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*} \i u_t+u_{xx}+\gamma\delta u+\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{ \frac{\gamma}{2\sqrt{\Omega}} } }^{+\infty} \sech^{\frac{4}{p-1}}\sts{y}\d y = { \frac{\gamma}{ 2\sqrt{\Omega} } }\sts{ 1-\frac{\gamma^2}{4\Omega} }^{-\frac{p-3}{p-1}} \Longleftrightarrow \mathbf{d}''(\Omega) =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 $\Mcal(Q_{\Omega})$ 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.