Linear Asymptotic Stability of the Smooth 1-Solitons for the Degasperis-Procesi Equation

TL;DR

This paper proves the linear asymptotic stability of smooth 1-solitons in the Degasperis-Procesi equation using spectral analysis and integrability properties, with implications for nonlinear stability.

Contribution

It establishes the spectral stability and exponential decay estimates for the linearized operator around solitons, advancing understanding of their stability in the DP equation.

Findings

Spectral gap exists for the linearized operator in weighted spaces.

Exponential decay of the semigroup implies linear asymptotic stability.

Analytical challenges remain for extending results to nonlinear stability.

Abstract

In this paper, we study the asymptotic stability of smooth 1-solitons in the Degasperis-Procesi (DP) equation. Such solutions necessarily exist on a non-zero background, and their spectral and orbital stability has previously been verified by Li, Liu & Wu and by Lafortune & Pelinovsky. Using the complete integrability of the DP equation to establish the strong spectral stability of smooth solitary waves, namely that the origin is the only eigenvalue of the associated linearized operator acting on $L^2(\mathbb{R})$ and that, moreover, in appropriate exponentially weighted spaces the non-zero spectrum for the linearized operator admits a spectral gap away from the imaginary axis. This spectral gap result {{is then}} upgraded to an exponential decay estimate on the semigroup associated with the linearized operator, establishing a linear asymptotic stability result in exponentially weighted spaces. Finally, we outline analytical challenges with extending our result to the nonlinear level.