Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Scattering for a 3D coupled nonlinear Schr\"odinger system   We consider the three-dimensional cubic nonlinear Schr\"odinger system
  \begin{equation*} \begin{cases} i\partial_tu+\Delta u+(|u|^2+\beta
|v|^2)u=0,\\ i\partial_tv+\Delta v+(|v|^2+\beta |u|^2)v=0. \end{cases}
\end{equation*} Let $(P,Q)$ be any ground state solution of the above
Schr\"odinger system. We show that for any initial data $(u_0,v_0)$ in
$H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$ satisfying
$M(u_0,v_0)A(u_0,v_0)<M(P,Q)A(P,Q)$ and $M(u_0,v_0)E(u_0,v_0)<M(P,Q)E(P,Q)$,
where $M(u,v)$ and $E(u,v)$ are the mass and energy (invariant quantities)
associated to the system, the corresponding solution is global in
$H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$ and scatters. Our approach is in
the same spirit of Duyckaerts-Holmer-Roudenko, where the authors considered the
3D cubic nonlinear Schr\"odinger equation.

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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th