Travelling Waves for the Gross-Pitaevskii Equation II

Abstract

The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results (see [4,3,8]) where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.and such that u p is solution to the minimization problemRemark 2. In [4], existence of solutions is obtained for small prescribed speeds c. Instead of using minimization under constraint, the idea there is to introduce, for given c, the Lagrangianwhose critical points are solutions to (TWc), and then to apply a mountain-pass argument.Although it is likely that the solutions obtained in [4] correspond to the solutions obtained in Theorem 1 for large p, we have no proof of this fact.Remark 3. Theorem 1 shows in particular that there exist travelling wave solutions of arbitrary small energy. This suggests that scattering in the energy space is not likely to hold.

Traveling vortex helices for Schrödinger map equations

“…By a completely different method, F. Lin and J. Wei [34] obtained the results of existence of solutions, similar as those of [8], [7], for the Landau-Lifshitz equation ( The main results of the paper [34] are the following Theorem 1.1. [34] Let N ≥ 2 and sufficiently small there is an axially symmetric solution m = m(|s |, s N ) ∈ C ∞ (R N , S 2 ) of (1.6) such that…”

“…In two dimensional plane, F. Bethuel and J. Saut constructed a traveling wave with two vortices of degree ±1 in [8]. In higher dimension, by minimizing the energy, F. Bethuel, G. Orlandi and D. Smets constructed solutions with a vortex ring [7], see also [5].…”

“…See also [10] for another proof by Mountain Pass Lemma. The reader can refer to the review paper [6] by F. Bethuel, P. Gravejat and J. Saut and the references therein.…”

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

“…Setting the problem. This short article is concerned with the socalled Gross-Pitaevskii equations (1) iu t + ∆u + u(1 − |u| 2 ) = 0, supplemented with non-standard boundary conditions that read |u(t, x)| → 1 as ||x|| → +∞. We also supplement this equation with initial condition u 0 that will be specified in the sequel.…”

“…Here the unknown u maps R t × R D x into C. These equations with this non-standard boundary conditions occur in several physical contexts, as the Bose-Einstein condensation for suprafluids (see [3] and the references therein). The mathematical study of solitary waves for these equations was initiated in the pioneering work [1]. Here we are interested in two questions related to the Cauchy problem for (1).…”