The Hamiltonian perturbation approach of two interacting nonlinear waves or solitary pulses in an optical coupler

The traveling-wave solutions dynamics of two linearly coupled nonlinear Schrِdinger equations, describing the wave propagation in a dual-core optical fiber, is studied in the framework of the Hamiltonian perturbation theory. The surface of section of the phase space for each wave is obtained as a contour plot of an approximate invariant of the system and the phase space distortions due to the interaction are analyzed. Depending on the magnitude of the coupling strength, the parametric resonance involved leads to weak, strong and chaotic interactions. The coupling strength threshold above which the interaction between two nonlinear waves of given amplitude becomes chaotic is also estimated. On the other hand, the interaction of a solitary pulse with a nonlinear periodic wave is studied using the Melnikov method for homoclinic orbits. This interaction causes stochastization of the homoclinic orbit, which corresponds to the solitary pulse and the separatrix splitting as well as the width of the stochastic layer near this orbit is given in terms of the Melnikov function. The latter is applied in the case of two interacting solitary pulses as well.