Soliton resolution for nonlinear Schr\"odinger type equations in the radial case

Abstract

We consider the Schr\"odinger equation with a general interaction term, which is localized in space, for radially symmetric initial data in $n$ dimensions, $n\geq5$. 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 $\mathcal{L}^2_x(\mathbb{R}^n)$ norm. Examples of such equations include inter-critical and super-critical nonlinear Schr\"odinger equations, saturated nonlinearities, time-dependent potentials, and combinations of these terms.