Scattering for the Gross-Pitaevskii equation. (English) Zbl 1119.35084

Gustafson, Stephen; Nakanishi, Kenji; Tsai, Tai-Peng

Math. Res. Lett. 13, No. 2-3, 273-285 (2006).

The authors deal with the asymptotic behaviour at time infinity of solutions close to a nonzero constant equilibrium for \[ i\partial_t\psi= -\Delta\psi+ (|\psi|^2- 1)\psi,\tag{1} \]
\[ \psi(0,\cdot)= \psi_0,\tag{2} \] where \(\psi:\mathbb{R}\times \mathbb{R}^d\to \mathbb{C}\). As a first step the authors here consider asymptotic stability of the vacuum solution \[ \psi(t,x)\equiv e^{i\theta_0}= \text{constant }\in S^1.\tag{3} \] The main goal of the authors is to investigate the asymptotic stability of the vacuum in terms of the scattering theory around the constant solution (3) of (1)–(2). The authors prove that, in dimension larger than 3, small perturbations can be approximated at time infinity by the linearized evolution, and the wave operators are homeomorphic around zero in certain Sobolev spaces.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35P25	Scattering theory for PDEs
35B35	Stability in context of PDEs

Keywords:

vacuum solution; asymptotic stability