Periodic waves of the modified KdV equation as minimizers of a new variational problem

TL;DR

This paper introduces a new constrained variational framework for periodic waves of the modified KdV equation, identifying stable minimizers and analyzing bifurcations with analytical and numerical methods.

Contribution

It presents a novel variational problem with stability properties for periodic mKdV waves, including analytical representations and bifurcation analysis.

Findings

Stable minimizers correspond to non-degenerate local minima.

A broken pitchfork bifurcation involves three solution families.

Numerical computation of critical points confirms analytical results.

Abstract

Periodic waves of the modified Korteweg-de Vries (mKdV) equation are identified in the context of a new variational problem with two constraints. The advantage of this variational problem is that its non-degenerate local minimizers are stable in the time evolution of the mKdV equation, whereas the saddle points are unstable. We explore the analytical representation of periodic waves given by Jacobi elliptic functions and compute numerically critical points of the constrained variational problem. A broken pitchfork bifurcation of three smooth solution families is found. Two families represent (stable) minimizers of the constrained variational problem and one family represents (unstable) saddle points.