Stability of the multi-solitons of the modified Korteweg-de Vries equation

TL;DR

This paper proves the nonlinear stability of multi-soliton solutions of the modified Korteweg-de Vries (mKdV) equation, showing they are local minima of conserved quantities and analyzing spectral properties of linearized operators.

Contribution

It introduces a variational characterization of N-solitons for mKdV and develops new operator identities and spectral analysis techniques for stability proof.

Findings

N-solitons are stable and behave as sums of 1-solitons at infinity.

N-solitons minimize a conserved quantity under fixed constraints.

Spectral properties of linearized operators are preserved over time.

Abstract

We establish the nonlinear stability of $N$-soliton solutions of the modified Korteweg-de Vries (mKdV) equation. The $N$-soliton solutions are global solutions of mKdV behaving at (positive and negative) time infinity as sums of $1$-solitons with speeds $0<c_1<\cdots< c_N$.The proof relies on the variational characterization of $N$-solitons. We show that the $N$-solitons realize the local minimum of the $(N+1)$-th mKdV conserved quantity subject to fixed constraints on the $N$ first conserved quantities.To this aim, we construct a functional for which $N$-solitons are critical points, we prove that the spectral properties of the linearization of this functional around a $N$-soliton are preserved on the extended timeline, and we analyze the spectrum at infinity of linearized operators around $1$-solitons. The main new ingredients in our analysis are a new operator identity based on a generalized Sylvester law of inertia and recursion operators for the mKdV equation.