Title

Bifurcations Of Plane Wave (Cw) Solutions In The Complex Cubic-Quintic Ginzburg-Landau Equation

Keywords

Cubic-quintic Ginzburg-Landau equation (CGLE); Hysteresis; Isola

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 bifurcations 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. © 2006 IMACS.

Publication Date

3-30-2007

Publication Title

Mathematics and Computers in Simulation

Volume

74

Issue

4-5

Number of Pages

266-280

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.matcom.2006.10.009

Socpus ID

33847161551 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/33847161551

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