Geometric Characterization of Solitons

The aim of this work is to describe a geometric definition of localized solutions of NLS. In the linear case we have the RAGE Theorem, which relates localized solutions to the pure point spectrum of the Hamiltonian: Localized solutions of the linear Schroedinger equation are linear combinations of L eigenfunctions of the Hamiltonian. In particular, they are almost periodic functions of time. For the nonlinear case see [Sig, Sof, Tao]. The question arises as to what is the analog of the bound states of a linear equation, in the nonlinear case. Here, I will show that solitons appear naturally from geometric considerations. It lends support to the conjecture that all generic outgoing states of NLS are solitons and free waves. That is, I will show that if the solution of NLS is purely incoming, up to L(dt) corrections, then the solution converges to a soliton, in any compact region around the origin. The method of proof is based on and motivated by the hydrodynamic reformulation of the Schroedinger equation. The incoming wave condition is then written in terms of the notion of flux through surfaces around the origin. It is then shown how to rigorously use the hydrodynamic formulation, by restricting the analysis to topologically trivial domains of space-time where the solution is nonvanishing. The solution in such regions can then be uniquely written in the polar form, with continuous phase function. This, together with the a-priori H bound is then