Existence and properties of travelling waves for the Gross-Pitaevskii equation

Abstract

This paper presents recent results concerning the existence and qualitative properties of travelling wave solutions to the Gross-Pitaevskii equation posed on the whole space R N . Unlike the defocusing nonlinear Schrödinger equations with null condition at infinity, the presence of non-zero conditions at infinity yields a rather rich and delicate dynamics. We focus on the case N = 2 and N = 3, and also briefly review some classical results on the one-dimensional case. The works we survey provide rigorous justifications to the impressive series of results which Jones, Putterman and Roberts [46,45] established by formal and numerical arguments.

Traveling vortex helices for Schrödinger map equations

“…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.…”

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

“…GP equation appears in many physical problems, such as: Bose-Einstein condensation, superconductivity, nonlinear optics, etc. We refer the reader to [4] and the references therein for more details about this subject. On a formal level, the Gross-Pitaevskii equation is Hamiltonian.…”

“…More recently, the program of Jones, Putterman and Roberts [14,15] has been provided with rigorous mathematical proofs (see the article [4] to review some of these progresses). For example, the non-existence of non-constant supersonic traveling waves (i.e., c > √ 2) of finite energy in dimension N ≥ 2 in [9], the non-existence of non-constant sonic traveling waves (i.e., c = √ 2) of finite energy in dimension two in [11] and the existence of non-constant axisymmetric traveling waves in [2,1,3,5,17,16].…”

Orbital stability of the black soliton for the Gross-Pitaevskii equation

“…The sound velocity c s = √ 2 appears naturally in the hydrodynamics formulation of (GP) (see [2]). If |c| ≥ √ 2, u is a constant of modulus one, whereas if − √ 2 < c < √ 2, then, up to a multiplication by a constant of modulus one and a translation, u is either identically equal to 1, or to…”

“…Orbital stability of travelling waves for any − √ 2 < c < √ 2, was established in [10] using a method of [8], and later in [2] relying on a variational principle, combined with several conservation laws. As a matter of fact, travelling wave solutions can be identified with critical points of the energy, keeping the momentum fixed.…”