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

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.