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We consider the dynamical degenerate four-wave mixing (FWM) model in a cubicnonlinear medium including both the time relaxation of the induced nonlinearityand the nonlocal coupling. The initial ten-dimensional FWM system can berewritten as a three-variable intrinsic system (namely the intensity pattern,the amplitude of the nonlinearity and the total net gain) which is very closeto the pumped Maxwell-Bloch system. In the case of a purely nonlocal responsethe initial system reduces to a real damped sine-Gordon (SG) equation. Weobtain a new solution of this equation in the form of a sech function with atime-dependent coefficient. By applying the reductive perturbation method tothis damped SG equation, we obtain exactly the cubic complex Ginzburg Landauequation (CGL3), but with a time dependence in the loss/gain coefficient. TheCGL3 describes the properties of the spatially localized interference patternformed by the FWM.

Ginzburg-Landau equation for dynamical four-wave mixing in gain nonlinear media with relaxation