Nonlinear stability of higher order mKdV breathers

TL;DR

This paper proves the stability of higher order mKdV breathers in $H^2( )$ and shows they satisfy the same stationary elliptic equation as classical breathers, extending known stability results to more complex equations.

Contribution

It establishes the stability of 5th, 7th, and 9th order mKdV breathers and demonstrates they satisfy a common stationary elliptic equation, generalizing classical results.

Findings

Higher order mKdV breathers are stable in $H^2( )$.

Breathers satisfy the same stationary elliptic equation regardless of order.

Stability results extend classical mKdV breather stability to higher orders.

Abstract

We are interested in stability results for breather solutions of the 5th, 7th and 9th order mKdV equations. We show that these higher order mKdV breathers are stable in $H^2(\R)$, in the same way as \emph{classical} mKdV breathers. We also show that breather solutions of the 5th, 7th and 9th order mKdV equations satisfy the same stationary fourth order nonlinear elliptic equation as the mKdV breather, independently of the order, 5th, 7th or 9th, considered.