Mathematical Research Letters

Volume 29 (2022)

Number 1

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

Pages: 285 – 322

DOI: https://dx.doi.org/10.4310/MRL.2022.v29.n1.a9

Authors

Xingdong Tang (School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China)

Guixiang Xu (School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, China)

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 $(\gamma \gt 0)$\[\mathrm{i} u_t + u_{xx} + \gamma \delta u + {\lvert u \rvert}^{p-1} u = 0, p \gt 5 \; \textrm{,}\]where $\Omega =\Omega (p,\gamma) \gt \frac{\gamma^2}{4}$ is the critical oscillation number and determined by\[\frac{p-5}{p-1} \int^{+\infty}_{\operatorname{arctanh} \left( \frac{\gamma}{2\sqrt{\Omega}} \right) } \operatorname{sech}^{\frac{4}{p-1}} (y) d_y = \frac{\gamma}{2\sqrt{\Omega}}{\left( 1-\frac{\gamma^2}{4\Omega} \right)}^{-\frac{p-3}{p-1}} \Longleftrightarrow d^{\prime\prime} (\Omega)=0 \; \textrm{.}\]The classical convex method and Grillakis–Shatah–Strauss’s stability approach in [2, 10] doesn’t work in this degenerate case, and the argument here is motivated by those in [5, 16, 17, 22, 23]. 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 $\mathcal{M} (Q_\Omega)$ according to the degenerate structure of the Hamiltonian and to construct a 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 [8] in the degenerate case.

Received 23 October 2019

Accepted 22 October 2020

Published 6 September 2022