The Cauchy Problem for Defocusing Nonlinear Schrödinger Equations with Non-Vanishing Initial Data at Infinity

Abstract

For rather general nonlinearities, we prove that defocusing nonlinear Schrödinger equations in n (n ≤ 4), with non-vanishing initial data at infinity u 0 , are globally well-posed in u 0 + H 1 . The same result holds in an exterior domain in n , n = 2 3.

On the linear wave regime of the Gross-Pitaevskii equation

“…In fact, it turns out that for any smooth nonnegative function α compactly supported in R N and satisfying α = 1 the function U := Ψ * α belongs to V and Ψ 0 − U belongs to H s+1 (R N ) (see e.g. [16] …”

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

“…After this work was completed, we learned that P. Gérard (see [5]) proved the global well-posedness for these equations, in the cases D = 2 or D = 3, in the energy space [4] for where the author uses similar framework to handle more general nonlinearities and problem in exterior domains. For the sake of completeness, we would like to point out also the articles [6], [7] where the authors solve a parabolic regularization of the Gross-Pitaevskii equations, the so-called complex Ginzburg-Landau equations, in local Sobolev spaces.…”

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

“…} and proved well-posedness results for (1.1) in X k (R N ), in the case of general nonlinearities, provided that k is large enough (see [Ga04], [Ga06], [Ga08]). Finally, P. Gérard ([G06]; see also the survey paper [G08]) studied the Cauchy problem for the Gross-Pitaevskii equation in the energy space E = {ψ ∈ H…”