Decay for travelling waves in the Gross–Pitaevskii equation

Abstract

We study the limit at infinity of the travelling waves of finite energy in the Gross-Pitaevskii equation in dimension larger than two: their uniform convergence to a constant of modulus one and their asymptotic decay. RésuméNous étudions la limite à l'infini des ondes progressives d'énergie finie pour les équations de Gross-Pitaevskii en dimension supérieure ou égale à deux : leur convergence uniforme vers une constante de module un et leur comportement asymptotique.

Asymptotic Axisymmetry of the Subsonic Traveling Waves to the Gross–pitaevskii Equation

“…To solve these problems the study of asymptotic behaviours of traveling waves plays an important role. As showed in [10] and [11] the asymptotic behaviour of sonic traveling waves is much involved than in the subsonic case. In the following we only consider subsonic traveling waves of finite energy.…”

“…By the proof of Corollary 1 in [12], the stretched dipole coefficient α given in (13) is non-negative.…”

“…In a previous paper [12] Gravejat has conjectured that function v ∞ will have the form (12) (see Conjecture 1 of [12]) and proved it in dimension two and in the axisymmetric case.…”

“…Recently Gravejat [12] proved rigorously formulae (7), (8), (9) and (10) in the axisymmetric case or in dimension two. Later on he [13] gave their first order asymptotics at infinity in the general case, which is the start point of our result.…”

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

“…The asymptotic behavior of traveling waves as |x| −→ ∞ has been accurately described by the physicists in the seventies by using formal computations. More recently, this behavior has been rigorously established in a series of works by P. Gravejat (see [Gr04a], [Gr04b] , [Gr05] [Gr06]) in the case of the Gross-Pitaevskii equation. Probably his proofs can be adapted to more general nonlinearities.…”