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# 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

#### 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