Soliton resolution for nonlinear Schrödinger type equations in the radial case

Abstract

We consider the Schrödinger equation with a general interaction term, which is localized in space, for radially symmetric initial data in $n$ dimensions, $n\ge 5$. The interaction term may be space-time dependent and nonlinear. Assuming that the solution is bounded in $H^1({\mathbb{R}}^n)$ uniformly in time, we prove soliton resolution conjecture (Asymptotic completeness) by demonstrating that the solution resolves into a smooth and localized part and a free radiation in ${\mathcal{L}}_x^2({\mathbb{R}}^n)$ norm. Examples of such equations include inter-critical and super-critical nonlinear Schrödinger equations, saturated nonlinearities, time-dependent potentials, and combinations of these terms.

50 pages. Comments welcome!