Existence and stability of singular heteroclinic orbits for the Ginzburg - Landau equation

The existence and stability of travelling waves of the Ginzburg - Landau equation is considered. The waves in question exist on a slow manifold in phase space and connect two stable plane waves with different wave numbers. The existence of these waves is proven via the use of the methods of geometric singular perturbation theory. Topological methods are used to prove the linear stability of the waves. The waves are shown to be nonlinearly stable in polynomially weighted spaces. Even though the Ginzburg - Landau equation possesses both a rotational invariance and a spatial translation invariance, small perturbations of the wave decay to the wave itself, and not to a translate of the wave. 35K57, 35Q35, 58F19