Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long-Time Asymptotics of Solutions to the Cauchy Problem for the
  Defocusing Non-Linear Schr\"{o}dinger Equation with Finite Density Initial
  Data. I. Solitonless Sector   The methodology of the Riemann-Hilbert (RH) factorisation approach for
Lax-pair isospectral deformations is used to derive, in the solitonless sector,
the leading-order asymptotics as $t \to \pm \infty$ $(x/t \sim \mathcal{O}(1))$
of solutions to the Cauchy problem for the defocusing non-linear
Schr\"{o}dinger equation (D${}_{f}$NLSE), $\mi \partial_{t}u
+\partial_{x}^{2}u-2(| u |^{2}-1)u=0$, with (finite-density) initial data
$u(x,0)=_{x \to \pm \infty} \exp (\tfrac{\mi (1 \mp 1) \theta}{2})(1+ o(1))$,
$\theta \in [0,2 \pi)$. A limiting case of these asymptotics related to the RH
problem for the Painlev\'{e} II equation, or one of its special reductions, is
also identified.

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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
2
The categories of this paper are: math
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-bio
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The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: cmp-lg