Singular tori as attractors of four-wave-interaction systems

We study the spatiotemporal dynamics of the Hamiltonian four-wave interaction in its counterpropagating configuration. The numerical simulations reveal that, under rather general conditions, the four-wave system exhibits a relaxation process toward a stationary state. Considering the Hamiltonian system associated to the stationary state, we provide a global geometrical view of all the stationary solutions of the system. The analysis reveals that the stationary state converges exponentially toward a pinched torus of the Hamiltonian system in the limit of an infinite nonlinear medium. The singular torus thus plays the role of an attractor for the spatiotemporal wave system. The topological properties of the singular torus confer a robust character to the stationary solution when the boundary conditions or the length of the nonlinear medium are modified. Furthermore, an adiabatic approach of the boundary conditions reveals that singular tori also play a major role for the description of the spatiotemporal dynamics of the wave system.