The Modulation of Multiple Phases Leading to the Modified KdV Equation

TL;DR

This paper introduces a novel modulation approach to derive the modified KdV equation from systems with two-parameter symmetry, linking coefficients to conservation laws and illustrating with physical examples.

Contribution

It presents a new method for deriving the mKdV equation from Lagrangian systems with symmetries, simplifying coefficient determination and broadening analytical tools.

Findings

Derived mKdV from abstract Lagrangians with symmetry

Developed a simplified adaptation of Kuramoto's method

Applied theory to hydrodynamics and nonlinear Schrödinger models

Abstract

This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified hydrodynamics and another using a coupled Nonlinear Schr\"odinger model, to illustrate how the criterion for the mKdV equation to emerge may be assessed and its coefficients generated.