Scattering theory for the Gross-Pitaevskii equation in three dimensions

Gustafson, S.
Nakanishi, K.
Tsai, T. -P.
Description
We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. Translated to the defocusing nonlinear Schr\"odinger equation, this implies asymptotic stability of all plane wave solutions for such disturbances. We also prove that every linearized solution with finite energy has a nonlinear solution which is asymptotic to it. The key ingredients are: (1) some quadratic transforms of the solutions, which effectively linearize the nonlinear energy space, (2) a bilinear Fourier multiplier estimate, which allows irregular denominators due to a degenerate non-resonance property of the quadratic interactions, and (3) geometric investigation of the degeneracy in the Fourier space to minimize its influence.
Comment: 44 pages. Preprint of an article submitted for consideration in Communications in Contemporary Mathematics
Keywords
Mathematics - Analysis of PDEs, 35Q55