’s averaged Lagrangian method extended with the multiple-scale formalism to derive a sixth-order nonlinear Schrodinger equation for the complex amplitude of the envelope of the slowly modulated wave trains whose evolution is governed by the nonlinear Klein–Gordon equation with polynomial nonlinearity. Such a high-order nonlinear Schrodinger equation is obtained as a variational problem for the Lagrangian averaged over the rapid oscillations of the carrier wave train. As compared to classical Whitham’s approach, we take into account the derivatives of the complex amplitude with respect to slow timescales and long coordinate in the averaged Lagrangian. The coefficients of the high-order nonlinear Schrodinger equation derived in this work identically coincide with those derived by us earlier by the method of multiple scales applied directly to the original nonlinear Klein–Gordon equation. Finally, we consider the sine-Gordon equation as a partial case of the nonlinear Klein–Gordon equation and demonstrate one example of the numerical solution to the corresponding sixth-order nonlinear Schrodinger equation in the form of the envelope quasi-soliton.