Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weakly nonlinear stochastic CGL equations   We consider the linear Schr\"odinger equation under periodic boundary
condition, driven by a random force and damped by a quasilinear damping: $$
\frac{d}{dt}u+i\big(-\Delta+V(x)\big) u=\nu \Big(\Delta u-\gr |u|^{2p}u-i\gi
|u|^{2q}u \Big) +\sqrt\nu\, \eta(t,x).\qquad (*) $$ The force $\eta$ is white
in time and smooth in $x$. We are concerned with the limiting, as $\nu\to0$,
behaviour of its solutions on long time-intervals $0\le t\le\nu^{-1}T$, and
with behaviour of these solutions under the double limit $t\to\infty$ and
$\nu\to0$. We show that these two limiting behaviours may be described in terms
of solutions for the {\it system of effective equations for $(*)$} which is a
well posed semilinear stochastic heat equation with a non-local nonlinearity
and a smooth additive noise, written in Fourier coefficients. The effective
equations do not depend on the Hamiltonian part of the perturbation
$-i\gi|u|^{2q}u$ (but depend on the dissipative part $-\gr|u|^{2p}u$). If $p$
is an integer, they may be written explicitly.

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The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th