Global solutions and attractors of a Maxwell-Bloch Raman laser system in two transverse dimensions

We study a Maxwell-Bloch system describing the dynamics of single-longitudinal Raman lasers in the two transverse space dimensions. Raman lasing is generated by a coherent external pump laser, as a three-wave interaction involving two optical and one material wave. On the other hand, a two-level laser involves incoherent external pumping through, for example, an electrical discharge or flashlamp. Raman lasers have the advantage of being tunable and display a novel explicit nonlinear detuning between the pump and laser emission frequencies. Consequently these lasers exhibit much richer nonlinear dynamics. We establish the global existence of classical H² solutions, and show that for periodic domains the dynamics is governed by a global C^∞ smooth attractor of finite dimensions. We explain the structures of nonlinear interactions and couplings that lead to the time asymptotic smoothing. We also construct mild solutions with the dispersive Strichartz inequality for rough but spatially decaying data (in H¹' x (L²∩L^p)², p ∈ (4, ∞)) on the whole plane, which physically corresponds to the absorbing boundary condition.