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On the asymptotic stability of ground states of the pure power NLS on the line at 3rd and 4th order Fermi Golden Rule

Scipio Cuccagna, Masaya Maeda

Abstract

Assuming as hypotheses the results proved numerically by Chang et al. \cite{Chang} for the exponent $p\in (3,5)$, we prove that some of the ground states of the nonlinear Schrödinger equation (NLS) with pure power nonlinearity of exponent $p$ in the line are asymptotically stable for a certain set of values of the exponent $p$ where the FGR occurs by means of a discrete mode 3rd or 4th order power interaction with the continuous mode. For the 3rd the result is true for generic $p$ while for the 4th order case we assume that there are $p$'s satisfying Fermi Golden rule and the non-resonance condition of the threshold of the continuous spectrum of the linearization. The argument is similar to our recent result valid for $p$ near 3 contained in \cite{CM24D1}.

On the asymptotic stability of ground states of the pure power NLS on the line at 3rd and 4th order Fermi Golden Rule

Abstract

Assuming as hypotheses the results proved numerically by Chang et al. \cite{Chang} for the exponent , we prove that some of the ground states of the nonlinear Schrödinger equation (NLS) with pure power nonlinearity of exponent in the line are asymptotically stable for a certain set of values of the exponent where the FGR occurs by means of a discrete mode 3rd or 4th order power interaction with the continuous mode. For the 3rd the result is true for generic while for the 4th order case we assume that there are 's satisfying Fermi Golden rule and the non-resonance condition of the threshold of the continuous spectrum of the linearization. The argument is similar to our recent result valid for near 3 contained in \cite{CM24D1}.
Paper Structure (4 sections, 9 theorems, 49 equations)

This paper contains 4 sections, 9 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Linearization
  3. Refined profile
  4. Refined profile in the 3rd order FGR

Key Result

Theorem 1.1

Let $p\in (1,5)$ and let $\omega _0 >0$. Then for any $\epsilon >0$ there exists a $\delta >0$ such that for any initial value $u_0\in \mathcal{U} (\omega _0,\delta )$, the corresponding solution satisfies $u\in C^0$ R , U (ω _0,ϵ ) $$.

Theorems & Definitions (15)

  • Theorem 1.1: Orbital Stability
  • Proposition 1.2
  • proof
  • Proposition 1.3: No Threshold Resonance
  • Remark 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Proposition 2.3
  • ...and 5 more