Travelling, Non-Periodic Patterns in Nonlinear Stability Problems

Abstract

In this short note we present results on the existence of several classes of travelling, non- periodic solutions of the complex Ginzburg-Landau equation. First we give a very short introduction to the G-L equation and show its importance in nonlinear stability theory. We then study the G-L equation with complex coefficients and establish the existence of a 2-parameter family of quasi-periodic solutions and two different types of one-parameter families of heteroclinic orbits; all members of these families travel with a well-defined wave-speed. The heteroclinic solutions correspond to (travelling) soliton-like ‘localized structures’ which connect different (stable) periodic patterns. Mathematically, these families of travelling solutions (quasi-periodic and heteroclinic) are continuations into the complex case of the stationary solutions of the real G-L equation.