On the asymptotic behavior of large radial data for a focusing non-linear Schr\"odinger equation

TL;DR

This paper analyzes the long-term behavior of large radial solutions to the focusing nonlinear Schrödinger equation in three dimensions, showing they decompose into radiation, a localized smooth function, and an error, supporting soliton resolution conjecture.

Contribution

It provides a detailed asymptotic decomposition of large radial solutions in the energy space, including conditions for scattering and soliton-like behavior, advancing understanding of the soliton resolution conjecture.

Findings

Solutions split into radiation, localized smooth function, and error as t→±∞

Localized function either vanishes or has non-zero mass and energy

Results support the soliton resolution conjecture in the focusing NLS context

Abstract

We study the asymptotic behavior of large data radial solutions to the focusing Schr\"odinger equation $i u_t + \Delta u = -|u|^2 u$ in $\R^3$, assuming globally bounded $H^1(\R^3)$ norm (i.e. no blowup in the energy space). We show that as $t \to \pm \infty$, these solutions split into the sum of three terms: a radiation term that evolves according to the linear Schr\"odinger equation, a smooth function localized near the origin, and an error that goes to zero in the $\dot H^1(\R^3)$ norm. Furthermore, the smooth function near the origin is either zero (in which case one has scattering to a free solution), or has mass and energy bounded strictly away from zero, and obeys an asymptotic Pohozaev identity. These results are consistent with the conjecture of soliton resolution.