Statistics of noise-driven coupled nonlinear oscillators: applications to systems with Kerr nonlinearity (Q556389)

The work starts with the equation for a nonlinear oscillator with cubic nonlinearity, which is driven by an external delta-correlated random force: \(du/dz=i| u| ^2u +\eta(z), \) where \(\eta(z)\) is the randomly varying drive. This complex equation is considered as a system of two coupled Langevin equations for the real and imaginary parts of \(u(z)\). The corresponding Fokker-Planck equation for the distribution function is then derived. The Green's function for the latter function is constructed in the form of an infinite series, each term being expressed in terms of the modified Bessel functions. Subsequently, a formal perturbation theory is developed on the basis of the Green's function (propagator) for various extra terms added to the regular part of the Langevin equations. The perturbations are chiefly motivated by applications to nonlinear optics, and account for such effects as second-order dispersion that gives rise to a weak linear coupling in a system of the stochastically driven nonlinear oscillators, polarization and birefringence effects (which produce a nonlinear coupling), nonlinear four-wave mixing, etc.