Scattering for the Gross-Pitaevskii equation

Abstract

We investigate the asymptotic behavior at time infinity of solutions close to a nonzero constant equilibrium for the Gross-Pitaevskii (or Ginzburg-Landau-Schrödinger) equation. We prove that, in dimensions larger than 3, small perturbations can be approximated at time infinity by the linearized evolution, and the wave operators are homeomorphic around 0 in certain Sobolev spaces.

On small amplitude solutions to the generalized Boussinesq equations

“…4 The authors heard that recently, Gustafson, Nakanishi and Tsai showed a similar result for the phase ω 1 by another approach in [14].…”

On the linear wave regime of the Gross-Pitaevskii equation

“…Let us emphasize that related estimates have been used by the second author in [14] for the study of slightly compressible fluids and by S. Gustafson, K. Nakanishi and T.P. Tsai in [19] for the Gross-Pitaevskii equation. These estimates allow to improve the control on the term (Db, Dz) L ∞ appearing in the key inequality of Proposition 1.…”

Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems

“…On the other hand, Gustafson, Nakanishi and Tsai established in [GNT06], [GNT09] the scattering theory of small solutions to the Gross-Pitaevskii equation in space dimension three and four. In dimension two, where a scattering theory is excluded due to the existence of small energy traveling-waves, they constructed dispersive solutions with some prescribed data at infinity (see [GNT07]).…”