Finite-dimensional attractor for the laser equations

The laser equations of Risken and Nummedal (1968) govern the dynamics of a ring laser cavity. They form a system of hyperbolic, semilinear, damped and driven partial differential equations with periodic boundary conditions. The Lorenz system of ordinary differential equations is an invariant subsystem of the laser equations corresponding to solutions without spatial dependence. The authors prove that the laser system admits global weak solutions for arbitrary L² initial data. Despite the absence of parabolic diffusion they prove that the laser system enjoys a remarkable property of hyperbolic smoothing for t to infinity : the universal attractor for the L² evolution consists of C^infinity functions. They show, moreover, that the universal attractor is finite dimensional and estimate its dimension.