Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Scattering for the quadratic nonlinear Schr\"{o}dinger system in
  $\mathbb{R}^5$ without mass-resonance condition   We consider the quadratic nonlinear Schr\"{o}dinger system (NLS system)
\begin{align*}\begin{cases} i\partial_t u + \Delta u = v \overline{u}, \\
i\partial_t v+\kappa \Delta v = u^2, \end{cases} \text{ on } I \times
\mathbb{R}^5, \end{align*} where $\kappa>0$. The scattering below the standing
wave solutions for NLS system was obtained by the first author when $\kappa =
1/2$. The condition of $\kappa=1/2$ is called mass-resonance. In this paper, we
prove scattering below the standing wave solutions when $\kappa \neq 1/2$ under
the radially symmetric assumption. Our proof is based on the concentration
compactness and the rigidity by Kenig--Merle. Moreover, we discuss the
concentration compactness and the rigidity for non-radial solutions.

8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph