A (concentration-) compact attractor for high-dimensional non-linear Schrödinger equations

Abstract

We study the asymptotic behavior of large data solutions to Schrödinger equations iut + Δu = F(u) in Rd, assuming globally bounded Hx1(Rd) norm (i.e. no blow-up in the energy space), in high dimensions d ≥ 5 and with nonlinearity which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as t →+ ∞, these solutions split into a radiation term that evolves according to the linear Schrödinger equation, and a remainder which converges in Hx1(Rd) to a compact attractor, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in Hx1(Rd). This is despite the total lack of any dissipation in the equation. This statement can be viewed as weak form of the "soliton resolution conjecture". We also obtain a more complicated analogue of this result for the non-spherically-symmetric case. As a corollary we obtain the "petite conjecture" of Soffer in the high dimensional non-critical case. © 2007 International Press.