Leapfrogging Vortex Rings for the Three Dimensional Gross-Pitaevskii Equation

Abstract

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.

Appendices

Vector Potential of Loop Currents

In the introduction we have considered the inhomogeneous Poisson equation

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\mathrm{div} \left( \frac{1}{r} \nabla \left( r A_a\right) \right) = 2\pi \delta _{a} &{}\qquad \text {in } \mathbb {H},\\ A_a = 0 &{} \qquad \text {on } \mathbb {H}. \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} A_a(r,z) = \frac{r(a)}{2} \int _0^{2\pi } \frac{\cos (t)}{\sqrt{r(a)^2+r^2 + (z-z(a))^2-2r(a)r\cos (t)}}\, dt, \end{aligned}$$

which in turn simplifies to

$$\begin{aligned} A_a(r,z) = \sqrt{\frac{r(a)}{r}} \frac{1}{k} \left[ (2-k^2)K(k^2)-2E(k^2)\right] \end{aligned}$$

where

$$\begin{aligned} k^2 = \frac{4r(a) r}{ r(a)^2 + r^2 + (z-z(a))^2 + 2 r(a) r} \end{aligned}$$

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 $s\rightarrow 1$ :

$$\begin{aligned} K(s)= & {} -\frac{1}{2}\log (1-s)\left( 1 + \frac{1-s}{4}\right) + \log (4) + O(1-s),\\ E(s )= & {} 1 - \log (1-s)\frac{1-s}{4} + O(1-s), \end{aligned}$$

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

$$\begin{aligned} A_a(r,z) = \left( \log \left( \tfrac{r(a)}{\rho }\right) + 3 \log (2) -2\right) + O\left( \tfrac{\rho }{r(a)} \left| \log \left( \tfrac{\rho }{r(a)}\right) \right| \right) \qquad \text {as } \tfrac{\rho }{r(a)} \rightarrow 0,\nonumber \\ \end{aligned}$$

(124)

and

$$\begin{aligned} \partial _{\rho } A_a = -\frac{1}{\rho } + O\left( \tfrac{1}{r(a)} \right) \qquad \text {as } \tfrac{\rho }{r(a)} \rightarrow 0, \end{aligned}$$

(125)

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

Concerning the asymptotic close to $r=0,$ we have

$$\begin{aligned} A_a(r,z) \simeq \frac{rr(a)^2}{r(a)^3+|z|^3} \qquad \text {as } \frac{r}{r(a)} \rightarrow 0. \end{aligned}$$

(126)

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

$$\begin{aligned} \Psi ^*_a = \sum _{i=1}^n A_{a_i}, \end{aligned}$$

so that

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\mathrm{div} \left( \frac{1}{r} \nabla (r \Psi ^*_a)\right) = 2\pi \sum _{i=1}^n \delta _{a_i} &{}\qquad \text {on } \mathbb {H},\\ \displaystyle \Psi ^*_a = 0 &{} \qquad \text {on } \partial \mathbb {H}. \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} r (iu^*_a,\nabla u^*_a) = rj(u^*_a) = -\nabla ^\perp (r\Psi ^*_a). \end{aligned}$$

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

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \mathrm{div}(rj(u^*_a)) &{} = 0\\ \displaystyle \mathrm{curl}(j(u^*_a)) &{} = 2\pi \sum _{i=1}^n \delta _{a_i}. \end{array} \right. \end{aligned}$$

Let

$$\begin{aligned} \rho _a :=\frac{1}{4}\min \Big ( \min _{i\ne j}|a_i-a_j|, \min _i r(a_i)\Big ), \end{aligned}$$

for $\rho \le \rho _a$ we set

$$\begin{aligned}\mathbb {H}_{a,\rho } := \mathbb {H}\setminus \cup _{i=1}^n B(a_i,\rho ). \end{aligned}$$

Lemma A.1

Under the above assumptions we have

$$\begin{aligned} \int _{\mathbb {H}_{a,\rho }} \frac{|j(u^*_a)|^2}{2} r\,drdz= & {} \pi \sum _{i=1}^n r(a_i) \times \Big [\log \big (\tfrac{r(a_i)}{\rho }\big )+ \sum _{j\ne i}A_{a_j}(a_i)\\&+ \big (3\log (2)-2\big ) + O\big (\tfrac{\rho }{\rho _a}\log ^2\big (\tfrac{\rho }{\rho _a}\big ) \big )\Big ]. \end{aligned}$$

Proof

We have the pointwise equality

$$\begin{aligned} |j(u^*_a)|^2 r = \frac{1}{r}|\nabla ^\perp (r\Psi ^*_a)|^2 = \frac{1}{r}|\nabla (r\Psi ^*_a)|^2, \end{aligned}$$

so that after integration by parts

$$\begin{aligned} \int _{\mathbb {H}_{a,\rho }} \frac{|j(u^*_a)|^2}{2} r\,drdz= & {} - \frac{1}{2} \int _{\mathbb {H}_{a,\rho }} \mathrm{div}\left( \frac{1}{r}\nabla (r\Psi ^*_a)\right) r\Psi ^*_a \,drdz\\&+ \frac{1}{2} \int _{\partial \mathbb {H}_{a,\rho }} \Psi ^*_a \nabla (r\Psi ^*_a)\cdot {\vec {n}}, \end{aligned}$$

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

$$\begin{aligned} \frac{1}{2} \int _{\partial \mathbb {H}_{a,\rho }} \Psi ^*_a \nabla (r\Psi ^*_a)\cdot {\vec {n}} = \frac{1}{2} \sum _{i,j,k=1}^n \int _{\partial B(a_i,\rho )} A_{a_j} \nabla (rA_{a_k})\cdot \vec {n}, \end{aligned}$$

and for fixed i, j, k we write

$$\begin{aligned} A_{a_j} \nabla (rA_{a_k})\cdot \vec {n} = \left( -A_{a_j} \partial _\rho A_{a_k} r + A_{a_j}A_{a_k}n_r\right) . \end{aligned}$$

Using (124), we have

$$\begin{aligned} \left| \int _{\partial B(a_i,\rho )} A_{a_j}A_{a_k}n_r \right| \le r(a_i)O\Big (\tfrac{\rho }{\rho _a}\log ^2(\tfrac{\rho }{\rho _a})\Big ). \end{aligned}$$

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

$$\begin{aligned} -\frac{1}{2} \int _{\partial B(a_i,\rho )} A_{a_j} \partial _\rho A_{a_k} r = \pi r(a_i) \left( \log \left( \frac{r(a_i)}{\rho }\right) + 3\log (2) -2 + O\Big (\tfrac{\rho }{\rho _a}\log (\tfrac{\rho }{\rho _a})\Big )\right) \end{aligned}$$

while when $i=k\ne j$ we have

$$\begin{aligned} -\frac{1}{2} \int _{\partial B(a_i,\rho )} A_{a_j} \partial _\rho A_{a_k} r = \pi r(a_i) \left( A_{a_j}(a_i) + O\Big (\tfrac{\rho }{\rho _a}\Big )\right) . \end{aligned}$$

Finally, when $i\ne k$ we have

$$\begin{aligned} \left| \frac{1}{2} \int _{\partial B(a_i,\rho )} A_{a_j} \partial _\rho A_{a_k} r \right| \le r(a_i) O\Big ((\tfrac{\rho }{\rho _a})^2 \log (\tfrac{\rho }{\rho _a})\Big ). \end{aligned}$$

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

$$\begin{aligned} a_{i} := \Big (r_0+ \frac{r(b_i)}{\sqrt{|\!\log \varepsilon |}} , z_0 + \frac{z(b_i)}{\sqrt{|\!\log \varepsilon |}} \Big ),\qquad i=1,\ldots ,n, \end{aligned}$$

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

$$\begin{aligned} \max _i |b_i| \le K_0,\qquad \text {and}\qquad \min _{i\ne j} \mathrm{dist}(b_i,b_j)\ge \frac{1}{K_0}, \end{aligned}$$

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

Lemma A.2

Under the above assumptions we have

$$\begin{aligned}&\int _{\mathbb {H}_{a,\rho }} \frac{|j(u^*_a )|^2}{2} r\,drdz\nonumber \\&\quad =\ \pi n r_0 \Big ( |\log \rho | + n\log r_0 + n\big (3\log (2)-2\big ) + \tfrac{n-1}{2}\log |\log \varepsilon |\Big ) \nonumber \\&\quad \quad + \pi r_0 \Big ( \sum _{i} \frac{r(b_i)}{r_0}\frac{|\log \rho |}{\sqrt{|\log \varepsilon |}} - \sum _{i\ne j} \log |b_i-b_j|\Big )\nonumber \\&\quad \quad + O_{K_0,r_0} \Big ( \frac{1}{\sqrt{|\log \varepsilon |}}\Big ). \end{aligned}$$

(127)

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

$$\begin{aligned} {\mathcal E}_{\varepsilon }(u,B_r) \ge \pi \log \frac{r}{\varepsilon } + \gamma -\frac{C}{r} \left( \varepsilon \sqrt{\log \tfrac{r}{\varepsilon }} + \Vert Ju-\pi \delta _0\Vert _{\dot{W}^{-1,1}(B_r)}\right) . \end{aligned}$$

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

$$\begin{aligned} \Big \Vert Ju \Big \Vert _{\dot{W}^{-1,1}(\Omega )} \le \varepsilon C {\mathcal E}_{\varepsilon }(u,\Omega )\exp \Big ( \frac{1}{\pi }{\mathcal E}_{\varepsilon }(u,\Omega )\Big ). \end{aligned}$$

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 $\varepsilon \in (0,1)$,

$$\begin{aligned} \Big | \int _\Omega \varphi Ju \, dx \Big | \le \pi d_\lambda \Vert \varphi \Vert _\infty + \Vert \varphi \Vert _{\mathcal {C}^{0,1}}h^\varepsilon (\varphi ,u,\lambda ) \end{aligned}$$

where

$$\begin{aligned} d_\lambda = \Big \lfloor \frac{\lambda }{\pi } \frac{{\mathcal E}_{\varepsilon }(u,\mathrm{spt}(\varphi ))}{|\!\log \varepsilon |} \Big \rfloor , \end{aligned}$$

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

$$\begin{aligned} h^\varepsilon (\varphi ,u,\lambda ) \le C \varepsilon ^\frac{\lambda -1}{12\lambda }\big (1+\frac{{\mathcal E}_{\varepsilon }(u,\mathrm{spt}(\varphi ))}{|\!\log \varepsilon |}\big ) \big (1+\mathcal {L}^2(\mathrm{spt}(\varphi )\big ). \end{aligned}$$

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

$$\begin{aligned}\Vert Ju-\pi \delta _0\Vert _{\dot{W}^{-1,1}(B_r)} < r/4,\end{aligned}$$

if we write

$$\begin{aligned} \Xi = {\mathcal E}_{\varepsilon }(u,B_r) - \pi \log \frac{r}{\varepsilon } \end{aligned}$$

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

$$\begin{aligned} \left\| Ju - \pi \delta _\xi \right\| _{\dot{W}^{-1,1}(B_r)} \le \varepsilon C(C+\Xi )\left[ (C+\Xi )e^{\Xi /\pi } + \sqrt{\log \frac{r}{\varepsilon }}\right] . \end{aligned}$$

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

$$\begin{aligned} \Vert Ju-\pi \sum _{i=1}^n d_i \delta _{a_i}\Vert _{\dot{W}^{-1,1}(\Omega )} \le \frac{\rho _a}{Kn^5}, \end{aligned}$$

where

$$\begin{aligned} \rho _a := \frac{1}{4}\mathrm{min}_i\left\{ \mathrm{min}_{j\ne i}|a_i-a_j|, \mathrm{dist}(a_i,\partial \Omega )\right\} , \end{aligned}$$

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

$$\begin{aligned} \frac{n^5}{\rho _a}{\mathcal E}_{\varepsilon }(u,\Omega ) + \frac{n^{10}}{\rho _a^2}\sqrt{{\mathcal E}_{\varepsilon }(u,\Omega )} \le \frac{1}{\varepsilon }, \end{aligned}$$

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

$$\begin{aligned}&\left\| Ju - \pi \sum _{i=1}^n d_i \delta _{\xi _i} \right\| _{\dot{W}^{-1,1}(\Omega )}\\&\quad \le C\varepsilon \left[ n(C+\Xi _\Omega ^\varepsilon )^2e^{\Xi _\Omega ^\varepsilon /\pi } + (C+\Xi _\Omega ^\varepsilon )\frac{n^5}{\rho _a} + {\mathcal E}_{\varepsilon }(u,\Omega )\right] , \end{aligned}$$

where

$$\begin{aligned} \Xi _\Omega ^\varepsilon:= & {} {\mathcal E}_{\varepsilon }(u,\Omega )-n(\pi \log \frac{1}{\varepsilon } + \gamma ) \\&+\,- \pi \Big ( \sum _{i\ne j} d_id_j \log |a_i-a_j| + \sum _{i,j} d_id_j H_\Omega (a_i,a_j)\Big )\end{aligned}$$

and $H_\Omega$ is the Robin function of $\Omega .$