Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp thresholds for stability and instability of standing waves in a
  double power nonlinear Schr\"odinger equation   We study the stability/instability of standing waves for the one dimensional
nonlinear Schr\"odinger equation with double power nonlinearities:
\begin{align*} &i\partial_t u +\partial_x^2 u -|u|^{p-1}u +|u|^{q-1}u=0, \quad
(t,x)\in \mathbb{R}\times\mathbb{R} ,~1<p<q. \end{align*} When $q<5$, the
stability properties of standing waves $e^{i\omega t}\phi_{\omega}$ may change
for the frequency $\omega$. A sufficient condition for yielding instability for
small frequencies are obtained in previous results, but it has not been known
what the sharp condition is. In this paper we completely calculate the explicit
formula of $\lim_{\omega\to0}\partial_{\omega}\|\phi_{\omega}\|_{L^2}^2$, which
is independent of interest, and establish the sharp thresholds for stability
and instability of standing waves.

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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat