Stability of $n$-soliton solutions for the Intermediate Long Wave equation

TL;DR

This paper proves the stability of multi-soliton solutions for the ILW equation, a model for internal waves, using variational methods, inverse scattering, and Hamiltonian structures, with results on both orbital and dynamical stability.

Contribution

It introduces a novel stability analysis for n-solitons of the ILW equation using a combination of variational, inverse scattering, and Hamiltonian techniques.

Findings

n-solitons are non-isolated constrained minimizers

n-solitons are dynamically stable in H^{n/2}

Double solitons are orbitally stable in H^1

Abstract

In this work, we focus on the stability of $n$-soliton solutions ($n\in \mathbb{N}, n\geq 1$) to the completely integrable intermediate long wave equation (ILW), which models long internal gravity waves in a stratified fluid of finite depth. We show that the $n$-soliton solutions of the ILW equation form non-isolated constrained minimizers of a variational problem associated with a non-local elliptic equation. To establish this result, we construct a suitable Lyapunov functional and utilize the inverse scattering transform to relate the infinite sequence of conservation laws to the scattering data. Furthermore, we employ the recursion operator derived from the bi-Hamiltonian structure to optimize our analysis. Our analysis demonstrates that the $n$-soliton solutions of the ILW equation are dynamically stable in the space $H^{\frac{n}{2}}(\mathbb{R})$ ($n\in \mathbb{N}, n\geq 1$). Additionally, we establish the orbital stability of double soliton solutions in $H^1(\mathbb{R})$.