Spatial heteroclinic bifurcation of time periodic solutions to the Ginzburg-Landau equation

Abstract

Abstract.In this paper we study the spatial structure of the time periodic solutions to the Ginzburg-Landau equation in various configurations (supercritical and subcritical). We show that spatially periodic bursting solutions or spatially heteroclinic solutions can occur depending upon the values of the coefficients. As a consequence of this study, we obtain an exact solution to the nonlinear system of Kapitula and Maier-Paape [16]. We then show that near a spatially heteroclinic solution there is an extremely intricate complex of spatially unstable solutions and KAM surfaces.

Relative Periodic Solutions of the Complex Ginzburg--Landau Equation

“…Previous studies of time-periodic solutions of the CGLE (1.1) have centered on two types of solutions. The first are single-frequency solutions of the form A(x, t) = B(x)e iωt (see, for example, [15,20,30]); these are referred to as stationary solutions. The second type are generalized traveling waves, also called coherent structures, for which A(x, t) = ρ(x − vt)e iφ(x−vt) e iωt , where ρ and φ are real-valued functions and ω is some frequency (see, for example, [1,7,44,58]).…”