## Scattering theory for the Gross-Pitaevskii equation in three dimensions

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Gustafson, S.

Nakanishi, K.

Tsai, T. -P.

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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

Comment: 44 pages. Preprint of an article submitted for consideration in Communications in Contemporary Mathematics

##### Keywords

Mathematics - Analysis of PDEs, 35Q55