Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Conservation laws, exact travelling waves and modulation instability for
  an extended nonlinear Schr\"odinger equation   We study various properties of solutions of an extended nonlinear
Schr\"{o}dinger (ENLS) equation, which arises in the context of geometric
evolution problems -- including vortex filament dynamics -- and governs
propagation of short pulses in optical fibers and nonlinear metamaterials. For
the periodic initial-boundary value problem, we derive conservation laws
satisfied by local in time, weak $H^2$ (distributional) solutions, and
establish global existence of such weak solutions. The derivation is obtained
by a regularization scheme under a balance condition on the coefficients of the
linear and nonlinear terms -- namely, the Hirota limit of the considered ENLS
model. Next, we investigate conditions for the existence of traveling wave
solutions, focusing on the case of bright and dark solitons. The balance
condition on the coefficients is found to be essential for the existence of
exact analytical soliton solutions; furthermore, we obtain conditions which
define parameter regimes for the existence of traveling solitons for various
linear dispersion strengths. Finally, we study the modulational instability of
plane waves of the ENLS equation, and identify important differences between
the ENLS case and the corresponding NLS counterpart. The analytical results are
corroborated by numerical simulations, which reveal notable differences between
the bright and the dark soliton propagation dynamics, and are in excellent
agreement with the analytical predictions of the modulation instability
analysis.

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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
1
The categories of this paper are: cs
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
2
The categories of this paper are: math
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th