Asymptotic stability of the Toda m-soliton

TL;DR

This paper proves the asymptotic stability of multi-soliton solutions in the Toda lattice using a novel approach that avoids traditional inverse spectral methods, focusing instead on the linearization of Bäcklund transformations.

Contribution

It introduces a new proof technique for stability of Toda m-solitons that does not rely on inverse spectral methods or Lax pairs, using Bäcklund transformations instead.

Findings

Multi-soliton solutions are linearly stable.

Multi-soliton solutions are nonlinearly stable.

The method simplifies stability analysis without inverse spectral methods.

Abstract

We prove that multi-soliton solutions of the Toda lattice are both linearly and nonlinearly stable. Our proof uses neither the inverse spectral method nor the Lax pair of the model but instead studies the linearization of the B\"acklund} transformation which links the ($m-1$)-soliton solution to the $m$-soliton solution. We use this to construct a conjugation between the Toda flow linearized about an $m$-solition solution and the Toda flow linearized about the zero solution, whose stability properties can be determined by explicit calculation.