Two-Parameter Method for Describing the Nonlinear Evolution of Narrow-Band Wave Trains

TL;DR

This paper introduces a novel two-parameter perturbative method to model the nonlinear evolution of narrow-band wave trains, resulting in a high-order nonlinear Schrödinger equation applicable to various physical systems.

Contribution

It develops a new perturbative technique that uses two independent small parameters, avoiding secular terms and improving modeling of wave train evolution.

Findings

Derivation of a high-order nonlinear Schrödinger equation

Application to ultrashort pulse propagation in optical fibers

Potential use in fluid surface waves and plasma waves

Abstract

We consider the evolution of narrow-band wave trains of finite amplitude in a nonlinear dispersive system which is described by the Klein--Gordon equation with arbitrary polynomial nonlinearity. We use a new perturbative technique which allows the original wave equation to be reduced to a model equation for the wave train envelope (high-order nonlinear Schr\"{o}dinger equation). The time derivative is expanded into an asymptotic series in two independent parameters which characterize the smallness of amplitudes ($\varepsilon$) and the slowness of their spatial variations ($\mu$). In contrast to other perturbative methods in which these two parameters are taken equal (e.g., the multiple scale method), the two-parameter method produces no secular terms. The results of this study can be applied to investigating the propagation of ultrashort (femtosecond) pulses in optical fibers, to studying the wave events on a fluid surface, and to describing the Langmuir waves in hot plasmas.