On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation

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We study the asymptotic behavior of large data radial solutions to the focusing Schrödinger equation i uₜ + Δu = -|u|² u in ³, assuming globally bounded H¹(³) norm (i.e. no blowup in the energy space). We show that as t → ± ∞, these solutions split into the sum of three terms: a radiation term that evolves according to the linear Schrödinger equation, a smooth function localized near the origin, and an error that goes to zero in the H¹(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.