Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exponential stability for the nonlinear Schr\"odinger equation with
  locally distributed damping   In this paper, we study the defocusing nonlinear Schr\"{o}dinger equation
with a locally distributed damping on a smooth bounded domain as well as on the
whole space and on an exterior domain. We first construct approximate solutions
using the theory of monotone operators. We show that approximate solutions
decay exponentially fast in the $L^2$-sense by using the multiplier technique
and a unique continuation property. Then, we prove the global existence as well
as the $L^2$-decay of solutions for the original model by passing to the limit
and using a weak lower semicontinuity argument, respectively. The distinctive
feature of the paper is the monotonicity approach, which makes the analysis
independent from the commonly used Strichartz estimates and allows us to work
without artificial smoothing terms inserted into the main equation. We, in
addition, implement a precise and efficient algorithm for studying the
exponential decay established in the first part of the paper numerically. Our
simulations illustrate the efficacy of the proposed control design.

18
The categories of this paper are: econ
19
The categories of this paper are: eess
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg