Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The nonlinear Schr\"odinger equation with combined power-type
  nonlinearities   We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$
i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where
$u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$,
$\lambda_1$ and $\lambda_2$ are nonzero real constants, and $0<p_1<p_2\le \frac
4{n-2}$. We address questions related to local and global well-posedness,
finite time blowup, and asymptotic behaviour. Scattering is considered both in
the energy space $H^1(\R^n)$ and in the pseudoconformal space $\Sigma:=\{f\in
H^1(\R^n); xf\in L^2(\R^n)\}$. Of particular interest is the case when both
nonlinearities are defocusing and correspond to the $L_x^2$-critical,
respectively $\dot H^1_x$-critical NLS, that is, $\lambda_1, \lambda_2>0$ and
$p_1=\frac{4}{n}$, $p_2=\frac{4}{n-2}$. The results at the endpoint $p_1 =
\frac{4}{n}$ are conditional on a conjectured global existence and spacetime
estimate for the $L^2_x$-critical nonlinear Schr\"odinger equation.
  As an off-shoot of our analysis, we also obtain a new, simpler proof of
scattering in $H^1_x$ for solutions to the nonlinear Schr\"odinger equation $$
i u_t +\Delta u = |u|^{p} u, $$ with $\frac{4}{n}<p<\frac{4}{n-2}$, which was
first obtained by J. Ginibre and G. Velo, \cite{gv:scatter}.

15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
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The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
24
The categories of this paper are: dg-ga