MULTIPULSE JUMPING ORBITS AND HOMOCLINIC TREES IN A MODAL TRUNCATION OF THE DAMPED-FORCED NONLINEAR SCHRODINGER-EQUATION

In this paper we prove the existence of multi-pulse orbits homoclinic to a slow manifold in a two-mode truncation of the damped-forced nonlinear Schrodinger equation (first suggested by Bishop et al.). These orbits are jumping, i.e., the corresponding solutions keep switching in time between neighborhoods of the two characteristic ''breathers'' of the integrable limit. In the case of no damping, we find multi-pulse Smale horseshoes in the two-mode model, while in the dissipative case we establish the existence of structurally stable, multi-pulse, heteroclinic connections between two unstable equilibria. The orbits we construct are not amenable to Melnikov-type perturbation methods. In both the Hamiltonian and the dissipative case we find homoclinic trees, which describe the repeated bifurcations of multi-pulse solutions. To illustrate the theoretical predictions, we also present visualizations of these complicated structures.