Long term dynamics of nonintegrable dispersive equations

In 2010 Nakanishi and the speaker introduced a "one-pass theorem" which precludes almost homoclinic orbits in certain infinite dimensional dynamical systems. More specifically, solutions to nonlinear wave equations starting from a small neighborhood of a ground state soliton and which return to another such neighborhood after some finite time. Such statements are non-perturbative and require truly global arguments based on monotone quantities. In the original setting invariant manifolds played an essential role, especially the one-dimensional unstable manifold associated with the steady-state solution. I will review these results, and discuss some of their ramifications over the past ten years. In particular, I will present a recent theorem by Jendrej and Lawrie on two soliton dynamics for energy critical equivariant wave maps into the 2-sphere (Inventiones 2018). The main tool in this classification result is a subtle one-pass type theorem based on the nonlinear inelastic collision of two dynamically rescaled harmonic maps.