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# Dynamics of Partial Differential Equations

## Volume 1 (2004)

### Number 1

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

Pages: 1 – 47

DOI: http://dx.doi.org/10.4310/DPDE.2004.v1.n1.a1

#### Author

#### Abstract

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