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

We study the asymptotic behavior of large data radial solutions to the focusing Schr"odinger equation $i{u}_t+\mathrm{\Delta }u=-\mathrm{\mid }u{\mathrm{\mid }}^{2}u$ in ${\mathbb{R}}^3$, assuming globally bounded $H^1({\mathbb{R}}^3)$ norm (i.e. no blowup in the energy space). We show that as $t\to \pm \mathrm{\infty }$, 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 ${\dot{H}}^1({\mathbb{R}}^3)$ 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.