Smale horseshoes and symbolic dynamics in perturbed nonlinear Schrodinger equations

In [12], we gave an intensive study on the level sets of the integrable cubic nonlinear Schrodinger (NLS) equation. Based upon that study, the existence of a symmetric pair of homoclinic orbits in certain perturbed NLS systems was established in [11]. [Stated in Theorem 1.3 below.] In this paper, the existence of Smale horseshoes and symbolic dynamics is established in the neighborhood of the symmetric pair of homoclinic orbits, under certain conditions (A1)-(A3), which are "except one point"-type conditions. [Stated in Theorem 8.1.] More specifically, a list of compact Canter sets is constructed through a study of the Conley-Moser conditions, each of which consists of points, and is invariant under the Poincare map induced by the flow. More importantly, the Poincare map restricted to each of the Canter sets is topologically conjugate to the shift automorphism on four symbols. This gives rise to deterministic chaos, which offers an interpretation of the numerical observation on the perturbed NLS system: chaotic center-wing jumping, of course under the "except one point"-type conditions (A1)-(A3). This study is a generalization of the finite-dimensional study [14] to infinite-dimensional perturbed NLS systems.