Explicit solutions of the four - wave mixing model ∗

The dynamical degenerate four-wave mixing is studied analytically in detail. By removing the unessential freedom, we first characterize this system by a lowerdimensional 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 singlevalued solutions may exist, according to the two essential parameters of the system: the real relaxation time τ , the complex response constant γ. In addition to the stationary case, the only two integrable cases occur for a purely nonlocal response (R(γ) = 0), these are the complex unpumped Maxwell-Bloch system and another one, which is explicitly integrated with elliptic functions. For a generic response (R(γ) 6= 0), we display strong similarities with the cubic complex GinzburgLandau equation.