Explicit solutions of the four-wave mixing model

Abstract

Dynamical degenerate four-wave mixing is studied analytically in detail. By removing the unessential freedom, we first characterize this system by a lower-dimensional closed subsystem of a deformed Maxwell–Bloch type, involving only three physical variables: the intensity pattern, the dynamical grating amplitude, the relative net gain. We then classify by the Painlevé test all the cases when single-valued solutions may exist, according to the two essential parameters of the system: the real relaxation time τ, and the complex response constant γ. In addition to the stationary case, the only two integrable cases occur for a purely nonlocal response (Re(γ) = 0), these are the complex unpumped Maxwell–Bloch system and another one, which is explicitly integrated with elliptic functions. For a generic response (Re(γ) ≠ 0), we display strong similarities with the cubic complex Ginzburg–Landau equation.