Breaking the hidden symmetry in the Ginzburg-Landau equation

Abstract

In this paper we study localised, traveling, solutions to a Ginzburg-Landau equation to which we have added a small, O ( " ), 0 < "? 1, quintic term. We consider this term as a model for the higher order nonlinearities which appear in the derivation of the Ginzburg-Landau equation. By a combination of a geometrical approach and an explicit perturbation analysis we are able to relate the family of Bekki & Nozaki solutions of the cubic equation [1] to a curve of co-dimension 2 homoclinic bifurcations in parameter space. Thus, we are able to interpret the hidden symmetry { which has been conjectured to explain the existence of the Bekki & Nozaki solutions { from the point of view of bifurcation theory. We show that the quintic term breaks this hidden symmetry and that the one-parameter family of Bekki & Nozaki solutions is embedded in a two-parameter family of homoclinic solutions which exist at a co-dimension 1 homoclinic bifurcation. Furthermore, we show, mainly by geometrical arguments, that the addition of the small quintic term can create large families of traveling localised structures that cannot exist in the cubic case. These solutions exist in open subsets of the parameter space and correspond to structurally stable multi- or N -circuit heteroclinic orbits in an ODE reduction and have a monotonically decreasing/increasing amplitude, except for a number, N , of (relatively fast) `jumps'.