Stability of mKdV breathers on the half-line

TL;DR

This paper proves the strong stability of leftward moving mKdV breathers on the half-line under homogeneous boundary conditions, using a Lyapunov functional approach to control boundary effects.

Contribution

It demonstrates the stability of mKdV breathers on the half-line and introduces a Lyapunov functional method to handle boundary terms.

Findings

Leftward moving breathers are strongly stable on the half-line.

Stability is established under homogeneous boundary conditions.

A Lyapunov functional effectively controls boundary effects.

Abstract

In this paper we study the stability problem for mKdV breathers on the left half-line. We are able to show that leftwards moving breathers, initially located far away from the origin, are strongly stable for the problem posed on the left half-line, when assuming homogeneous boundary conditions. The proof involves a Lyapunov functional which is almost conserved by the mKdV flow once we control some boundary terms which naturally arise.