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

Simon Deng, Mathew A. Johnson, Stéphane Lafortune

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.

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

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 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.

Paper Structure

This paper contains 19 sections, 25 theorems, 184 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary Properties of the DP Equation
  3. Existence of Smooth Solitary Waves
  4. Conservation Laws & Hamiltonian Structures
  5. The Linearization
  6. Analysis of the Essential Spectrum
  7. The Essential Spectrum on $L^2({\mathbb{R}})$
  8. The Essential Spectrum on $L^2_{\alpha}({\mathbb{R}})$
  9. Analysis of the Point Spectrum
  10. Weighted vs. Unweighted Point Spectrum
  11. Lax Pair & Squared Eigenfunctions
  12. Absence of Non-Zero Point Spectrum
  13. Generalized Kernel in the space $L^2_\alpha({\mathbb{R}})$
  14. Semigroup Decay Estimates & Linear Asymptotic Stability
  15. Free Evolution & Resolvent Estimates
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $u_0(\cdot;k,c)$ be a smooth solitary wave solution of the DP equation DP, as constructed in Lemma L1, and let $0<\alpha<\sqrt{(c-4k)/(c-k)}$ be fixed and $\eta>0$ be such that Further, let $\Pi:L^2_\alpha({\mathbb{R}})\to {\rm gKer}\left(\mathcal{A}[u_0]\right)$ be the rank-2 spectral projection onto the generalized kernel of $\mathcal{A}[u_0]$ on $L^2_\alpha({\mathbb{R}})$. Then there exist

Figures (2)

  • Figure 1: Graph of the essential spectrum of $\mathcal{A}_\alpha[u_0]$, parametrized by $\sigma$ as given in \ref{['RI']} for $0<\alpha<1$. For these graphs, the underlying wave corresponds to $k=0.1$ and $c=1$, in which case \ref{['conddef1']} reduces to $0<\alpha^2<2/3$. The plot on the left takes $\alpha=0.5$, thus satisfying \ref{['conddef1']}, indicating stability of the essential spectrum. The middle plot takes $\alpha=\sqrt{2/3}$, thus showing an essential spectrum touching the imaginary axis corresponding to marginal stability. The plot on the right takes $\alpha=0.9>\sqrt{2/3}$, thus displaying an unstable spectrum. The plot on the bottom right takes $\alpha=1.2$ and hence, since $\alpha>1$, also indicating stability of the essential spectrum.
  • Figure 2: Using the same $k$ and $c$ values from Figure \ref{['fig1']}, this is a graph of the essential spectrum of $\mathcal{A}_\alpha[u_0]$ in the case where $\alpha=1.2$, indicating the expected stability of the essential spectrum.

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Theorem 2.7: Li2019
  • Remark 2.8
  • ...and 45 more