Orbital Stability of Localized Structures via Backlund Transformations

TL;DR

This paper presents a new dynamical perspective on Backlund Transformations, proving an abstract orbital stability theorem and applying it to the sine-Gordon equation and Toda lattice to analyze localized structures.

Contribution

It introduces a dynamical framework for Backlund Transformations and establishes an orbital stability theorem applicable to integrable systems.

Findings

Proved an abstract orbital stability theorem.

Applied the theorem to sine-Gordon and Toda lattice.

Demonstrated stability of localized structures in these systems.

Abstract

The Backlund Transform, first developed in the context of differential geometry, has been classically used to obtain multi-soliton states in completely integrable infinite dimensional dynamical systems. It has recently been used to study the stability of these special solutions. We offer here a dynamical perspective on the Backlund Transform, prove an abstract orbital stability theorem, and demonstrate its utility by applying it to the sine-Gordon equation and the Toda lattice.