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

We study the asymptotic behavior of large data radial solutions to the focusing Schrödinger 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 Schrödinger 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.

2010 Mathematics Subject Classification

35Q55

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Published 1 January 2004