Global attraction to solitary waves in nonlinear dispersive Hamiltonian systems Vladimir Buslaev (St. Petersburg University), Andrew Comech (Texas A&M University), Alexander Komech (University of Vienna), Boris Vainberg (UNC - Charlotte) May 20, 2008 – May 30, 2008 1 Overview of the Field The long time asymptotics for nonlinear wave equations have been the subject of intensive research. The modern history of nonlinear wave equations starts in nonlinear meson theories [Sch51a, Sch51b]. The well- posedness was addressed in early sixties by J¨ orgens [J¨ or61] and Segal [Seg63a]. Later Segal [Seg63a, Seg63b], Strauss [Str68], and Morawetz and Strauss [MS72], where the nonlinear scattering and local at- traction to zero were considered. Global attraction (for large initial data) to zero may not hold if there are stationary or quasistationary solitary wave solutions of the form ψ(x, t)= φ(x)e −iωt , with ω ∈ R, lim |x|→∞ φ(x)=0. (1) We will call such solutions solitary waves. Other appropriate names are nonlinear eigenfunctions and quan- tum stationary states (the solution (1) is not exactly stationary, but certain observable quantities, such as the charge and current densities, are time-independent indeed). According to “Derrick’s theorem” [Der64], time-independent soliton-like solutions to Hamiltonian sys- tems, under rather general assumptions, are unstable. On the other hand, quasistationary solutions may be stable. This is caused by the additional conservation laws, which may prevent a slightly perturbed solitary wave from tumbling in the direction of lower energy states. This stimulated the study of the existence and stability properties of solitary waves in the Hamiltonian systems with symmetries. Existence of solitary waves was addressed by Strauss in [Str77], and then the orbital stability of solitary waves in a general case was proved in [GSS87]. The asymptotic stability of solitary waves was considered by Soffer and Weinstein [SW90, SW92], Buslaev and Perelman [BP93, BP95], 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. 1