Bifurcations of plane wave (CW) solutions in the complex cubic–quintic Ginzburg–Landau equation

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.

Periodic and quasi-periodic solutions for the complex Ginzburg–Landau equation

“…where U(x) = cos nx + O( 2) is a periodic function independent of t. Although we cannot prove that solution (4) in theorem 1.1 is not of the rotation wave type, it is indeed different from solution (5) in view of the fact that…”

“…As for (1) and other related equations, the existence and stability of periodic solutions of travelling wave type have been extensively investigated in many papers, for example [4,5,7,[12][13][14]17]. In this paper, we will focus our attention on the periodic and quasi-periodic solutions which are not travelling waves.…”

“…More recently, Mancas and Choudhury [5] investigated some dynamical behaviour, such as stability and bifurcation diagrams, of the periodic solutions of the form…”

Response solution to complex Ginzburg–Landau equation with quasi-periodic forcing of Liouvillean frequency

“…The existence and stability of periodic or quasi-periodic solutions to (1.1) have been extensively investigated in many papers, for example [2,[5][6][7][8]. When x ∈ T d := (R/2πZ) d , there are some papers concerning the existence of KAM-type tori for (1.1).…”

Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$

“…The existence and stability of periodic solutions of traveling-wave type for (1) have been extensively investigated in many papers. See [3,5,6,7], for example, for more physical and mathematical backgrounds. However, there are fewer papers concerning the existence of KAM tori and quasi-periodic solutions for (1), although numerical evidence has shown the existence of 2-and 3-tori.…”