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Instability of the solitary waves for the 1d NLS with an attrictive delta potential in the degenerate case

Xingdong Tang, Guixiang Xu

Abstract

In this paper, we show the orbital instability of the solitary waves $Q_Ωe^{iΩt}$ of the 1d NLS with an attractive delta potential ($γ>0$) \begin{equation*} ıu_t+u_{xx}+γδu+\abs{u}^{p-1}u=0, \; p>5, \end{equation*} where $Ω=Ω(p,γ)>\frac{γ^2}{4}$ is the critical oscillation number and determined by \begin{equation*} \frac{p-5}{p-1} \int_{ \arctanh\sts{ \fracγ{2\sqrtΩ} } }^{+\infty} \sech^{\frac{4}{p-1}}\sts{y}\d y = { \fracγ{ 2\sqrtΩ } }\sts{ 1-\frac{γ^2}{4Ω} }^{-\frac{p-3}{p-1}} \Longleftrightarrow \mathbf{d}''(Ω) =0. \end{equation*} The classical convex method and Grillakis-Shatah-Strauss's stability approach in \cite{A2009Stab, GSS1987JFA1} don't work in this degenerate case, and the argument here is motivated by those in \cite{CP2003CPAM, MM2001GAFA, M2012JFA, MTX2018, O2011JFA}. The main ingredients are to construct the unstable second order approximation near the solitary wave $Q_Ωe^{iΩt}$ on the level set $\Mcal(Q_Ω)$ accoding to the degenerate structure of the Hamiltonian and to construct the refined Virial identity to show the orbital instability of the solitary waves $Q_Ωe^{iΩt}$ in the energy space. Our result is the complement of the results in \cite{FOO2008AIHP} in the degenerate case.

Instability of the solitary waves for the 1d NLS with an attrictive delta potential in the degenerate case

Abstract

In this paper, we show the orbital instability of the solitary waves of the 1d NLS with an attractive delta potential () \begin{equation*} ıu_t+u_{xx}+γδu+\abs{u}^{p-1}u=0, \; p>5, \end{equation*} where is the critical oscillation number and determined by \begin{equation*} \frac{p-5}{p-1} \int_{ \arctanh\sts{ \fracγ{2\sqrtΩ} } }^{+\infty} \sech^{\frac{4}{p-1}}\sts{y}\d y = { \fracγ{ 2\sqrtΩ } }\sts{ 1-\frac{γ^2}{4Ω} }^{-\frac{p-3}{p-1}} \Longleftrightarrow \mathbf{d}''(Ω) =0. \end{equation*} The classical convex method and Grillakis-Shatah-Strauss's stability approach in \cite{A2009Stab, GSS1987JFA1} don't work in this degenerate case, and the argument here is motivated by those in \cite{CP2003CPAM, MM2001GAFA, M2012JFA, MTX2018, O2011JFA}. The main ingredients are to construct the unstable second order approximation near the solitary wave on the level set accoding to the degenerate structure of the Hamiltonian and to construct the refined Virial identity to show the orbital instability of the solitary waves in the energy space. Our result is the complement of the results in \cite{FOO2008AIHP} in the degenerate case.

Paper Structure

This paper contains 9 sections, 18 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear Schrödinger operator with a delta potential
  4. Basic properties of the action functional ${\mathcal{S}}_{\omega}$ and $d\left(\omega\right)$
  5. Geometric decomposition of $u$ and landscape of ${\mathcal{S}}_{\Omega}$ near $Q_{\Omega}$
  6. Properties of the linearized operator ${\mathcal{S}}_{\Omega}^{\prime\prime}\left( Q_{\Omega} \right)$
  7. The $\varepsilon$-variable equation and the dynamics of the parameters
  8. Proof of Theorem \ref{['mainthm']}
  9. Proof of \ref{['lem:d3rdd']}

Key Result

Proposition 1.1

For any $u_0\in H^1\left({\mathbb R}\right)$, there exists $T_{\max}$ with $0<T_{\max}\leqslant +\infty$ and a unqie solution $u\in{\mathcal{C}}\left( \left[0, T_{\max}\right), H^1\left({\mathbb R}\right) \right)$ for dnls satisfying Moreover, the mass and the energy are conserved under the flow generated by dnls, i.e., for any $t\in \left[0, T_{\max}\right)$, we have

Theorems & Definitions (34)

  • Proposition 1.1: Local well-posedness in $H^1\left({\mathbb R}\right)$
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 24 more