Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $H^1$-scattering for systems of $N$-defocusing weakly coupled NLS
  equations in low space dimensions   We prove that the scattering operators and wave operators are well-defined in
the energy space for the system of defocusing Schr\"odinger equations
  $$ \begin{cases} i\partial_t u_\mu + \Delta u_\mu - \sum_{\mu,\nu=1 }^N
\beta_{\mu\nu}|u_\nu|^{p+1}|u_\mu|^{p-1}u_\mu=0, \quad\quad
\mu=1,\dots,N,\\(u_\mu(0,\cdot))_{\mu=1}^N= (u_{\mu,0})_{\mu=1}^N \in
H^1(\mathbb R^d)^N. \end{cases} $$ with $N\geq 2$, $\beta_{\mu\nu} \geq 0$,
$\beta_{\mu\mu}\neq 0$ for $p>2 $ if $d=1$, $p>1 $ if $d=2$ and $ 1 \leq p < 2$
if $d=3$.

28
The categories of this paper are: alg-geom
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol