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Conditional asymptotic stability of solitary waves of the Euler-Poisson system on the line

Junsik Bae, Scipio Cuccagna, Masaya Maeda

Abstract

We apply the idea of using a combination of virial inequalities and Kato smoothing, previously applied to NLS and generalized KdV pure power equations to Euler-Poisson: we assume that a solution remains very close for all times to a soliton in an appropriate space and then we prove an asymptotic convergence to a soliton for $t\to +\infty$.

Conditional asymptotic stability of solitary waves of the Euler-Poisson system on the line

Abstract

We apply the idea of using a combination of virial inequalities and Kato smoothing, previously applied to NLS and generalized KdV pure power equations to Euler-Poisson: we assume that a solution remains very close for all times to a soliton in an appropriate space and then we prove an asymptotic convergence to a soliton for .
Paper Structure (11 sections, 14 theorems, 159 equations)

This paper contains 11 sections, 14 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Notation
  4. Solitary waves, Linearization and modulation
  5. Solitary waves
  6. Linearized operator
  7. Modulation
  8. Main estimates
  9. Estimates on some Schrödinger operators
  10. Proof of Lemma \ref{['lem:lemdscrt']}
  11. Virial inequalities: proof of Lemma \ref{['prop:1virial']}

Key Result

Theorem 1.1

There exists a $\varepsilon _0\in (0, c_K-\mathsf{V})$ such that for any $a>0$, $c_0\in (\mathsf{V}, \mathsf{V} + \varepsilon _0)$ and $\epsilon >0$ there exists a $\delta >0$ such that for any solution of EP which satisfies there exist a function $(c, D) \in C^1$ [0,+∞ ) , (c_0-ϵ , c_0+ϵ) R $$ and a $c_+>0$ such that

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 3.1: Theorem 1.2 of BK19JDE
  • Remark 3.2
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • Lemma 3.5: Modulation
  • Proposition 4.1
  • Lemma 4.2
  • ...and 10 more