No abstract

In the previous paper [4], we constructed a smooth branch of travelling waves for the 2 dimensional Gross-Pitaevskii equation. Here, we continue the study of this branch. We show some coercivity results, and we deduce from them the kernel of the linearized operator, a spectral stability result, as well as a uniqueness result in the energy space., the decay in space being critical, to use a Hardy type inequality. We emphasize that here ψ is compactly supported away from 0, it is in particular needed since ψ 1,D is not necessarily finite otherwise.Proposition 1.3 is a consequence of Proposition 3.2 and Lemma 3.3, proven in subsection 3.2. The proofs in subsection 3.2 are mostly slight variations or improvements of proofs given in [5]. Coercivity for the travelling wave for test functionsThe main part of this paper consists of coercivity results for the family of travelling waves constructed in Theorem 1.1. We will show it on Q c defined in Theorem 1.1, and with (1.1), it extends to all speed values c of small norm. We recall the linearized operator around Q c :We start by a coercivity result for test functionsThere are some difficulties in the definition of the quadratic form and the coercivity norm for functions in the energy space, related to the justification of some integration by parts. They will be discussed later on.From Proposition 1.2, we know that Q c has only two zeros. We will write the quadratic form)) as the quadratic form for one vortex (computed in Proposition 1.3), up to some small error. We will therefore infer a coercivity result under four orthogonality conditions near the zeros of Q c (two for each zero). Then, we shall show that far from the zeros of Q c , the coercivity holds, without any additional orthogonality conditions.Proposition 1.4 There exists c 0 > 0 such that, for 0 < c c 0 , if one defines Ṽ±1 to be the vortices centered aroundProposition 1.5 There exists c 0 > 0 such that, for 0 < c c 0 , if one defines Ṽ±1 to be the vortices centered around ± d c − → e 1 ( d c is defined in Proposition 1.2) and R > 0 from Proposition 1.4, there exist K > 0 such that forare satisfied, then, forthe following coercivity result holds: Properties of the branch of travelling wavesThis section is devoted to the proof of Proposition 1.2. In subsection 2.1, we recall some estimates on Q c defined in Theorem 1.1 from previous works ([2], [4], [9] and [13]). In subsection 2.2, we compute some equalities and equivalents when c → 0 on the energy, momentum and the four particular directions (Finally, the properties of the zeros of Q c are studied in subsection 2.3. Decay estimates Estimates on vorticesWe recall that vortices are stationary solutions of (GP) of degrees n ∈ * (see [2]):V n (x) = ρ n (r)e inθ , where x = re iθ , solvingWe regroup here estimates on quantities involving vortices. We start with estimates on V ±1 .

We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg-Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg-Landau energy is well-defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross-Pitaevskii equation, the natural Hamiltonian evolution equation associated to the Ginzburg-Landau energy.

We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg–Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg–Landau energy is well defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross–Pitaevskii equation, the natural Hamiltonian evolution equation associated to the Ginzburg–Landau energy.

For the Nonlinear Schrödinger equation in dimension 2, the existence of a global minimizer of the energy at fixed momentum has been established by Bethuel-Gravejat-Saut [7] (see also [14]). This minimizer is a travelling wave for the Nonlinear Schrödinger equation. For large momentums, the propagation speed is small and the minimizer behaves like two well separated vortices. In that limit, we show the uniqueness of this minimizer, up to the invariances of the problem, hence proving the orbital stability of this travelling wave. This work is a follow up to two previous papers [16], [15], where we constructed and studied a particular travelling wave of the equation. We show a uniqueness result on this travelling wave in a class of functions that contains in particular all possible minimizers of the energy.Besides the energy, the momentum is another quantity formally conserved by the (NLS) flow, and is associated with the invariance by translation of (NLS). Formally, the momentum of u is 1 2 Ê 2 Re(i∇uū) ∈ Ê 2 , but its precise definition requires some care in the energy space due to the condition at infinity (see [37] in dimension larger than two and [14] ) for instance, then the expression of the momentum reduces toRe i∇u(ū − 1) .In addition to the translation invariance, the (NLS) equation is also phase shift invariant, that is invariant by multiplication by a complex of modulus one, and rotation invariant. Travelling waves for (NLS)Following the works in the physical literature of Jones and Roberts (see [32], [31]), there has been a large amount of mathematical works on the question of existence and properties of travelling waves solutions in the (NLS) equation, that is solution offor some c > 0, corresponding to particular solutions of (NLS) of the form Ψ(t, x) = u(x 1 , x 2 + ct) (due to the rotational invariance, we may always assume that the traveling wave moves along the direction − − → e 2 ). We refer to [6] for an overview on these problems in several dimensions. A natural approach is to look at the minimizing problem for p > 0It was shown by Bethuel-Gravejat-Saut in [7] that there exists a minimizer to this problem.Theorem 1.2 ([7]) For any p > 0, there exists a non constant function u p ∈ E and c(u p ) > 0 such that P 2 (u p ) = p, u p is a solution to (TW c(up) )(u p ) = 0 and E(u p ) = E min (p).Furthermore, any minimizer for E min (p) is, up to a translation in x 1 , even in x 1 .The strategy is to look at the corresponding minimization problem on tori (this avoids the problems with the definition of the momentum) larger and larger, and then pass to the limit. For the minimizing problem E min (p), the compactness of minimizing sequences has been shown later on in [14] for the natural semi-distance onTheorem 1.3 ([14]) For any p > 0, and any minimizing sequence (u n ) n∈AE for E min (p), there exists a subsequence (u nj ) j∈AE , a sequence of translations (y j ) j∈AE and a non constant function u p ∈ E such that D 0 (u nj , u p ) → 0, P 2 (u nj ) → P 2 (u p ) = p and E(u nj ) → E(u p ) = E min (p) as ...