Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solitary pulses and periodic waves in the parametrically driven complex
  Ginzburg-Landau equation   A one-dimensional model of a dispersive medium with intrinsic loss,
compensated by a parametric drive, is proposed. It is a combination of the
well-known parametrically driven nonlinear Schr\"{o}dinger (NLS) and complex
cubic Ginzburg-Landau equations, and has various physical applications (in
particular, to optical systems). For the case when the zero background is
stable, we elaborate an analytical approximation for solitary-pulse (SP)
states. The analytical results are found to be in good agreement with numerical
findings. Unlike the driven NLS equation, in the present model SPs feature a
nontrivial phase structure. Combining the analytical and numerical methods, we
identify a stability region for the SP solutions in the model's parameter
space. Generally, the increase of the diffusion and nonlinear-loss parameters,
which differ the present model from its driven-NLS counterpart, lead to
shrinkage of the stability domain. At one border of the stability region, the
SP is destabilized by the Hopf bifurcation, which converts it into a localized
breather. Subsequent period doublings make internal vibrations of the breather
chaotic. In the case when the zero background is unstable, hence SPs are
irrelevant, we construct stationary periodic solutions, for which a very
accurate analytical approximation is developed too. Stability of the periodic
waves is tested by direct simulations.

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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ph
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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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: stat
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph