TY - JOUR

T1 - Ramped-induced states in the parametrically driven Ginzburg-Landau model

AU - Malomed, Boris A.

AU - Rotstein, Horacio G.

PY - 2001/5/21

Y1 - 2001/5/21

N2 - 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 x = ±∞. 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. However, 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).

AB - 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 x = ±∞. 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. However, 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).

UR - http://www.scopus.com/inward/record.url?scp=0035926849&partnerID=8YFLogxK

U2 - 10.1016/S0375-9601(01)00215-8

DO - 10.1016/S0375-9601(01)00215-8

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AN - SCOPUS:0035926849

SN - 0375-9601

VL - 283

SP - 327

EP - 334

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

IS - 5-6

ER -