Global Attraction to Solitary Waves

The long time asymptotics for nonlinear wave equations have been the subject of intensive research, starting with the pioneering papers by Segal, Strauss, and Morawetz, where the nonlinear scattering and local attraction to zero were considered. Global attraction (for large initial data) to zero may not hold if there are quasistationary solitary wave solutions. We will call such solutions "solitary waves". Other appropriate names are "nonlinear eigenfunctions" and "quantum stationary states". Existence of such solitary waves was addressed by Strauss, and then the orbital stability of solitary waves in a general case has been considered by Grillakis, Shatah, and Strauss. The asymptotic stability of solitary waves has been obtained by Soffer and Weinstein, Buslaev and Perelman, and then by others. The existing results suggest that the set of orbitally stable solitary waves typically forms a local attractor, that is, attracts any finite energy solutions that were initially close to it. Moreover, a natural hypothesis is that the set of all solitary waves forms a global attractor of all finite energy solutions. This question is addressed in this paper. We develop required techniques and prove global attraction to solitary waves in several models. More precisely, for several U(1)-invariant Hamiltonian systems based on the Klein-Gordon equation, we prove that under certain generic assumptions the global attractor of all finite energy solutions is finite-dimensional and coincides with the set of all solitary waves. We prove the convergence to the global attractor in the metric which is just slightly weaker than the convergence in the local energy seminorms.