Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Instability of ground states for the NLS equation with potential on the
  star graph   We study the nonlinear Schr\"odinger equation with an arbitrary real
potential $V(x)\in (L^1+L^\infty)(\Gamma)$ on a star graph $\Gamma$. At the
vertex an interaction occurs described by the generalized Kirchhoff condition
with strength $-\gamma<0$. We show the existence of ground states
$\varphi_{\omega}(x)$ as minimizers of the action functional on the Nehari
manifold under additional negativity and decay conditions on $V(x)$. Moreover,
for $V(x)=-\dfrac{\beta}{x^\alpha}$, in the supercritical case, we prove that
the standing waves $e^{i\omega t}\varphi_{\omega}(x)$ are orbitally unstable in
$H^{1}(\Gamma)$ when $\omega$ is large enough. Analogous result holds for an
arbitrary $\gamma\in\mathbb{R}$ when the standing waves have symmetric profile.

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The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
3
The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc