STABILITY OF TRAVELING OSCILLATING FRONTS IN COMPLEX GINZBURG LANDAU EQUATIONS.

Traveling oscillating fronts (TOFs) are specific one-dimensional waves of the form U (x, t) = e^-iωV (x ct) with a complex-valued profile V that decays at -∞ but approaches a nonzero limit at +∞. In particular, we consider TOFs which connect two stable homogeneous rest states and which occur generically in complex Ginzburg Landau equations of the type U_t = \alpha Uxx + G(| U| ²)U. In this paper we prove a theorem on the asymptotic stability of TOFs, where we allow the initial perturbation to be the sum of an exponentially localized part and a front-like part which approaches a small but nonzero limit at +∞. The underlying assumptions guarantee that the operator, obtained from linearizing about the TOF in a co-moving and co-rotating frame, has essential spectrum touching the imaginary axis in a quadratic fashion and that further isolated eigenvalues are bounded away from the imaginary axis. The basic idea of the proof is to consider the problem in an extended phase space which couples the wave dynamics on the real line to the ODE dynamics at +∞. Using slowly decaying exponential weights, the framework allows one to derive appropriate resolvent estimates, semigroup techniques, and Gronwall estimates. [ABSTRACT FROM AUTHOR]