Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Vortices and ring dark solitons in nonlinear amplifying waveguides   We consider the generation and propagation of (2+1)-dimensional beams in a
nonlinear waveguide with the linear gain. Simple self-similar evolution of the
beams is achieved at the asymptotic stage, if the input beams represent the
fundamental mode. On the contrary, if they carry vorticity or amplitude nodes
(or phase slips), vortex tori and ring dark solitons (RDSs) are generated,
featuring another type of the self-similar evolution, with an exponentially
shrinking vortex core or notch of the RDS. Numerical and analytical
considerations reveal that these self-similar structures are robust entities in
amplifying waveguides, being \emph{stable} against azimuthal perturbations.

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The categories of this paper are: cond-mat
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con