Leapfrogging Vortex Rings for the Three Dimensional Gross-Pitaevskii Equation

Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation.

1. For \(y\in \mathbb {C}\) and \(z=(z_1,\ldots ,z_k) \in \mathbb {C}^k\) we write \((y,z):=\big (\mathrm{Re}(y\bar{z}_1),\ldots ,\mathrm{Re}(y\bar{z}_k)\big ) \in \mathbb {R}^k\) and \(y\times z := (iy,z)\).

2. We refer to [5, 17] for some attempts in that direction, and an account of the difficulties.

3. The integration is actually simpler in the original cartesian coordinates. A classical reference is the book of Jackson [10], an extended analysis can be found in the 1893 paper of Dyson [6]. See Appendix A for some details.

4. Exact traveling waves having the form of vortex rings have been constructed in [4], these are very similar in shape but not exactly equal to reference vortex rings.

5. A similar situation is described by Hicks [9] for a simplified vortex model introduced by Love [16] in 1894.

6. We stress that this holds at the level of the system \((\text {LF})_\varepsilon \), we do not know whether such special solutions exist at the level of equation \((\text {GP})_\varepsilon ^{c}\).

7. Another work on the 2D inhomogeneous GP equation is a recent preprint of Kurzke et al. [15], which studies a situation where the inhomogeneity and its derivatives are of order \(|\!\log \varepsilon |^{-1}.\) This is critical in the sense that interaction of vortices with the background potential and with each other are of the same order of magnitude. In the present work, by contrast, critical coupling occurs in hard-to-resolve corrections to the leading-order dynamics.

The research of RLJ was partially supported by the National Science and Engineering Council of Canada under operating Grant 261955. The research of DS was partially supported by the Agence Nationale de la Recherche through the Project ANR-14-CE25-0009-01. Both authors wish to warmly thank a referee for his very careful reading of the manuscript and his judicious remarks.

Authors and Affiliations

1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

   Robert L. Jerrard

2. Laboratoire Jacques-Louis Lions, Université Pierre and Marie Curie, 4 place Jussieu BC 187, 75252, Paris Cedex 05, France

   Didier Smets

Corresponding author

Correspondence to Didier Smets.

Vector Potential of Loop Currents

In the introduction we have considered the inhomogeneous Poisson equation

Its integration is classical (see e.g. [10]) and yields

which in turn simplifies to

where

and where E and K denote the complete elliptic integrals of first and second kind respectively (see e.g. [1]). Note that \(A_{\lambda a}(\lambda r, \lambda z)= A_a(r,z)\) for any \(\lambda >0\) and that we have the asymptotic expansions [1] of the complete elliptic integrals as \(s\rightarrow 1\) :

and similarly for their derivatives. For \((r,z) \in \mathbb {H}\setminus \{a\},\) direct computations therefore yield

and

where \(\rho := |a-(r,z)|.\)

Concerning the asymptotic close to \(r=0,\) we have

Singular Unimodular Maps

When \(a = \{a_1,\ldots , a_n\}\) is a family of n distinct points in \(\mathbb {H}\), we define the function \(\Psi ^*_a\) on \(\mathbb {H}_{a}:=\mathbb {H}\setminus a\) by

so that

Up to a constant phase shift, there exists a unique unimodular map \(u^*_a \in \mathcal {C}^\infty (\mathbb {H}_a,S^1)\cap W^{1,1}_\mathrm{loc}(\mathbb {H},S^1)\) such that

In the sense of distributions in \(\mathbb {H}\), we have

Let

for \(\rho \le \rho _a\) we set

Lemma A.1

Under the above assumptions we have

Proof

We have the pointwise equality

so that after integration by parts

and the first integral of the right-hand side in the previous identity vanishes by definition of \(\Psi ^*_a\) and \(\mathbb {H}_{a,\rho }.\) We next decompose the boundary integral as

and for fixed i, j, k we write

Using (124), we have

When \(i=j=k,\) we have by (124) and (125)

while when \(i=k\ne j\) we have

Finally, when \(i\ne k\) we have

The conclusion follows by summation. \(\square \)

If we next fix some constant \(K_0>0\) and we assume that the points \(a_i\) are of the form

for some \(r_0>0\), \(z_0\in \mathbb {R}\) and n points \(\{b_1,\ldots ,b_n\} \in \mathbb {R}^2\) which satisfy

we directly deduce from Lemma A.1, (124) and (125):

Lemma A.2

Under the above assumptions we have

Jacobian and Excess for 2D Ginzburg-Landau Functional

For the ease of reading, we recall in this appendix a few results from [12, 13] and [14] which we use in our work.

Theorem B.1

(Thm 1.3 in [13]—Lower energy bound) There exists an absolute constant \(C>0\) such that for any \(u \in H^1(B_r,\mathbb {C})\) satisfying \(\Vert Ju-\pi \delta _0\Vert _{\dot{W}^{-1,1}(B_r)} < r/4\) we have

Theorem B.2

(from Thm 1.1 in [13]—Jacobian estimate without vortices) There exists an absolute constant \(C>0\) with the following property. If \(\Omega \) is a bounded domain, \(u \in H^1(\Omega ,\mathbb {C})\), \(\varepsilon \in (0,1]\) and \({\mathcal E}_{\varepsilon }(u,\Omega )< \pi |\!\log \varepsilon |\), then

Theorem B.3

(Thm 2.1 in [12]—Jacobian estimate with vortices) There exists an absolute constant \(C>0\) with the following property. If \(\Omega \) is a bounded domain, \(u \in H^1(\Omega , \mathbb {C})\), and \(\varphi \in \mathcal {C}^{0,1}_c(\Omega )\), then for any \(\lambda \in (1,2]\) and any \(\varepsilon \in (0,1)\),

where

\(\lfloor x \rfloor \) denotes the greatest integer less than or equal to x, and

Theorem B.4

(Thm 1.2’ in [14]—Jacobian localization for a vortex in a ball) There exists an absolute constant \(C>0\), such that for any \(u \in H^1(B_r,\mathbb {C})\) satisfying

if we write

then there exists a point \(\xi \in B_{r/2}\) such that

Theorem B.5

(Thm 3 in [14]—Jacobian localization for many vortices) Let \(\Omega \) be a bounded, open, simply connected subset of \(\mathbb {R}^2\) with \(\mathcal {C}^1\) boundary. There exists constants C and K, depending on \(\mathrm{diam}(\Omega )\), with the following property: For any \(u \in H^1(\Omega ,\mathbb {C})\), if there exists \(n\ge 0\) distinct points \(a_1,\ldots ,a_n\) in \(\Omega \) and \(d \in \{\pm 1\}^{n}\) such that

where

and if in addition \({\mathcal E}_{\varepsilon }(u,\Omega ) \ge 1\) and

then there exist \(\xi _1,\ldots ,\xi _d\) in \(\Omega \) such that

where

and \(H_\Omega \) is the Robin function of \(\Omega .\)

Reprints and permissions