title: Ginzburg-Landau equation for dynamical four-wave mixing in gain nonlinear media with relaxation.
abstract: We consider the dynamical degenerate four-wave mixing (FWM) model in a cubic nonlinear medium including both the time relaxation of the induced nonlinearity and the nonlocal coupling. The initial ten-dimensional FWM system can be rewritten as a three-variable intrinsic system (namely, the intensity pattern, the amplitude of the nonlinearity, and the total net gain) which is very close to the pumped Maxwell-Bloch system. In the case of a purely nonlocal response the initial system reduces to a real damped sine-Gordon (SG) equation. We obtain a solution of this equation in the form of a sech function with a time-dependent coefficient. By applying the reductive perturbation method to this damped SG equation, we obtain exactly the cubic complex Ginzburg Landau equation (CGL3) but with a time dependence in the loss or gain coefficient. The CGL3 describes the properties of the spatially localized interference pattern formed by the FWM.
authors.affiliation: Institute of Physics, National Academy of Sciences, 46 Prospect Nauki, Kiev, Ukraine. bugaich@iop.kiev.ua
authors.forename: Svitlana
authors.initials: S
authors.lastname: Bugaychuk
authors.forename: Robert
authors.initials: R
authors.lastname: Conte
mesh_terms: D000465:Algorithms; D001703:Biophysics / Q000379:methods*; D008027:Light; D015233:Models, Statistical; D008962:Models, Theoretical; D017711:Nonlinear Dynamics; D017785:Photons; D015203:Reproducibility of Results; D013997:Time Factors