Bifurcations and competing coherent structures in the cubic-quintic Ginzburg-Landau equation I: Plane wave (CW) solutions

    Authors

    S. Mancas;S. R. Choudhury

    Comments

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    Abbreviated Journal Title

    Chaos Solitons Fractals

    Keywords

    MODULATED AMPLITUDE WAVES; TIME-PERIODIC SOLUTIONS; SYSTEMS; FRONTS; PULSES; SINKS; Mathematics, Interdisciplinary Applications; Physics, Multidisciplinary; Physics, Mathematical

    Abstract

    Singularity theory is used to comprehensively investigate the bifurcations of the steady-states of the traveling wave ODEs of the cubic-quintic Ginzburg-Landau equation (CGLE). These correspond to plane waves of the PDE. In addition to the most general situation, we also derive the degeneracy conditions on the eight coefficients of the CGLE under which the equation for the steady states assumes each of the possible quartic (the quartic fold and an unnamed form), cubic (the pitchfork and the winged cusp), and quadratic (four possible cases) normal forms for singularities of codimension up to three. Since the actual governing equations are employed, all results are globally valid, and not just of local applicability. In each case, the recognition problem for the unfolded singularity is treated. The transition varieties, i.e. the hysteresis, isola, and double limit curves are presented for each normal form. For both the most general case, as well as for various combinations of coefficients relevant to the particular cases, the bifurcation curves are mapped out in the various regions of parameter space delimited by these varieties. The multiplicities and interactions of the plane wave solutions are then comprehensively deduced from the bifurcation plots in each regime, and include features such as regimes of hysteresis among co-existing states, domains featuring more than one interval of hysteresis, and isola behavior featuring dynamics unrelated to the primary solution branch in limited ranges of parameter space. (c) 2005 Elsevier Ltd. All rights reserved.

    Journal Title

    Chaos Solitons & Fractals

    Volume

    27

    Issue/Number

    5

    Publication Date

    1-1-2006

    Document Type

    Article

    Language

    English

    First Page

    1256

    Last Page

    1271

    WOS Identifier

    WOS:000232456100013

    ISSN

    0960-0779

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