Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2001-05-09
Nonlinear Sciences
Pattern Formation and Solitons
A latex text file and 8 ps files with figures. Physics Letters A, in press
Scientific paper
10.1016/S0375-9601(01)00215-8
We introduce a parametrically driven Ginzburg-Landau (GL) model, which admits a gradient representation, and is subcritical in the absence of the parametric 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 and imaginary constant solutions at +infinity and -infinity. In this situation, we find stationary solutions numerically, and also analytically for a particular case. They seem to be of two different types, viz., a pair of kinks in the real and imaginary components, or the same with an extra kink inserted into each component, but we show that both belong to a single continuous family of solutions. The family is parametrized by the coordinate of a point at which the extra kinks are inserted. Solutions with more than one kink inserted into each component do not exist. Simulations show that the former 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 remain virtually constant in time, rather than decaying or growing (they eventually decay, but extremely slowly).
Malomed Boris
Rotstein Horacio
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