Modulated modulations approach to the loss of stability of periodic solutions for the degenerate Ginzburg-Landau equation

Abstract

The degenerate GinzburgLandau equation gives a description of patterns which arise in the case of weakly unstable PDEs with a unbounded spatial direction when the Landau constant characterizing the inuence of the nonlinearity is small This equation possesses a family of periodic solutions moreover there exists a band of stable periodic solutions among them We give the full description of the possible behavior of the system just outside this stable band This is done through derivation of the socalled modulated modulation equations introduced in Harten

We also study some solutions of these equations among which stationary periodic and heteroclinic solutions moving solitons cnoidal waves and frontlike solutions are found