Scattering Theory for the Gross–pitaevskii Equation in Three Dimensions

Abstract

We study the 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 Schrödinger 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.

Absence of Shocks for One Dimensional Euler–Poisson System

“…Similar estimates are also used in [10,11]. The proof of this lemma is given in Lemma B.1 of the appendix.…”

“…For the L ∞ decay estimate, we employ the following refined linear decay estimate for the solution g (see Lemma A.1), 11) where w = e it ∂x g. It is important to note that −…”

From the Gross–Pitaevskii equation to the Euler Korteweg system, existence of global strong solutions with small irrotational initial data

“…Jones, S. J. Putterman and P. H. Roberts (see [34])).This would imply that there is no scattering even for small energy initial data when N = 2. Despite these issues, scattering has been obtained in a series of papers by Gustafson, Nakanishi and Tsai in [28,29,30]. For N ≥ 4 they proved scattering for small initial data in H N 2 −1 (R N ), the case N = 3 is much more intricate and requires the data to be small in weighted H 1 (R N ) spaces (to which, nevertheless, traveling waves belong).…”

“…It explains why the case N = 3 is difficult for (GP) since there are quadratic nonlinearities (see [29]). Before stating the ideas use in [29] in order to overcome this difficulty in the case of the GrossPitaevskii equation, let us briefly review the known results on the NLS for comparison in the case of a quadratic nonlinearity when N = 3. In this situation it is in particular necessary to take resonances into account.…”

“…The main feature is that our solutions have no vacuum for all time. Our construction uses in a crucial way some deep results on the scattering of the Gross-Pitaevskii equation due to Gustafson, Nakanishi and Tsai in [28,29,30]. An important part of the paper is devoted to explain the main technical issues of the scattering in [29] and we give a detailed proof in order to make it more accessible.…”