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

TL;DR

This paper derives a complex Ginzburg-Landau equation with time-dependent coefficients to model dynamical four-wave mixing in nonlinear media with relaxation, revealing new solutions and insights into interference pattern formation.

Contribution

It introduces a novel derivation of a time-dependent complex Ginzburg-Landau equation from a FWM model with relaxation and nonlocal effects, connecting it to the damped sine-Gordon equation.

Findings

Derived a new sech-shaped solution for the damped sine-Gordon equation.

Established the exact form of the complex Ginzburg-Landau equation with time-dependent coefficients.

Analyzed the properties of interference patterns in 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 new 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/gain coefficient. The CGL3 describes the properties of the spatially localized interference pattern formed by the FWM.