Homoclinic Tubes in Nonlinear Schrödinger Equation under Hamiltonian Perturbations

The concept of a homoclinic tube was introduced by Silnikov 14) in a study on the structure of the neighborhood of a homoclinic tube asymptotic to an invariant torus σ under a diffeomorphism F in a finite dimensional phase space. The asymptotic torus is of the saddle type. The homoclinic tube consists of a doubly infinite sequence of tori {σj , j = 0,±1,±2, · · ·} in the transversal intersection of the stable and unstable manifolds of σ, such that σj+1 = F ◦ σj for any j. This is a generalization of the concept of a transversal homoclinic orbit in which the points are replaced by tori. Silnikov obtained a theorem on the symbolic dynamics structures in the neighborhood of the homoclinic tube that is similar to Smale’s theorem for a transversal homoclinic orbit. 13), 12) We are interested in homoclinic tubes for several reasons: 1. Especially in high dimensions, it is often the case that the dynamics inside each invariant tube in the neighborhood of a homoclinic tube is also chaotic. We refer to such chaotic dynamics as “chaos in the small” and the symbolic dynamics of the invariant tubes as “chaos in the large”. Such cascade structures are more important than the structures in a neighborhood of a homoclinic orbit when high or infinite dimensional dynamical systems are studied. 2. Symbolic dynamics structures in the neighborhoods of homoclinic tubes are more observable than in the neighborhoods of homoclinic orbits in numerical and physical experiments. 3. When studying a high or infinite dimensional Hamiltonian system (for example, the cubic nonlinear Schrödinger equation under Hamiltonian perturbations), each invariant tube contains both KAM tori and stochastic layers (chaos in the small). Thus, not only is it the case that the dynamics inside each stochastic layer is chaotic, all these stochastic layers also move chaotically under Poincaré maps. In this paper, we are interested in studying Hamiltonian partial differential equations. Denote by W (c) a normally hyperbolic center manifold, by W (cu) and W (cs)