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 + ∆u = F (u) in R d , assuming globally bounded H 1x (R d ) norm (i.e.no blowup 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 H 1x (R d ) to a compact attractor, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in H 1x (R d ).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.

Locations

• Dynamics of Partial Differential Equations - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• Dynamics of Partial Differential Equations - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF