Ramped-induced states in the parametrically driven Ginzburg-Landau model

We introduce a parametrically driven Ginzburg-Landau (GL) model, which admi ts a gradient representation, and is subcritical in the absence of the para metric drive (PD). In the case when PD acts uniformly in space, this model has a stable kink solution. A nontrivial situation takes places when PD is itself subject to a kink-like spatial modulation, so that it selects real a nd imaginary constant solutions at x = +/- infinity. In this situation, we find stationary solutions numerically, and also analytically for a particul ar case. They seem to be of two different types, viz. a pair of kinks in th e real and imaginary components, or the same with an extra kink inserted in to each component, but we show that both belong to a single continuous fami ly of solutions. The family is parametrized by the coordinate of a point at which the extra kinks are inserted. However, solutions with more than one kink inserted into each component do not exist. Simulations show that the f ormer solution is always stable, and the latter one is, in a certain sense, neutrally stable, as there is a special type of small perturbations that r emain virtually constant in time, rather than decaying or growing (they eve ntually decay, but extremely slowly). (C) 2001 Elsevier Science B.V. All ri ghts reserved.