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 $Q_{\Omega }{e}^{i\Omega t}$ of the 1d NLS with an attractive delta potential ($\gamma >0$) \begin{equation*} ıu_t+u_{xx}+γδu+\abs{u}^{p-1}u=0, \; p>5, \end{equation*} where $\Omega =\Omega (p,\gamma )>\frac{{\gamma }^2}{4}$ 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 $Q_{\Omega }{e}^{i\Omega t}$ 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 $Q_{\Omega }{e}^{i\Omega t}$ in the energy space. Our result is the complement of the results in \cite{FOO2008AIHP} in the degenerate case.

27 pages. Any comment is welcome. arXiv admin note: text overlap with arXiv:1803.06451

References in corpus (3)

• Fast soliton scattering by delta impurities
• Stability of small solitary waves for the 1$d$ NLS with an attractive delta potential
• Modified scattering for the $1d$ cubic NLS with a repulsive delta potential