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We use Whitham’s averaged Lagrangian method extended with the multiple-scale formalism to derive a sixth-order nonlinear Schrödinger 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 Schrödinger equation is obtained as a variational problem...
We use the method of multiple scales to derive a sixth-order nonlinear Schrödinger 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...
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