Evolution equations for pulse propagation in nonlinear media

TL;DR

This paper analyzes the complex modified KdV and generalized nonlinear Schrödinger equations, showing their place in the AKNS hierarchy, deriving variational principles, and exploring conditions for solitary wave solutions.

Contribution

It introduces auxiliary fields to establish variational principles for nonintegrable equations and provides analytical solutions for solitary waves.

Findings

Both equations are part of the AKNS hierarchy.

Auxiliary fields enable variational formulations.

Conditions for bright and dark solitary waves are derived.

Abstract

We show that the complex modified KdV (cmKdV) equation and generalized nonlinear Schr\"odinger (GNLS) equation belong to the Ablowitz, Kaup, Newell and Segur or so-called AKNS hierarchy. Both equations do not follow from the action principle and are nonintegrable. By introducing some auxiliary fields we obtain the variational principle for them and study their canonical structures. We make use of a coupled amplitude-phase method to solve the equations analytically and derive conditions under which they can support bright and dark solitary wave solutions.