Sixth-order Nonlinear Schrodinger Equation Research Articles

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

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.

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

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.

Bilinear forms and solitons for a generalized sixth-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a generalized sixth-order nonlinear Schrodinger equation, which could describe the attosecond pulses in an optical fiber. Bilinear forms and soliton solutions are derived via the Hirota method. Dynamic behaviors of the solitons are also analyzed. Moreover, we advance a new method, at the heart of which lies the idea that we simplify the limitation of the complex functions to the real ones, to demonstrate that the interaction between the two solitons is elastic and present the mathematical expression of velocity and phase shift of each soliton simultaneously.