HOMOCLINIC JUMPING IN THE PERTURBED NONLINEAR SCHRODINGER-EQUATION

We study the damped, periodically forced, focusing NLS equation with e ven, periodic boundary conditions. We prove the existence of complicat ed solutions that repeatedly leave and come back to the vicinity of a quasi-periodic plane wave with two time scales. For pure forcing, we p rove the existence of a complicated, self-similar family of homoclinic bifurcations. For mode-independent damping, we construct ''jumping'' transients. For mode-dependent damping, we find generalized Silnikov-t ype solutions that connect a periodic plane wave to itself through rep eated jumps. We also study the breakdown of the unstable manifold of p lane waves through repeated jumping. Our results give a direct explana tion for the numerical observations of Bishop et al. (C) 1999 John Wil ey & Sons, Inc.