quantized vortex dynamics and interaction patterns … · quantized vortex dynamics and interaction...

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018096DYNAMICAL SYSTEMS SERIES BVolume 23, Number 6, August 2018 pp. 2265–2297

QUANTIZED VORTEX DYNAMICS AND INTERACTION

PATTERNS IN SUPERCONDUCTIVITY BASED ON THE

REDUCED DYNAMICAL LAW

Zhiguo Xu

School of Mathematics, Jilin University, Changchun 130012, China

Weizhu Bao

Department of Mathematics, National University of Singapore, 119076, Singapore

Shaoyun Shi

School of Mathematics, Jilin University, Changchun 130012, China

(Communicated by Xiao-Ping Wang)

Abstract. We study analytically and numerically stability and interaction

patterns of quantized vortex lattices governed by the reduced dynamical law –

a system of ordinary differential equations (ODEs) – in superconductivity. Byderiving several non-autonomous first integrals of the ODEs, we obtain qualita-

tively dynamical properties of a cluster of quantized vortices, including global

existence, finite time collision, equilibrium solution and invariant solution man-ifolds. For a vortex lattice with 3 vortices, we establish orbital stability when

they have the same winding number and find different collision patterns when

they have different winding numbers. In addition, under several special initialsetups, we can obtain analytical solutions for the nonlinear ODEs.

1. Introduction. In this paper, we study analytically and numerically stabilityand interaction patterns of the following system of ordinary differential equations(ODEs) describing the dynamics of N ≥ 2 quantized vortices in superconductivitybased on the reduced dynamical law [23, 12, 17, 30, 31]

xj(t) = 2mj

N∑k=1,k 6=j

mkxj(t)− xk(t)

|xj(t)− xk(t)|2, 1 ≤ j ≤ N, t > 0, (1.1)

with initial data

xj(0) = x0j = (x0

j , y0j )T ∈ R2, 1 ≤ j ≤ N. (1.2)

Here t is time, xj(t) = (xj(t), yj(t))T ∈ R2 is the center of the j-th (1 ≤ j ≤ N)

quantized vortex at time t, mj = +1 or −1 is the winding number or index orcirculation of the j-th (1 ≤ j ≤ N) quantized vortex. We always assume thatthe initial data satisfies X0 := (x0

1, . . . ,x0N ) ∈ R2×N

∗ := {X = (x1, . . . ,xN ) ∈

2010 Mathematics Subject Classification. Primary: 34C60, 34D05; Secondary: 34A33, 34D30,

65L07.Key words and phrases. Quantized vortex, reduced dynamical law, superconductivity, interac-

tion pattern, non-autonomous first integral, winding number, orbital stability, finite time collision,collision cluster.

2265

http://dx.doi.org/10.3934/dcdsb.2018096

2266 ZHIGUO XU, WEIZHU BAO AND SHAOYUN SHI

R2×N | xj 6= xl ∈ R2 for 1 ≤ j < l ≤ N} and denote its mass center as x0 :=1N

∑Nj=1 x0

j . Throughout this paper, we assume that N ≥ 2.

The ODEs (1.1) with (1.2) was derived asymptotically as a reduced dynamicallaw for the dynamics of N quantized vortices – particle-like or topological defects– in the Ginzburg-Landau equation [22, 12, 17]

∂tψ(x, t) = ∇2ψ(x, t) +1

ε2(1− |ψ(x, t)|2)ψ(x, t), x ∈ R2, t > 0, (1.3)

with initial condition

ψ(x, 0) = ψ0(x) = ΠNj=1φmj

(x− x0j ), x ∈ R2, (1.4)

for superconductivity when either ε = 1 and d0min := min1≤j<l≤N |x0

j − x0l | → ∞

[23] or for a given X0 ∈ R2×N∗ and ε → 0+ [12, 17]. Here x = (x, y)T ∈ R2 is

the Cartesian coordinates in two dimensions (2D), ψ := ψ(x, t) is a complex-valuedorder parameter, ε > 0 is a constant, and φm(x) = f(r)eimθ (m = +1 or −1) with(r, θ) the polar coordinates in 2D and f(r) satisfying [23, 12, 17, 30, 31]

1

r

d

dr

(rdf(r)

dr

)− 1

r2f(r) +

1

ε2(1− f2(r))f(r) = 0, 0 < r < +∞,

f(0) = 0, limr→+∞

f(r) = 1.

Here φm(x) is a typical quantized vortex in 2D, which is zero of the order parameterat the vortex center located at the origin and has localized phase singularity withinteger m topological charge usually called also as winding number or index orcirculation. In fact, quantized vortices have been widely observed in superconductor[11, 17, 4], liquid helium [21], Bose-Einstein condensates [26, 2, 13]; and they are keysignatures of superconductivity and superfluidity. The study of quantized vorticesand their dynamics is one of the most important and fundamental problems insuperconductivity and superfluidity [23, 3, 20, 28, 6, 8, 9, 10, 16, 18, 5, 24, 25].

Based on the reduced dynamical law, i.e. (1.1), for the quantized vortex dynamicsin superconductivity, when two quantized vortices have the same winding number(i.e. vortex pair), they undergo a repulsive interaction; and respectively, whenthey have opposite winding numbers (i.e. vortex dipole or vortex-antivortex), theyundergo an attractive interaction [23, 30, 31]. For N ≥ 2 and X0 ∈ R2×N

∗ , itis straightforward to obtain local existence of the ODEs (1.1) with (1.2) by thestandard theory of ODEs. Specifically, when N = 2, one can obtain explicitly theanalytical solution of (1.1) with (1.2): when m1 = m2 (i.e. vortex pair), the twovortices move outwards by repelling each other along the line passing through theirinitial locations x0

1 6= x02 and they never collide at finite time; and when m1 = −m2

(i.e. vortex dipole or vortex-antivortex), the two vortices move towards each otheralong the line passing through their initial locations x0

1 6= x02 and they will collide

at 12

(x0

1 + x02

)in finite time [23, 30, 31]. For analytical solutions of the ODEs (1.1)

with several special initial setups in (1.2), we refer to [30, 31] and references therein.In addition, define the mass center of the N vortices as

x(t) :=1

N

N∑j=1

xj(t), t ≥ 0, (1.5)

then it was proven that the mass center is conserved under the dynamics of (1.1)with (1.2) [30, 31]

x(t) ≡ x(0) = x0, t ≥ 0. (1.6)

QUANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 2267

Introduce

W (X) = −∑

1≤j 6=k≤N

mjmk ln |xj − xk|

= − ln∏

1≤j 6=k≤N

|xj − xk|mjmk , X ∈ R2×N∗ , (1.7)

then (1.1) can be reformulated as

X(t) = −∇XW (X), t > 0, (1.8)

which implies that

W (X(t2)) ≤W (X(t1)) ≤W (X(0)) = W (X0), 0 ≤ t1 ≤ t2. (1.9)

In addition, let zj(t) := xj(t) + iyj(t) ∈ C for 1 ≤ j ≤ N , then (1.1) can bereformulated as

zj(t) = 2mj

N∑k=1,k 6=j

mkzj(t)− zk(t)

|zj(t)− zk(t)|2

= 2mj

N∑k=1,k 6=j

mk

zj(t)− zk(t), 1 ≤ j ≤ N, t > 0, (1.10)

where z denotes the complex conjugate of z ∈ C.For rigorous mathematical justification of the derivation of the above reduced dy-

namical law (1.1) with (1.2) for superconductivity, we refer to [12, 17] and referencestherein, and respectively, for numerical comparison of quantized vortex center dy-namics under the Ginzburg-Landau equation (1.3) with (1.4) and its correspondingreduced dynamical law (1.1) with (1.2), we refer to [30, 31] and references therein.Based on the mathematical and numerical results [12, 17, 30, 31], the dynamicsof the N quantized vortex centers under the reduced dynamical law agrees qual-itatively (and quantitatively when they are well-separated) with that under theGinzburg-Landau equation. The main aim of this paper is to study analyticallyand numerically the dynamics and interaction patterns of the reduced dynamicallaw (1.1) with (1.2), which will generate important insights about quantized vortexdynamics and interaction patterns in superconductivity and is much simpler than tosolve the Ginzburg-Landau equation (1.3) with (1.4). We establish global existenceof the ODEs (1.1) when the N quantized vortices have the same winding numberand possible finite time collision when they have opposite winding numbers. ForN = 3, we prove orbital stability when they have the same winding number and finddifferent collision patterns when they have different winding numbers. Analyticalsolutions of the ODEs (1.1) are obtained under several initial setups with symmetry.

The paper is organized as follows. In section 2, we obtain some invariant solu-tion manifolds and several non-autonomous first integrals of the ODEs (1.1) andestablish its global existence when the N quantized vortices have the same windingnumber and possible finite time collision when they have opposite winding numbers.In section 3, we prove orbital stability when they have the same winding number andfind different collision patterns when they have different winding numbers for thedynamics of N = 3 vortices. Analytical solutions of the ODEs (1.1) are presentedunder several initial setups with symmetry in section 4. Finally, some conclusionsare drawn in section 5.


2. Dynamical properties of a cluster with N quantized vortices. In thissection, we establish dynamical properties of the system of ODEs (1.1) with theinitial data (1.2) for describing the dynamics – reduced dynamical law – of a clusterwith N quantized vortices in superconductivity.

For any two vortices xj(t) and xl(t) (1 ≤ j < l ≤ N), if there exists a finitetime 0 < Tc < +∞ such that djl(t) := |xj(t) − xl(t)| > 0 for 0 ≤ t < Tc anddjl(Tc) = 0, then we say that they will be finite time collision or annihilation (cf.Fig. 2.1a); otherwise, i.e. djl(t) > 0 for t ≥ 0, then we say that they will not collide.When N ≥ 2 and let I ⊆ {1, 2, . . . , N} be a set with at least 2 elements, if thereexists a finite time 0 < Tc < +∞ such that min1≤j<l≤N djl(t) > 0 for 0 ≤ t < Tc,limt→T−

cxj(t) = x0 ∈ R2 for j ∈ I with x0 a fixed point and minj∈I djl(t) > 0 for

0 ≤ t ≤ Tc and l ∈ J := {m | 1 ≤ m ≤ N,m /∈ I}, then we say that all vortices inthe set I will form a (finite time) collision cluster among the N vortices (cf. Fig.2.1b). Define

Tmax = sup {t ≥ 0 | xj(t) 6= xl(t), for all 1 ≤ j 6= l ≤ N} ,

it is easy to see that 0 < Tmax ≤ +∞ by noting (1.2). If Tmax < +∞, a finitetime collision happens among at least two vortices in the N vortices (or the ODEs(1.1) with (1.2) will blow-up at finite time); otherwise, i.e. Tmax = +∞, there is nocollision among all the N quantized vortices (or the ODEs (1.1) with (1.2) is globalwell-posed in time).

−1 0 1

−1

0

1

y

x

−

x03 x01

x02

(a)

Figure 2.1. Illustrations of a finite time collision of a vortex dipolein a vortex cluster with 3 vortices (a) and a (finite time) collisioncluster with 3 vortices in a vortex cluster with 5 vortices (b). Hereand in the following figures, ‘+’ and ‘−’ denote the initial vortexcenters with winding numbers m = +1 and m = −1, respectively;and ‘o’ denotes the finite time collision position.

2.1. Invariant solution manifolds. Let α > 0 be a positive constant, 0 ≤ θ0 <2π be a constant, x0 ∈ R2 be a given point and Q(θ) be the rotational matrixdefined as

Q(θ) =

(cos θ − sin θsin θ cos θ

), 0 ≤ θ < 2π.


Then it is easy to see that the ODEs (1.1) with (1.2) is translational and rota-tional invariant with the proof omitted here for brevity.

Lemma 2.1. Let X(t) = (x1(t),x2(t), . . . ,xN (t)) ∈ R2×N∗ be the solution of the

ODEs (1.1) with (1.2), then we have(i) If x0

j → x0j+x0 for 1 ≤ j ≤ N in (1.2), then xj(t)→ xj(t)+x0 for 1 ≤ j ≤ N .

(ii) If x0j → αx0

j for 1 ≤ j ≤ N in (1.2), then xj(t)→ αxj(t/α2) for 1 ≤ j ≤ N .

(iii) If x0j → Q(θ)x0

j for 1 ≤ j ≤ N in (1.2), then xj(t) → Q(θ)xj(t) for1 ≤ j ≤ N .

Denote Se(x0) :={X = (x1, . . . ,xN ) ∈ R2×N

∗ | |(xj − x0) · e| = |xj − x0| × |e|,1 ≤ j ≤ N

}, where e ∈ R2 is a given unit vector. In fact, Se(x0) is a line in 2D

passing the point x0 and parallel to the unit vector e. For X = (x1, . . . ,xN ) ∈R2×N∗ , if there exist x0 ∈ R2 and a unit vector e ∈ R2 such that X ∈ Se(x0), then

we say that X is collinear.

Lemma 2.2. If the initial data X0 ∈ R2×N∗ in (1.2) is collinear, i.e. there exist

x0 ∈ R2 and a unit vector e ∈ R2 such that X0 ∈ Se(x0), then the solution X(t) of(1.1)-(1.2) is collinear, i.e. X(t) ∈ Se(x0) for 0 ≤ t < Tmax.

Proof. From X0 ∈ Se(x0), there exist a0j ∈ R (1 ≤ j ≤ N) satisfying a0

j 6= a0l for

1 ≤ j < l ≤ N such that

x0j = x0 + a0

je, 1 ≤ j ≤ N. (2.1)

Noting the symmetric structure in (1.1) and (2.1), we can assume

xj(t) = x0 + aj(t)e, 1 ≤ j ≤ N, t ≥ 0. (2.2)

Plugging (2.2) into (1.1), we have

aj(t) = 2mj

N∑k=1,k 6=j

mkaj(t)− ak(t)

|aj(t)− ak(t)|2, 1 ≤ j ≤ N, t > 0, (2.3)

with the initial data by noting (2.1)

aj(0) = a0j , 1 ≤ j ≤ N. (2.4)

The ODEs (2.3) with (2.4) is locally well-posed. Thus X(t) ∈ Se(x0) for 0 ≤ t <Tmax.

Let e ∈ R2 be a unit vector, denote θjN := 2(j−1)πN and x

(0)j = Q(θjN + θ0)e for

1 ≤ j ≤ N and define

SNe (x0, θ0) :={

X0r = (x0 + rx

(0)1 , . . . ,x0 + rx

(0)N ) ∈ R2×N

∗ | r > 0},

SNe (x0) :=⋃

0≤θ0<2π

SNe (x0, θ0).

Lemma 2.3. Assume the N vortices have the same winding number, i.e. m1 =. . . = mN = ±1. If there exists a unit vector e ∈ R2, x0 ∈ R2 and 0 ≤ θ0 < 2πsuch that the initial data X0 ∈ SNe (x0, θ0) in (1.2), then the solution X(t) of (1.1)satisfies X(t) ∈ SNe (x0, θ0) for t ≥ 0.

Proof. Since X0 ∈ SNe (x0, θ0), there exists a r0 > 0 such that

x0j = x0 + r0Q(θjN + θ0)e, 1 ≤ j ≤ N. (2.5)


Noting the symmetric structure in (1.1) and (2.5), we can assume

xj(t) = x0 + r(t)Q(θjN + θ0)e, 1 ≤ j ≤ N, t ≥ 0. (2.6)

Plugging (2.6) into (1.1), noting m1 = . . . = mN and (2.5), we have [30, 31]

r(t) =N − 1

r(t), t > 0, r(0) = r0,

which implies r(t) =√r20 + 2(N − 1)t for t ≥ 0. Thus X(t) ∈ SNe (x0, θ0) for

t ≥ 0.

From the above two lemmas, for any θ0 ∈ R, x0 ∈ R2 and a unit vector e ∈R2, Se(x0) is an invariant solution manifold of the ODEs (1.1) with (1.2). Inaddition, when m1 = . . . = mN , then SNe (x0, θ0) is also an invariant solutionmanifold of the ODEs (1.1) with (1.2). Specifically, when X0 ∈ SNe (0, θ0) andm1 = . . . = mN , then the ODEs (1.1) with (1.2) admits the self-similar solution

X(t) =√r20 + 2(N − 1)tX0 with r0 = |x0

1| = . . . = |x0N | for t ≥ 0. For more self-

similar solutions of the ODEs (1.1) with special initial setups, we refer to [30, 31]and references therein.

2.2. Non-autonomous first integrals. In this subsection, we establish severlnon-autonomous first integrals of the ODEs (1.1). Here a nonconstant functionH(X, t) is called a non-autonomous first integral of the ODEs (1.1) if it is constantalong any solution curve X = X(t) of (1.1). In fact, first integral is an importantconcept in the qualitative analysis of dynamical system. The classical applicationsof first integral can be traced back to H. Poincare for studying the three bodyproblem [27] and S. Kowalevski for studying the Euler top [14] and so on. For morerecent applications, we refer to [15, 21, 29] and references therein.

Let N+ and N− be the number of vortices with winding number m = 1 andm = −1, respectively, then we have

0 ≤ N+ ≤ N, 0 ≤ N− ≤ N, N+ +N− = N.

In addition, it is easy to get

M0 =∑

1≤j<l≤N

mjml =1

2

N∑j,l=1,j 6=l

mjml

=N+(N+ − 1)

2+N−(N− − 1)

2−N+N− =

(N+ −N−)2 −N2

. (2.7)

Define

H1(X, t) = −4NM0t+∑

1≤j<l≤N

|xj − xl|2, (2.8)

H2(X, t) = −4M0t+

N∑j=1

|xj |2, X := (x1,x2, . . . ,xN ) ∈ R2×N , t ≥ 0, (2.9)

H3(X, t) = −4(N − 2)M0t+∑

1≤j<l≤N

|xj + xl|2. (2.10)

then we have


Lemma 2.4. Let X(t) = (x1(t),x2(t), . . . ,xN (t)) ∈ R2×N∗ be the solution of the

ODEs (1.1) with (1.2), then H1(X, t), H2(X, t) and H3(X, t) are non-autonomousfirst integrals of (1.1), i.e.

H1(X(t), t) ≡ H01 :=

∑1≤j<l≤N

|x0j − x0

l |2, (2.11)

H2(X(t), t) ≡ H02 :=

N∑j=1

|x0j |2, t ≥ 0, (2.12)

H3(X(t), t) ≡ H03 :=

∑1≤j<l≤N

|x0j + x0

l |2. (2.13)

Specifically, when M0 = 0, H1(X) := H1(X, t) =∑

1≤j<l≤N |xj−xl|2 and H2(X) :=

H2(X, t) =∑Nj=1 |xj |2 are two autonomous first integrals of (1.1); and when either

N = 2 or M0 = 0, H3(X) := H3(X, t) =∑

1≤j<l≤N |xj + xl|2 is an autonomous

first integral of (1.1).

Proof. Differentiating the left equation in (2.8) (with X = X(t)) with respect to t,we have for t ≥ 0

dH1(X(t), t)

dt= ∇XH1(X, t) · X +

∂H1(X, t)

∂t

∣∣∣∣X=X(t)

= −4NM0 + 2∑

1≤j<l≤N

(xj(t)− xl(t)) · (xj(t)− xl(t)). (2.14)

Using summation by parts and noting (2.7) and (1.1), we obtain

I := 2∑

1≤j<l≤N

(xj(t)− xl(t)) · (xj(t)− xl(t))

=

N∑j,l=1,j 6=l

(xj(t)− xl(t)) · (xj(t)− xl(t))

= 2

N∑j,l=1,j 6=l

(xj(t)− xl(t)) · N∑k=1,k 6=j

mjmkxj(t)− xk(t)

|xj(t)− xk(t)|2−

N∑k=1,k 6=l

mlmkxl(t)− xk(t)

|xl(t)− xk(t)|2

= 2

N∑j,l=1,j 6=l

2mjml|xj(t)− xl(t)|2

|xj(t)− xl(t)|2

+2

N∑j,l=1,j 6=l

(xj(t)− xl(t)) ·N∑

k=1,k 6=j,l

mjmkxj(t)− xk(t)

|xj(t)− xk(t)|2

−2

N∑j,l=1,j 6=l

(xj(t)− xl(t)) ·N∑

k=1,k 6=j,l

mlmkxl(t)− xk(t)

|xl(t)− xk(t)|2. (2.15)

I = 4

N∑j,l=1,j 6=l

mjml + 2

N∑j,k=1,j 6=k

mjmk

N∑l=1,l 6=j,k

(xj(t)− xl(t)) · (xj(t)− xk(t))

|xj(t)− xk(t)|2


−2

N∑l,k=1,l 6=k

mlmk

N∑j=1,l 6=l,k

(xj(t)− xl(t)) · (xl(t)− xk(t))

|xl(t)− xk(t)|2

= 8M0 + 2

N∑j,k=1,j 6=k

mjmk

N∑l=1,l 6=j,k

(xj(t)− xl(t)) · (xj(t)− xk(t))

|xj(t)− xk(t)|2

−2

N∑j,k=1,j 6=k

mjmk

N∑l=1,l 6=j,k

(xk(t)− xl(t)) · (xj(t)− xk(t))

|xj(t)− xk(t)|2

= 2

N∑j,k=1,j 6=k

mjmk

N∑l=1,l 6=j,k

(xj(t)− xk(t)) · [(xj(t)− xl(t))− (xk(t)− xl(t))]

|xj(t)− xk(t)|2

+8M0

= 8M0 + 2

N∑j,k=1,j 6=k

(N − 2)mjmk

= 8M0 + 4(N − 2)M0 = 4NM0, t ≥ 0. (2.16)

Plugging (2.15) and (2.16) into (2.14), we get

dH1(X(t), t)

dt= −4NM0 + 4NM0 = 0, t ≥ 0, (2.17)

which immediately implies (2.11) by noting the initial condition (1.2).Similarly, differentiating (2.10) (with X = X(t)) with respect to t, we get

dH3(X(t), t)

dt= ∇XH3(X, t) · X +

∂H3(X, t)

∂t

∣∣∣∣X=X(t)

= −4(N − 2)M0 + 2∑

1≤j<l≤N

(xj(t) + xl(t)) · (xj(t) + xl(t)).(2.18)

Similar to (2.15), noting (1.1) and (2.7), we get

II := 2∑

1≤j<l≤N

(xj(t) + xl(t)) · (xj(t) + xl(t))

=

N∑j,l=1,j 6=l

(xj(t) + xl(t)) · (xj(t) + xl(t))

= 2

N∑j,l=1,j 6=l

(xj(t) + xl(t)) · N∑k=1,k 6=j

mjmkxj(t)− xk(t)

|xj(t)− xk(t)|2+

N∑k=1,k 6=l

mlmkxl(t)− xk(t)

|xl(t)− xk(t)|2

. (2.19)

II = 2

N∑j,k=1,j 6=k

mjmk

N∑l=1,l 6=j,k

(xj(t) + xl(t)) · (xj(t)− xk(t))

|xj(t)− xk(t)|2

+2

N∑l,k=1,l 6=k

mlmk

N∑j=1,l 6=l,k

(xj(t) + xl(t)) · (xl(t)− xk(t))

|xl(t)− xk(t)|2


= 2

N∑j,k=1,j 6=k

mjmk

N∑l=1,l 6=j,k

(xj(t) + xl(t)) · (xj(t)− xk(t))

|xj(t)− xk(t)|2

−2N∑

j,k=1,j 6=k

mjmk

N∑l=1,l 6=j,k

(xk(t) + xl(t)) · (xj(t)− xk(t))

|xj(t)− xk(t)|2

= 2

N∑j,k=1,j 6=k

mjmk

N∑l=1,l 6=j,k

(xj(t)− xk(t)) · [(xj(t) + xl(t))− (xk(t) + xl(t))]

|xj(t)− xk(t)|2

= 2

N∑j,k=1,j 6=k

(N − 2)mjmk = 4(N − 2)M0, t ≥ 0. (2.20)

Plugging (2.19) and (2.20) into (2.18), we get

dH3(X(t), t)

dt= −4(N − 2)M0 + 4(N − 2)M0 = 0, t ≥ 0, (2.21)

which immediately implies (2.13) by noting the initial condition (1.2).From (2.8) and (2.10), it is easy to see that

H2(X(t), t) =1

2(N − 1)[H1(X(t), t) +H3(X(t), t)] , t ≥ 0. (2.22)

Differentiating (2.22) with respect to t, noticing (2.17) and (2.21), we have

dH2(X(t), t)

dt=

1

2(N − 1)

[dH1(X(t), t)

dt+dH3(X(t), t)

dt

]= 0, t ≥ 0,

which immediately implies (2.12) by noting the initial condition (1.2). ThereforeH1(X, t), H2(X, t) and H3(X, t) are three non-autonomous first integrals of theODEs (1.1).

2.3. Global existence in the case with the same winding number. Letm0 = +1 or −1 be fixed. When the N quantized vortices have the same windernumber, e.g. m0, we have

Theorem 2.1. Suppose the N vortices have the same winding number, i.e. mj =m0 for 1 ≤ j ≤ N in (1.1), then Tmax = +∞, i.e. there is no finite time collisionamong the N quantized vortices. In addition, at least two vortices move to infinityas t→ +∞.

Proof. The proof will be proceeded by the method of contradiction. Assume 0 <Tmax < +∞, i.e. there exist M (2 ≤ M ≤ N) vortices (without loss of generality,we assume here that they are x1, . . . ,xM ) that collide at a fixed point x0 ∈ R2 andthe rest N −M vortices are all away from this point. Taking t = Tmax in the leftequation in (2.11), noting (2.7), (2.8) and |N+ −N−| = N , we get

0 < H01 = H1(X(Tmax), Tmax)

= −4NM0Tmax +∑

1≤j<l≤N

|xj(Tmax)− xl(Tmax)|2

= −2N2(N − 1)Tmax +

N∑j=M+1

|xj(Tmax)− x0|2

+∑

M+1≤j<l≤N

|xj(Tmax)− xl(Tmax)|2,


which immediately implies that 2 ≤ M < N . Denote the non-empty sets I ={1, . . . ,M} and J = {M + 1, . . . , N}, and define

DI(t) =∑

1≤j<l≤M

d2jl(t), dI,J(t) = min

j∈I,l∈Jdjl(t), 0 ≤ t ≤ Tmax, (2.23)

where

djl(t) = |xj(t)− xl(t)|, t ≥ 0, 1 ≤ j < l ≤ N.Then we have

limt→T−

max

DI(t) = 0, limt→T−

max

dI,J(t) > 0,

which yields

d1 := min0≤t≤Tmax

dI,J(t) > 0.

Choose ε = Md13(N−M) > 0, then there exists a 0 < T1 < Tmax such that

0 ≤ DI(t) < ε, T1 ≤ t ≤ Tmax.

Differentiating (2.23) with respect to t, we obtain

DI(t) = 2∑

1≤j<l≤M

(xj(t)− xl(t)) · (xj(t)− xl(t))

=

M∑j,l=1,j 6=l

(xj(t)− xl(t)) · (xj(t)− xl(t))

= 2

M∑j,l=1,j 6=l

(xj(t)− xl(t)) ·

N∑k=1,k 6=j

xj(t)− xk(t)

d2jk(t)

−N∑

k=1,k 6=l

xl(t)− xk(t)

d2lk(t)

= 2

M∑j,l=1,j 6=l

(xj(t)− xl(t)) ·

M∑k=1,k 6=j

xj(t)− xk(t)

d2jk(t)

−M∑

k=1,k 6=l

xl(t)− xk(t)

d2lk(t)

+2

M∑j,l=1,j 6=l

(xj(t)− xl(t)) ·N∑

k=M+1

[xj(t)− xk(t)

d2jk(t)

−N∑

k=M+1

xl(t)− xk(t)

d2lk(t)

].

Similar to (2.14) via (2.15) (with details omitted here for brevity), we get

DI(t) = 4∑

1≤j<l≤M

[M + (xj(t)− xl(t)) ·

N∑k=M+1

(xj(t)− xk(t)

d2jk(t)

− xl(t)− xk(t)

d2lk(t)

)]

≥ 4∑

1≤j<l≤M

[M − djl(t)

N∑k=M+1

(1

djk(t)+

1

dlk(t)

)]

≥ 4∑

1≤j<l≤M

[M − 2

N∑k=M+1

ε

d1

]

= 2M(M − 1)

(M − 2(N −M)

ε

d1

)> 0, T1 ≤ t ≤ Tmax, (2.24)


which immediately implies that 0 = DI(T−max) ≥ DI(T1) > 0. This is a contra-

diction, and thus Tmax = +∞, i.e. there is no finite time collision among the Nquantized vortices.

Noticing M0 = 12N(N −1) > 0, combining (2.8) and (2.11), we get

∑Nj=1 |xj(t)|2

= limt→+∞H02 + 4M0t → +∞ when t → +∞. Hence there exists an 1 ≤ i0 ≤ N

such that |xi0(t)| → +∞ when t → +∞. Due to the conservation of mass center,

i.e. x(t) := 1N

∑Nj=1 xj(t) ≡ x(0), there exists at least another 1 ≤ j0 6= i0 ≤ N

such that |xj0(t)| → +∞ when t → +∞. Thus there exist at least two vorticesmove to infinity when t→ +∞.

Define

dmin(t) = min1≤j<l≤N

djl(t),

Dmin(t) = d2min(t), 1 ≤ j 6= l ≤ N, t ≥ 0,

Djl(t) = d2jl(t).

(2.25)

Then it is easy to see that dmin(t) and Dmin(t) are continuous and piecewise smoothfunctions. In addition, for 1 ≤ j < l ≤ N , noting (1.1), we have for t ≥ 0

Djl(t) = 2(xj(t)− xl(t)) · (xj(t)− xl(t))

= 4(xj(t)− xl(t)) ·

N∑k=1,k 6=j

xj(t)− xk(t)

d2jk(t)

−N∑

k=1,k 6=l

xl(t)− xk(t)

d2lk(t)

= 4

2 + (xj(t)− xl(t)) ·N∑

k=1,k 6=j,l

(xj(t)− xk(t)

d2jk(t)

− xl(t)− xk(t)

d2lk(t)

) . (2.26)

When the N vortices are initially collinear, we have

Theorem 2.2. Suppose the N vortices have the same winding number, i.e. mj =m0 for 1 ≤ j ≤ N in (1.1), and the initial data X0 in (1.2) is collinear, then dmin(t)and Dmin(t) are monotonically increasing functions.

Proof. Since X0 is collinear, there exist x0 ∈ R2 and a unit vector e ∈ R2 such thatX0 ∈ Se(x0), by Lemma 2.2, we know that X(t) ∈ Se(x0) for t ≥ 0. Thus thereexist aj(t) (1 ≤ j ≤ N) satisfying aj(t) 6= al(t) for 1 ≤ j < l ≤ N such that

xj(t) = x0 + aj(t)e, t ≥ 0, 1 ≤ j ≤ N. (2.27)

Taking 0 ≤ t0 < t1 such that dmin(t) is smooth on [t0, t1), without loss of generality,we assume that there exists 1 ≤ i0 ≤ N − 1 (otherwise by re-ordering) such that

a1(t) < a2(t) < . . . < ai0(t) < ai0+1(t) < . . . < aN (t),

dmin(t) = di0,i0+1(t), t0 ≤ t < t1.(2.28)

Plugging j = i0 and l = i0 + 1 into (2.26) and noting (2.28), (2.27) and (2.25), wegave

Dmin(t) = Di0,i0+1(t) = 4

2 + dmin(t)

N∑k=1,k 6=i0,i0+1

(1

di0k(t)− 1

di0+1,k(t)

)


= 4

[2−

i0−1∑k=1

d2min(t)

di0k(t)di0+1,k(t)−

N∑k=i0+2

d2min(t)

di0k(t)di0+1,k(t)

]

≥ 4

[2−

i0−1∑k=1

1

(i0 − k)(i0 + 1− k)−

N∑k=i0+2

1

(k − i0)(k − i0 − 1)

]

= 4

(1

i0+

1

N − i0

)> 0, t0 ≤ t < t1.

Here we useddjl(t)dmin(t) ≥ |j − l| for 1 ≤ j < l ≤ N by noting (2.28). Thus Dmin(t)

(and dmin(t)) is a monotonically increasing function over [t0, t1). Therefore, Dmin(t)(and dmin(t)) is a monotonically increasing function over its every piecewise smoothinterval. Due to that it is a continuous function, thus Dmin(t) (and dmin(t)) is amonotonically increasing function for t ≥ 0.

Similarly, when 2 ≤ N ≤ 4 and X0 ∈ R2×N∗ , we have

Theorem 2.3. Suppose 2 ≤ N ≤ 4 and the N vortices have the same windingnumber, i.e. mj = m0 for 1 ≤ j ≤ N in (1.1), then dmin(t) and Dmin(t) aremonotonically increasing functions.

Proof. Taking 0 ≤ t0 < t1 such that dmin(t) is smooth on [t0, t1), without lossof generality, we assume that dmin(t) = d12(t) for t0 ≤ t < t1 (otherwise by re-ordering). Taking j = 1 and l = 2 in (2.26), we get for t0 ≤ t < t1

Dmin(t) = D12(t)

= 4

2 + (x1(t)− x2(t)) ·N∑

k=1,k 6=1,2

(x1(t)− xk(t)

d21k(t)

− x2(t)− xk(t)

d22k(t)

) .When N = 2 or 3, noting 0 < d12(t) ≤ djl(t) for 1 ≤ j 6= l ≤ N , we get

Dmin(t) > 4

2−N∑

k=1,k 6=1,2

(d12(t)

d1k(t)+d12(t)

d2k(t)

)≥ 4 (2− 2(N − 2)) ≥ 0, t0 ≤ t < t1,

which implies that Dmin(t) and dmin(t) are monotonically increasing functions overt ∈ [t0, t1]. When N = 4, without loss of generality, we can assume

d12(t) ≤ d13(t) ≤ d23(t), d12(t) ≤ d14(t) ≤ d24(t), t0 ≤ t < t1,

then we get

Dmin(t) > 4

[2−

(d12(t)

d23(t)+d12(t)

d24(t)

)]≥ 0, t0 ≤ t < t1,

which implies that Dmin(t) and dmin(t) are monotonically increasing functions overt ∈ [t0, t1].

Remark 2.1. When N ≥ 5 and the initial data X0 ∈ R2×N∗ is not collinear, dmin(t)

might not be a monotonically increasing function, especially when 0 ≤ t� 1. Basedon our extensive numerical results, for any given X0 ∈ R2×N

∗ , there exits a constantT0 > 0 depending on X0 such that dmin(t) is a monotonically increasing functionwhen t ≥ T0. Rigorous mathematical justification is ongoing.


2.4. Finite time collision in the case with opposite winding numbers.When the N vortices have opposite winding numbers, we have

Theorem 2.4. Suppose the N vortices have opposite winding numbers, i.e. |N+−N−| < N , we have

(i) If M0 < 0, finite time collision happens, i.e. 0 < Tmax < +∞, and there

exists a collision cluster among the N vortices. In addition, Tmax ≤ Ta := − H01

4NM0.

(ii) If M0 = 0, then the solution of (1.1) is bounded, i.e.

|xj(t)| ≤√H0

2 =

√√√√ N∑j=1

|x0j |2, t ≥ 0, 1 ≤ j ≤ N. (2.29)

(iii) If M0 > 0 and there is no finite time collision, i.e. Tmax = +∞, then atleast two vortices move to infinity as t→ +∞.

(iv) Let I ⊆ {1, 2, . . . , N} be a set with M (2 ≤ M ≤ N) elements. If thecollective winding number of I defined as M1 := 1

2

∑j,l∈I,j 6=lmjml ≥ 0, then the

set of vortices {xj(t) | j ∈ I} cannot be a collision cluster among the N vortices for0 ≤ t ≤ Tmax.

Proof. (i) Combining (2.8) and (2.11), we get

0 ≤∑

1≤j<l≤N

|xj(t)− xl(t)|2 = 4NM0t+H01 , t ≥ 0. (2.30)

If M0 < 0, when t → Ta = − H01

4NM0, we have 4NM0t + H0

1 → 0. Thus finite timecollision happens at t = Tmax ≤ Ta < +∞.

(ii) If M0 = 0, combining (2.9) and (2.12), we get

0 ≤ |xj(t)|2 ≤N∑j=1

|xj(t)|2 ≡ H02 =

N∑j=1

|x0j |2, t ≥ 0,

which immediately implies (2.29).(iii) If M0 > 0 and Tmax = +∞, then there exists no finite time collision cluster

among the N vortices. The proof can be proceeded similarly as the last part inTheorem 2.1 and details are omitted here for brevity.

(iv) When M = N , for any given X0 ∈ R2×N∗ , we get H0

1 > 0. Noting (2.30) andM0 = M1 ≥ 0, we have∑

1≤j<l≤N

|xj(t)− xl(t)|2 = 4NM0t+H01 ≥ H0

1 > 0, 0 ≤ t ≤ Tmax,

which immediately implies that the N vortices cannot be a collision cluster whent ∈ [0, Tmax] for any given X0.

When 2 ≤ M < N and N ≥ 3, without loss of generality, we assume I ={1, . . . ,M} and denote J = {M + 1, . . . , N}. Thus M1 =

∑1≤j<l≤M mjml ≥ 0.

We will proceed the proof by the method of contradiction. Assume that the Mvortices x1, . . . ,xM collide at x0 ∈ R2 when t → Tc satisfying 0 < Tc ≤ Tmax, i.e.xj(t) → x0 when t → T−c for 1 ≤ j ≤ M and |xj(t) − x0| > 0 when t → T−c for

M + 1 ≤ j ≤ N . Denote d2 := minj∈J

min0≤t≤Tc

|xj(t)− x0|2 and we have d2 > 0. Since

limt→Tc

DI(t) = 0, there exists 0 < T1 < Tc, such that DI(t) <d22 and dI,J(t) > d2

2 for


t ∈ [T1, Tc). Choose T2 ∈ [T1, Tc), such that

0 < Tc − T2 <d2

8M(M − 1)(N −M).

Since DI(t) is a continuous function, there exists T3 ∈ [T2, Tc], such that DI(T3) =maxt∈[T2,Tc]DI(t) > 0. Similar to (2.24), we have

DI(T3) = DI(T3)−DI(Tc) = −∫ Tc

T3

d

dtDI(t)dt

= −4∫ Tc

T3

∑1≤j<l≤M

(xj(t)− xl(t)) ·

N∑k=M+1

mk

[mj

xj(t)− xk(t)

d2jk(t)−ml

xl(t)− xk(t)

d2lk(t)

]dt− 4MM1(Tc − T3)

≤ 4

∫ Tc

T3

∑1≤j<l≤M

N∑k=M+1

(djl(t)

djk(t)+

djl(t)

dlk(t)

)dt

≤ 16

∫ Tc

T3

∑1≤j<l≤M

N∑k=M+1

DI(T3)

d2dt

=8M(M − 1)(N −M)(Tc − T3)

d2DI(T3)

≤ 8M(M − 1)(N −M)(Tc − T2)

d2DI(T3)

< DI(T3).

This is a contradiction and thus the set of vortices {xj(t) | j ∈ I} cannot be acollision cluster among the N vortices for 0 ≤ t ≤ Tmax.

Proposition 2.1. If the N vortices be a collision cluster at 0 < Tmax < +∞ undera given initial data X0 ∈ R2×N

∗ , then we have

M0 < 0, H01 = NH0

2 , H03 = (N − 2)H0

2 . (2.31)

Proof. Due to the conservation of mass center and X0 ∈ R2×N∗ , we get

x(t) ≡ x0 =⇒ limt→T−

max

xj(t) = xj(Tmax) = x0, 1 ≤ j ≤ N. (2.32)

Plugging (2.32) into (2.8) and (2.11), we get

H1(X(Tmax), Tmax) =∑

1≤j<l≤N

|xj(Tmax)− xl(Tmax)|2 − 4NM0Tmax

= −4NM0Tmax = H01 . (2.33)

Similarly, we have

H2(X(Tmax), Tmax) = −4M0Tmax = H02 ,

H3(X(Tmax), Tmax) = −4(N − 2)M0Tmax = H03 .

(2.34)

Combining (2.33) and (2.34), we obtain (2.31).

Proposition 2.2. If the ODEs (1.1) admits an equilibrium solution, then N ≥ 4is a square of an integer, i.e. N = (N+ −N−)2 and

1 ≤ N+ =1

2

(N ±

√N)< N, 1 ≤ N− = N −N+ < N. (2.35)


Proof. Assume X(t) ≡ X0 ∈ R2×N∗ be an equilibrium solution of (1.1), noting (2.8)

and (2.11), we get

H1(X(t), t) = H01 − 4NM0t ≡ H0

1 , t ≥ 0. (2.36)

Thus M0 = 0. Noting (2.7), we have

4 ≤ N = (N+ −N−)2 = (2N+ −N)2 = (2N− −N)2. (2.37)

Thus N ≥ 4 is a square of an integer and we obtain (2.35) by solving (2.37).

Remark 2.2. When N = 4, an equilibrium solution of (1.1) was constructed in[30, 31] by taking m4 = −1, x0

4 = (0, 0)T and m1 = m2 = m3 = 1, x0j located in

the vertices of a right triangle centered at the origin. Here we want to remark thatany equilibrium solution of (1.1) is dynamically unstable.

3. Interaction patterns of a cluster with 3 quantized vortices. In this sec-tion, we assume N = 3 in (1.1) and (1.2).

3.1. Structural/obital stability in the case with the same winding num-ber. Assume that m1 = m2 = m3 and by Theorem 2.1, we know the ODEs (1.1)with (1.2) is globally well-posed, i.e. Tmax = +∞.

Lemma 3.1. If the initial data X0 ∈ R2×3∗ in (1.2) with N = 3 is collinear, then

one vortex moves to the mass center x0 and the other two vortices repel with eachother and move outwards to far field along the line when t→ +∞.

Proof. Since X0 ∈ R2×3∗ is collinear, there exist x0 ∈ R2 and a unit vector e ∈ R2

such thatx0j = x0 + a0

je, j = 1, 2, 3.

Without loss of generality, we assume that

a01 < a0

2 < a03, a0

1 < 0, a03 > 0, a0

1 + a02 + a0

3 = 0.

Based on the results in Lemma 2.2 and Theorem 2.1, we know that there exist aj(t)(j = 1, 2, 3) such that

xj(t) = x0 + aj(t)e, j = 1, 2, 3, (3.1)

satisfying

a1(t) < a2(t) < a3(t), a1(t) + a2(t) + a3(t) ≡ 0, t ≥ 0. (3.2)

Plugging (3.1) into (1.1) with N = 3 and m1 = m2 = m3, noting (3.2), we get

a1(t) = − 2

a2(t)− a1(t)− 2

a3(t)− a1(t)=

6a1(t)

[a2(t)− a1(t)][a3(t)− a1(t)]< 0,

a2(t) =2

a2(t)− a1(t)− 2

a3(t)− a2(t)=

−6a2(t)

[a2(t)− a1(t)][a3(t)− a2(t)], t > 0,

a3(t) =2

a3(t)− a1(t)+

2

a3(t)− a2(t)=

6a3(t)

[a3(t)− a1(t)][a3(t)− a2(t)]> 0,

with the initial dataaj(0) = a0

j , j = 1, 2, 3. (3.3)

Thus a1(t) is a monotonically decreasing function and a3(t) is a monotonicallyincreasing function for t ≥ 0. Let ρ2(t) = a2

2(t) ≥ 0, then we have

ρ2(t) =−12ρ2(t)

[a2(t)− a1(t)][a3(t)− a2(t)]< 0, t > 0,


which immediately implies that ρ2(t) is a monotonically decreasing function andlimt→+∞ ρ2(t) = 0. Thus we have

limt→+∞

a2(t) = 0 =⇒ limt→+∞

x2(t) = x0 =1

3

3∑j=1

x0j .

Thus the vortex x2(t) moves towards x0 along the line Se(x0). Based on the resultsin Theorem 2.1, we know that at least two vortices must move to infinity whent→ +∞. Thus we have

a1(t)→ −∞, a3(t)→ +∞ when t→ +∞.

Thus the other two vortices x1(t) and x3(t) repel with each other and move outwardsto far field along the line Se(x0) when t→ +∞.

Theorem 3.1. Assume the initial data X0 ∈ R2×3∗ in (1.2) with N = 3 is not

collinear, then there exists a unit vector e ∈ R2 such that

limt→+∞

dS(t) := infX∈S3

e(x0)‖X(t)−X‖2 = 0. (3.4)

Proof. Without loss of generality, as shown in Fig. 3.1a, we assume x0 = 0 andd0

12 := d12(0) ≤ d013 := d13(0) ≤ d0

23 := d23(0). Thus 0 < θ03 := θ3(0) ≤ θ0

2 :=θ2(0) ≤ θ0

1 := θ1(0) < π satisfying θ01 + θ0

2 + θ03 = π (cf. Fig. 3.1a). From (1.1) with

N = 3, we get

d12(t) =4

d12(t)+

2 cos(θ1(t))

d13(t)+

2 cos(θ2(t))

d23(t),

d13(t) =4

d13(t)+

2 cos(θ1(t))

d12(t)+

2 cos(θ3(t))

d23(t), t > 0,

d23(t) =4

d23(t)+

2 cos(θ2(t))

d12(t)+

2 cos(θ3(t))

d13(t),

(3.5)

and

θ3(t) = B(t)[d2

13(t) + d223(t)− 2d2

12(t)],

θ2(t) = B(t)[d2

12(t) + d223(t)− 2d2

13(t)], t > 0,

θ1(t) = B(t)[d2

12(t) + d213(t)− 2d2

23(t)],

(3.6)

whereB(t) := 4A(t)/(d12(t)d13(t)d23(t))2 withA(t) denoting the area of the trianglewith vertices x1(t), x2(t) and x3(t). Denote

ρ12(t) = d212(t), ρ13(t) = d2

13(t), ρ23(t) = d223(t), t ≥ 0.

From (3.5)-(3.6) and noting the initial data, we get π3 ≤ θ1(t) < π and 0 < θ3(t) ≤ π

3are monotonically decreasing and increasing functions, respectively, and

ρ12(t) ≤ ρ23(t), 0 < θ3(t) ≤ π

3≤ θ1(t) < π, t ≥ 0;

limt→+∞

θ3(t) = limt→+∞

θ1(t) =π

3.

(3.7)

Combining this with θ1(t) + θ2(t) + θ3(t) ≡ π for t ≥ 0, we get limt→+∞ θ2(t) = π3 ,

which immediately implies (3.4).

For θ ∈ R and X = (x1, . . . ,xN ) ∈ SNe (0, θ0), define Q(θ)X := (Q(θ)x1, . . . ,Q(θ)xN ).


θ2(t)

θ3(t)θ1(t)x1(t) x3(t)

x2(t)

d 12(t)

d13(t)

d23 (t)

(a)

θ1(t)

x1(t)x2(t)

x3(t)

θ2(t)

θ3(t)

φ1(t)

φ2(t)

r1(t)r 2

(t)

r 3(t)

O

(b)

θ2(t)

θ3(t)θ1(t)x1(t) x3(t)

x2(t)

d 12(t)

d13(t)

d23 (t)

(c)

Figure 3.1. Interaction of 3 vortices with the same winding num-ber (a and b) and opposite winding numbers (c).

Definition 3.1. For the self-similar solution X(t) =√r20 + 2(N − 1)t X0 with

X0 = (x01, . . . , x

0N ) ∈ SNe (0, θ0) and r0 = |x0

1| of the ODEs (1.1) with m1 = . . . =mN , if for any ε > 0, there exists δ > 0 such that, when the initial data X0 in (1.2)

satisfies ‖X0 − X0‖2 < δ, the solution X(t) of the ODEs (1.1) with (1.2) satisfies

supt≥0

infr>0, θ∈[0,2π)

∥∥∥X(t)− x0 − rQ(θ)X(t)∥∥∥ < ε,

then the self-similar solution X(t) is called as orbitally stable.

Theorem 3.2. For any θ0 ∈ R and X0 = (x01, x

02, x

03) ∈ S3

e(0, θ0), the solution

X(t) =√

4t+ r20 X0 with r0 = |x0

1| of the ODEs (1.1) with N = 3 and m1 = m2 =m3 is orbitally stable.

Proof. By using Lemma 2.1, without loss of generality, we can assume that θ0 = 0,r0 = 1 and x0 = 0. In addition, as shown in Fig. 3.1b, we assume

xj(t) = rj(t)(cos(θj(t)), sin(θj(t))T , j = 1, 2, 3, (3.8)

satisfying θ01 := θ1(0) < θ0

2 := θ2(0) < θ03 := θ3(0) < 2π and ‖X0 − X0‖2 ≤ δ with

0 < δ ≤ 112 sufficiently small and to be determined later. In fact, from


‖X0 − X0‖22 =

3∑j=1

(rj(0)− 1)2 +

3∑j=1

4rj(0) sin2(ϕ0j ), (3.9)

with ϕ0j =

θ0j2 −

(j−1)π3 for j = 1, 2, 3, we can get

|r1(0)− 1|+ |r2(0)− 1|+ |r3(0)− 1| ≤ 3δ <1

2,

|θ01|+

∣∣∣∣θ02 −

2π

3

∣∣∣∣+

∣∣∣∣θ03 −

4π

3

∣∣∣∣ ≤ 6δ <π

4.

(3.10)

Plugging (3.8) into (1.6), we get

r1(t) cos(θ1(t)) + r2(t) cos(θ2(t)) + r3(t) cos(θ3(t)) ≡ 0,

r1(t) sin(θ1(t)) + r2(t) sin(θ2(t)) + r3(t) sin(θ3(t)) ≡ 0, t ≥ 0.

Solving the above equations, we obtain

r2(t) = −r1(t)sin(θ3(t)− θ1(t))

sin(θ3(t)− θ2(t))= −r1(t)

sin (φ1(t) + φ2(t))

sin (φ2(t)), (3.11)

r3(t) = r1(t)sin(θ2(t)− θ1(t))

sin(θ3(t)− θ2(t))= r1(t)

sin (φ1(t))

sin (φ2(t)), t ≥ 0, (3.12)

where (cf. Fig. 3.1b)

φ1(t) = θ2(t)− θ1(t), φ2(t) = θ3(t)− θ2(t), t ≥ 0. (3.13)

By Lemma 2.4, we have

r21(t) + r2

2(t) + r23(t) = 12t+H0

3 , t ≥ 0, (3.14)

with H03 = r2

1(0) + r22(0) + r2

3(0). Substituting (3.11) and (3.12) into (3.14), we get

r1(t) =(12t+H0

3 )1/2 sin (φ2(t))

D1/2(t),

D(t) := sin2 (φ1(t)) + sin2 (φ2(t)) + sin2 (φ1(t) + φ2(t)) .

(3.15)

Plugging (3.8) into (1.1) with N = 3, noting (3.11)-(3.13) and (3.15), we have

Φ(t) =2

12t+H03

F (φ1(t), φ2(t)) =2

12t+H03

F (Φ(t)), t > 0, (3.16)

where Φ(t) := (φ1(t), φ2(t))T and F (Φ) = (f1(Φ), f2(Φ))T is defined as

f1(Φ) =sin (φ1)

sin (φ2) sin (φ1 + φ2)

[sin2 (φ1 + φ2)

D13(Φ)+

sin2 (φ2)

D23(Φ)

− sin2 (φ2) + sin2 (φ1 + φ2)

D12(Φ)

],

f2(Φ) =sin (φ2)

sin (φ1) sin (φ1 + φ2)

[sin2 (φ1 + φ2)

D13(Φ)+

sin2 (φ1)

D12(Φ)

− sin2 (φ1) + sin2 (φ1 + φ2)

D23(Φ)

],


with

D12(Φ) =1

D(Φ)

(sin2 (φ2) + sin2 (φ1 + φ2) + 2 sin (φ2) sin (φ1 + φ2) cos (φ1)

),

D13(Φ) =1

D(Φ)

(sin2 (φ1) + sin2 (φ2)− 2 sin (φ1) sin (φ2) cos (φ1 + φ2)

),

D23(Φ) =1

D(Φ)

(sin2 (φ1) + sin2 (φ1 + φ2) + 2 sin (φ1) sin (φ1 + φ2) cos (φ2)

),

P (Φ) =1

D(Φ)

[sin (φ2)− sin (φ1 + φ2) cos

(φ1 −

2π

3

)+ sin (φ1) cos

(φ1 + φ2 −

4π

3

)]2

;

and

θ1(t) =2

12t+H03

g(Φ(t)), t > 0, (3.17)

with

g(Φ) = g(φ1, φ2) =sin (φ1) sin (φ1 + φ2) (D13(Φ)−D12(Φ))

sin (φ2)D12(Φ)D13(Φ).

Let

s =1

4ln

(12t+H0

3

H03

), Ψ(s) = Φ(t)− (2π/3, 2π/3)T , s ≥ 0, (3.18)

then (3.16) can be re-written as

Ψ(s) = F(Ψ(s) + (2π/3, 2π/3)T

)= −2Ψ(s) +G(Ψ(s)), s > 0, (3.19)

where

G(Ψ) = F(Ψ + (2π/3, 2π/3)T

)+ 2Ψ.

It is easy to verify that Ψ(s) ≡ 0 is an equilibrium solution of (3.19). By thevariation-of-constant formula, we have

Ψ(s) = e−2sΨ(0) +

∫ s

0

e−2(s−τ)G(Ψ(τ))dτ, s ≥ 0. (3.20)

By using the Taylor expansion, there exist constants Kj > 0 (j = 1, 2, 3) and0 < δ1 < 1 such that

‖G(Ψ)‖2 ≤ ‖Ψ‖2, ‖G(Ψ)‖2 ≤ K1‖Ψ‖22,|g(Φ)| = |g(Ψ + (2π/3, 2π/3)T )| ≤ K2‖Ψ‖2,|3− P (Φ)| =

∣∣3− P (Ψ + (2π/3, 2π/3)T )∣∣ ≤ K3‖Ψ‖22, when ‖Ψ‖2 < δ1.

(3.21)

For any 0 < δ2 ≤ δ1, when ‖Ψ(0)‖2 ≤ δ22 (⇔ ‖Φ(0) − (2π/3, 2π/3)T ‖2 ≤ δ2

2 ) andlet S > 0 such that ‖Ψ(s)‖2 ≤ δ2 for 0 ≤ s ≤ S, noting (3.20) and (3.21) and usingthe triangle inequality, we have

‖Ψ(s)‖2 ≤ e−2s‖Ψ(0)‖2 +

∫ s

0

e−2(s−τ)‖Ψ(τ)‖2 dτ, 0 ≤ s ≤ S,

which is equivalent to

e2s‖Ψ(s)‖2 = ‖Ψ0‖2 +

∫ s

0

e2τ‖Ψ(τ)‖2 dτ, 0 ≤ s ≤ S.


Using the Gronwall’s inequality, we get

‖Ψ(s)‖2 ≤ ‖Ψ(0)‖2 e−s ≤ ‖Ψ(0)‖2 ≤δ22, 0 ≤ s ≤ S. (3.22)

From (3.22) and using the standard extension theorem for ODEs, we can obtain

‖Ψ(s)‖2 ≤ ‖Ψ(0)‖2 e−s, 0 ≤ s < +∞. (3.23)

Combining (3.23) and (3.20), using the triangle inequality, we obtain

‖Ψ(s)‖2 ≤ e−2s‖Ψ(0)‖2 + e−2s

∫ s

0

e2τ‖G(Ψ(τ))‖2 dτ

≤ e−2s‖Ψ(0)‖2 + e−2s

∫ s

0

e2τK1‖Ψ(τ)‖22 dτ

≤ e−2s‖Ψ(0)‖2 + e−2s

∫ s

0

e2τK1‖Ψ(0)‖22e−2τ dτ

≤[‖Ψ(0)‖2 +K1‖Ψ(0)‖22

]e−2s

≤ (1 +K1)δ2e−2s, 0 ≤ s < +∞,

which immediately implies

‖Φ(t)− (2π/3, 2π/3)T ‖2 < (1 +K1)δ2

√H0

3

12t+H03

, 0 ≤ t < +∞.

Noting (3.18) and (3.21), we have

|g(Φ)| = |g(Ψ + (π/3, π/3)T )| ≤ K2‖Ψ‖2

≤ K2(1 +K1)δ2

√H0

3

12t+H03

, 0 ≤ t < +∞.

This implies that the ODE (3.17) is globally solvable, and the solution can bewritten as

θ1(t) = θ1(0) +

∫ t

0

3

12s+H03

g(φ1(s), φ2(s)) ds, 0 ≤ t ≤ +∞.

Denote θ∞1 = limt→∞ θ1(t) and θ(t) = θ1(t)− θ∞1 , then we have

infr>0‖X(t)− rQ(θ(t))X(t)‖2 = inf

r>0

{12t+H0

3 + 3r2 − 2r d(t)}

= 12t+H03 −

1

3d2(t)

=12t+H0

3

3(3− P (Φ(t))) , t ≥ 0, (3.24)

where

d(t) := r1(t) + r2(t) cos(φ1(t)− 2π/3) + r3(t) cos(φ1(t) + φ2(t)− 4π/3).

Noting (3.21), we have

|3− P (Φ(t))| =∣∣3− P (Ψ(t) + (2π/3, 2π/3)T )

∣∣ ≤ K3‖Ψ(t)‖22

≤ K3(1 +K1)2δ2H03

12t+H03

, t ≥ 0. (3.25)


For any ε > 0, taking δ3 = ε2K3(1+K1)2H0

3and 0 < δ = min

{112 ,

δ112 , δ3

}, when

‖X0 − X0‖2 < δ, noting (3.25) and (3.24), we get

supt≥0

infr>0‖X(t)− rQ(θ(t))X(t)‖2 ≤ K3(1 +K1)2H0

3δ < ε, (3.26)

which completes the proof by taking δ2 = δ in the above proof.

3.2. Collision patterns in the case with opposite winding numbers. With-out loss of generality, we assume m1 = m3 = +1 and m2 = −1 in (1.1) with N = 3.Then we have M0 = 1

2 [(N+ − N−)2 − N ] = 12 (12 − 3) = −1 < 0, thus finite time

collision must happen.

Theorem 3.3. For any given initial data X0 ∈ R2×3∗ in (1.2) with N = 3, we have

(i) If |x01 − x0

2| = |x02 − x0

3|, then the three vortices be a collision cluster and they

will collide at x0 when t→ T−max =H0

1

12 with H01 =

∑1≤j<l≤3 |x0

j − x0l |2.

(ii) If |x01 − x0

2| < |x02 − x0

3|, then only x1 and x2 form a collision cluster, andrespectively, if |x0

1 − x02| > |x0

2 − x03|, then only x2 and x3 form a collision cluster.

Moreover, the collision time 0 < Tmax <H0

1

12 .

Proof. (i) If the initial data X0 ∈ R2×3∗ in (1.2) with N = 3 is collinear, i.e. there

exists a unit vector e ∈ R2 such that

x0j = x0

2 + a0je, j = 1, 2, 3,

satisfying a02 = 0, a0

1 < a03 and 0 < |a0

1| ≤ |a03| (without loss of generality, otherwise

we need only switch x1 and x3). Based on the results in Lemma 2.2 and Theorem2.1, we know that there exist aj(t) (j = 1, 2, 3) such that

xj(t) = x02 + aj(t)e, j = 1, 2, 3,

satisfying a1(t) + a2(t) + a3(t) ≡ a01 + a0

2 + a03 and a1(t) < a3(t) for 0 ≤ t < Tmax

and

a1(t) = − 2

a1(t)− a2(t)+

2

a1(t)− a3(t)=

2[a3(t)− a2(t)]

[a2(t)− a1(t)][a3(t)− a1(t)],

a2(t) = − 2

a2(t)− a1(t)− 2

a2(t)− a3(t)=

2[3a2(t)− (a01 + a0

2 + a03)]

[a2(t)− a1(t)][a3(t)− a2(t)], t > 0,

a3(t) =2

a3(t)− a1(t)− 2

a3(t)− a2(t)=

2[a1(t)− a2(t)]

[a3(t)− a1(t)][a3(t)− a2(t)],

with the initial data (3.3).If |a0

1| = |a03|, i.e. a0

3 = −a01 > 0, then the above ODEs with (3.3) admits the

unique solution as

a1(t) = −√

(a01)2 − 2t,

a2(t) ≡ 0, 0 ≤ t < Tmax :=1

2(a0

3)2,

a3(t) =√

(a03)2 − 2t,

(3.27)

which immediately implies that the three vortices be a collision cluster and they

collide at x0 = x02 when t→ T−max =

H01

12 with H01 =

∑1≤j<l≤3 |x0

j − x0l |2 = 6(a0

3)2.

If |a01| < |a0

3|, then a03 > 0. If 0 = a0

2 < a01 < a0

3, then we can show thata2(t) < a1(t) < a3(t) for 0 ≤ t < Tmax and a1(t), a2(t) and a3(t) are monotonically


decreasing, increasing and increasing functions over t ∈ [0, Tmax), respectively. Thusonly x1 and x2 form a collision cluster among the 3 vortices. On the other hand,if a0

1 < 0 = a02 < a0

3, then we can show that a1(t) < a2(t) < a3(t) for 0 ≤ t < Tmax

and a1(t) and a2(t) are monotonically increasing and decreasing functions overt ∈ [0, Tmax), respectively. In addition we have a1(Tmax) ≤ a2(Tmax) < 0 anda3(Tmax) = a0

3 + a02 + a0

1 − a1(Tmax) − a2(Tmax) > 0, therefore, again only x1 andx2 form a collision cluster among the 3 vortices.

(ii) If the initial data X0 ∈ R2×3∗ in (1.2) with N = 3 is not collinear, i.e. the

initial locations of the 3 vortices form a triangle. Without loss of generality, asshown in Fig. 3.1c, we assume x0 = 0 and d0

12 := d12(0) ≤ d023 := d23(0). Thus

0 < θ03 := θ3(0) ≤ θ0

1 := θ1(0) < π satisfying θ01 + θ0

2 + θ03 = π (cf. Fig. 3.1b). From

(1.1) with N = 3, we get

d12(t) = − 4

d12(t)+

2 cos(θ1(t))

d13(t)− 2 cos(θ2(t))

d23(t),

d13(t) =4

d13(t)− 2 cos(θ1(t))

d12(t)− 2 cos(θ3(t))

d23(t), t > 0,

d23(t) = − 4

d23(t)− 2 cos(θ2(t))

d12(t)+

2 cos(θ3(t))

d13(t),

(3.28)

and

θ3(t) = −B(t)[d2

13(t) + d223(t)

]< 0,

θ2(t) = B(t)[d2

12(t) + d223(t) + 2d2

13(t)]> 0, t > 0,

θ1(t) = −B(t)[d2

12(t) + d213(t)

]< 0.

(3.29)

If d012 = d0

23, then 0 < θ03 = θ0

1 <π2 (cf. Fig. 3.1c), this together with (3.28)-(3.29)

yields

d12(t) = d23(t), 0 < θ3(t) = θ1(t) <π

2, 0 ≤ t < Tmax;

limt→T−

max

θ3(t) = limt→T−

max

θ1(t) = 0,(3.30)

which immediately implies that the three vortices are forming a collision cluster.By using Theorem 2.4, we get Tmax = H0

1/12.If 0 < d0

12 < d023, then 0 < θ0

3 < θ01 <

π2 (cf. Fig. 3.1b). From (3.28) and (3.29),

we have

d23(t)− d12(t) =[4− 2 cos(θ2(t))](d23(t)− d12(t))

d12(t)d23(t)

+2[cos(θ3(t))− cos(θ1(t))]

d13(t)> 0, t > 0, (3.31)

θ1(t)− θ3(t) = B(t)[d2

23(t)− d212(t)

]> 0.

Then we have

d23(t) ≥ d12(t) + d023 − d0

12, θ1(t) ≥ θ3(t) + θ01 − θ0

3, 0 ≤ t ≤ Tmax.

This, together with that θ1(t) and θ3(t) are monotonically decreasing functions,0 < θ2(t) = π − θ1(t)− θ3(t) < π is a monotonically increasing functions and finitetime collision must happen (cf. Fig. 3.1c), we get that limt→T−

maxd12(t) = 0 and

limt→T−max

θ3(t) = 0. Thus only the two vortices x1(t) and x2(t) form a collision


cluster among the 3 vortices. By using Theorem 2.4, we get the collision time

0 < Tmax <H0

1

12 .

4. Analytical solutions under special initial setups. Let 0 ≤ θ0 < 2π be aconstant, n ≥ 2 be an integer, 0 < a1 < a2 be two constants, C1 := 1

2

(a2

1 + a22

),

C2 := 12

(a2

2 − a21

), and m0 = +1 or −1. Denote

θjn =2(j − 1)π

n+ θ0, αjn =

2(j − 1)π

n+π

n+ θ0, 1 ≤ j ≤ n.

4.1. For the interaction of two clusters. Here we take N = 2n with n ≥ 2.

Proposition 4.1. Taking mj = m0 for 1 ≤ j ≤ N = 2n and the initial data X0 in(1.2) as

x0j = a1

(cos(θjn), sin(θjn)

)T, x0

n+j = a2


)T, 1 ≤ j ≤ n, (4.1)

then the solution of the ODEs (1.1) with (4.1) can be given as

xj(t) =√ρ1(t)


)T,

xn+j(t) =√ρ2(t)


)T, 1 ≤ j ≤ n, t ≥ 0,

(4.2)

where when n = 2,

ρ1(t) = C1 + 6t−√C2

2 + 8C1t+ 24t2,

ρ2(t) = C1 + 6t+√C2

2 + 8C1t+ 24t2, t ≥ 0;(4.3)

and when n ≥ 3,

ρ1(t) ∼ α1t, ρ2(t) ∼ β1t, t� 1, (4.4)

with α1 and β1 being two positive constants satisfying

0 < α1 < β1, α1 + β1 = 8n− 4, β1 − α1 = 4nβn/21 + α

n/21

βn/21 − αn/21

. (4.5)

Specifically, when n� 1, we have

α1 ≈ 2n− 2, β1 ≈ 6n− 2. (4.6)

Proof. Noting the symmetry of the ODEs (1.1) with the initial data (4.1), we cantake the solution ansatz (4.2). Substituting (4.2) into (1.1) and (1.2), we obtain

ρ1(t) = 4

n∑j=2

n(θ1n) ·

(n(θ1

n)− n(θjn))∣∣∣n(θ1

n)− n(θjn)∣∣∣2

+4

n∑j=1

n(θ1n) ·

(ρ1(t)n(θ1

n)−√ρ1(t)ρ2(t)n(θjn)

)∣∣∣√ρ1(t)n(θ1

n)−√ρ2(t)n(θjn)

∣∣∣2= 2n− 2 + 4

n∑j=1

ρ1(t)−√ρ1(t)ρ2(t) cos(θ1

n − θjn)

ρ1(t) + ρ2(t)− 2√ρ1(t)ρ2(t) cos(θ1

n − θjn), t > 0, (4.7)


ρ2(t) = 4

n∑j=1

n(θ1n) ·

(ρ2(t)n(θ1

n)−√ρ1(t)ρ2(t)n(θjn)

)∣∣∣√ρ2(t)n(θ1

n)−√ρ1(t)n(θjn)

∣∣∣2+4

n∑j=2

n(θ1n) ·

(n(θ1

n)− n(θjn))∣∣∣n(θ1

n)− n(θjn)∣∣∣2

= 2n− 2 + 4

n∑l=1


n − θjn)

ρ1(t) + ρ2(t)− 2√ρ1(t)ρ2(t) cos(θ1

n − θjn), t > 0, (4.8)

where

n(θ) = (cos(θ), sin(θ))T , θ ∈ R. (4.9)

Summing (4.7) and (4.8), we have

ρ1(t) + ρ2(t) = 8n− 4, t > 0, (4.10)

Subtracting (4.7) from (4.8), we get

ρ2(t)− ρ1(t) = 4nρn/22 (t) + ρ

n/21 (t)

ρn/22 (t)− ρn/21 (t)

= 4n+8nρ

n/21 (t)

ρn/22 (t)− ρn/21 (t)

, t > 0. (4.11)

Here we use the equalityn∑j=1

x2 − 1

x2 + 1− 2x cos(θ1n − θ

jn)

= nxn + 1

xn − 1, 1 < x ∈ R.

Combining (4.10) and (4.11), we obtain

ρ1(t) = 2n− 2− 4nρn/21 (t)

ρn/22 (t)− ρn/21 (t)

,

ρ2(t) = 6n− 2 +4nρ

n/21 (t)

ρn/22 (t)− ρn/21 (t)

, t ≥ 0,

(4.12)

with the initial data

ρ1(0) = ρ01 := a2

1 < ρ2(0) = ρ02 := a2

2. (4.13)

When n = 2, we can solve (4.12) with (4.13) analytically and obtain the solution(4.3) immediately. When n ≥ 3, noting that all the vortices have the same windingnumber, i.e. Tmax = +∞ by using Theorem 2.1, we get ρ1(t) < ρ2(t) for t ≥ 0 andthus

ρ2(t) > 0, ρ2(t)− ρ1(t) > 0, t ≥ 0.

Therefore, we conclude that ρ2(t) and ρ2(t) − ρ1(t) are monotonically increasingfunctions when t ≥ 0 and lim

t→+∞ρ2(t) = +∞ by noting Theorem 2.4. From (4.12),

we can conclude that there exist two positive constants 0 < α1 < β1 such that (4.4)is valid. Plugging (4.4) into (4.10), we get (4.5) immediately. When n � 1, (4.5)yields

α1 + β1 = 8n− 4, β1 − α1 ≈ 4n,

which immediately implies (4.6). In addition, Figure 4.1 depicts the solution ρ1(t)and ρ2(t) of (4.12) obtained numerically with ρ1(0) = 1 and ρ2(0) = 4 for differentn ≥ 2.


0 1 2 3 4 5 61

10

20

30

40

ρ 1(t)

t

n=2

n=3

n=4

n=5

0 0.1 0.2

1

2

0 1 2 3 4 5 64

50

100

150

ρ 2(t)

t

n=2

n=3

n=4

n=5

0.1 0.2 0.36

8

10

12

Figure 4.1. Time evolution of ρ1(t) (left) and ρ2(t) (right) of(4.12) with ρ0

1 = 1 and ρ02 = 4 for different n ≥ 2.

Proposition 4.2. Taking mj = m0 for 1 ≤ j ≤ N = 2n and the initial data X0 in(1.2) as

x0j = a1


)T, x0

n+j = a2

(cos(αjn), sin(αjn)

)T, 1 ≤ j ≤ n, (4.14)


xj(t) =√ρ1(t)


)T,

xn+j(t) =√ρ2(t)


)T, 1 ≤ j ≤ n, t ≥ 0,

(4.15)

where when n = 2,

ρ1(t) = C1 + 6t− C2

(1 +

6t

C1

)2/3

,

ρ2(t) = C1 + 6t+ C2

(1 +

6t

C1

)2/3

, t ≥ 0;

and when n ≥ 3,

ρ1(t) ∼ α2t, ρ2(t) ∼ β2t, t� 1,


0 < α2 < β2, α2 + β2 = 8n− 4, β2 − α2 = 4nβn/22 − αn/22

βn/22 + α

n/22

.

Specifically, when n� 1, α2 ≈ 2n− 2 and β2 ≈ 6n− 2.

Proof. The proof is analogue to that of Proposition 4.1 and thus it is omitted herefor brevity.

Proposition 4.3. Taking mj = m0 and mn+j = −m0 for 1 ≤ j ≤ n and the initialdata X0 in (1.2) as (4.14), then the solution of the ODEs (1.1) with (4.14) can begiven as (4.15), where

ρ1(t) > 0, ρ2(t) > 0, 0 ≤ t < Tc :=1

4(a2

1+a22), lim

t→T−c

ρ1(t) = limt→T−

c

ρ2(t) = 0,

which implies that the N = 2n vortices will be a (finite time) collision cluster.


Proof. Similar to the proof of Proposition 4.1, noting the symmetry of the ODEs(1.1) with the initial data (4.14), we can take the solution ansatz (4.15). In addition,plugging (4.15) into (1.1) and (1.2), we get

ρ1(t) = 2n− 2− 4

n∑l=1


n − αln)

ρ1(t) + ρ2(t)− 2√ρ1(t)ρ2(t) cos(θ1

n − αln), (4.16)

ρ2(t) = 2n− 2− 4

n∑l=1

ρ2(t)−√ρ1(t)ρ2(t) cos(α1

n − θln)

ρ1(t) + ρ2(t)− 2√ρ1(t)ρ2(t) cos(α1

n − θln), (4.17)

with the initial data (4.13).Summing (4.16) and (4.17), we obtain

ρ1(t) + ρ2(t) = 4n− 4− 4

n∑l=1

1 = 4n− 4− 4n = −4, t ≥ 0. (4.18)

Subtracting (4.16) from (4.17), we get

ρ2(t)− ρ1(t) = −4nρn/22 (t)− ρn/21 (t)

ρn/22 (t) + ρ

n/21 (t)

= −4n+8nρ

n/21 (t)

ρn/22 (t) + ρ

n/21 (t)

, t > 0. (4.19)

Here we use the equality

n∑j=1

x2 − 1

x2 + 1− 2x cos(θ1n − θ

jn + π

n )= n

xn − 1

xn + 1, 1 < x ∈ R.

Combining (4.18) and (4.19), we obtain

ρ1(t) = 2n− 2− 4nρn/21 (t)

ρn/22 (t) + ρ

n/21 (t)

,

ρ2(t) = −2n− 2 +4nρ

n/21 (t)

ρn/22 (t) + ρ

n/21 (t)

, t ≥ 0,

(4.20)

with the initial data (4.13).Solving (4.18) by noting (4.13), we get

ρ1(t) + ρ2(t) = −4t+ a21 + a2

2, 0 ≤ t < Tc :=1

4(a2

1 + a22).

Noticing N+ = N− = n = N2 , thus M0 = −N2 = −n < 0 by noting (2.7). From

Theorem 2.4, finite time collision must happen among the N = 2n vortices. Thusthere exist 1 ≤ j0 ≤ n and 1 ≤ l0 ≤ n such that the vortex dipole xj0 and xn+l0

will collide at t = Tc, i.e. ρ1(Tc) = ρ2(Tc) = 0. Therefore, the N = 2n vortices willbe a (finite time) collision cluster. In addition, Figure 4.2 depicts the solution ρ1(t)and ρ2(t) of (4.20) obtained numerically with ρ1(0) = 1 and ρ2(0) = 4 for differentn ≥ 2.

Remark 4.1. When a1 = a2, i.e. ρ01 = ρ0

2, we can get

ρ1(t) = ρ2(t) = −2t+ a21,

which also implies the N = 2n vortices will be a (finite time) collision cluster.


0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2ρ 1(t

)

t

n= 2

n= 3

n= 4

n= 5

0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

ρ 2(t)

t

n= 2

n= 3

n= 4

n= 5



4.2. For the interaction of two clusters and a single vortex. Here we takeN = 2n+ 1 with n ≥ 2.

Proposition 4.4. Taking mj = m0 for 1 ≤ j ≤ N = 2n + 1 and the initial dataX0 in (1.2) as

x0N = 0, x0

j = a1


)T,

x0n+j = a2


)T, 1 ≤ j ≤ n,

(4.21)


xN (t) ≡ 0, xj(t) =√ρ1(t)


)T,

xn+j(t) =√ρ2(t)


)T, 1 ≤ j ≤ n,

(4.22)

where when n = 2,

ρ1(t) = C1 + 10t−√C2

2 + 8C1t+ 40t2,

ρ2(t) = C1 + 10t+√C2

2 + 8C1t+ 40t2, t ≥ 0;

and when n ≥ 3,ρ1(t) ∼ α3t, ρ2(t) ∼ β3t, t� 1,


0 < α3 < β3, α3 + β3 = 8n+ 4, β3 − α3 = 4nβn/23 + α

n/23

βn/23 − αn/23

.

Specifically, when n� 1, we have α3 ≈ 2n+ 2, and β3 ≈ 6n+ 2.

Proof. Due to symmetry, we get xN (t) ≡ 0 for t ≥ 0. The rest of the proof isanalogue to that of Proposition 4.1 and thus it is omitted here for brevity.

Proposition 4.5. Taking mj = m0 for 1 ≤ j ≤ N = 2n + 1 and the initial dataX0 in (1.2) as

x0N = 0, x0

j = a1


)T,

x0n+j = a2


)T, 1 ≤ j ≤ n,

(4.23)



xN (t) ≡ 0, xj(t) =√ρ1(t)


)T,

xn+j(t) =√ρ2(t)


)T, 1 ≤ j ≤ n,

(4.24)

where when n = 2,

ρ1(t) = C1 + 10t− C2

(1 +

10t

C1

)2/5

,

ρ2(t) = C1 + 10t+ C2

(1 +

10t

C1

)2/5

, t ≥ 0;

and when n ≥ 3,ρ1(t) ∼ α4t, ρ2(t) ∼ β4t, t� 1,


0 < α4 < β4, α4 + β4 = 8n+ 4, β4 − α4 = 4nβn/24 − αn/24

βn/24 + α

n/24

.

Specifically, when n� 1, α4 ≈ 2n+ 2 and β4 ≈ 6n+ 2.

Proof. Due to symmetry, we get xN (t) ≡ 0 for t ≥ 0, and the rest of the proof isanalogue to that of Proposition 4.1 and 4.3, thus it is omitted here for brevity.

Proposition 4.6. Taking mN = −m0, mj = m0 for 1 ≤ j ≤ 2n = N − 1 and theinitial data X0 in (1.2) as (4.21), then the solution of the ODEs (1.1) with (4.21)can be given as (4.22), where

(i) when n = 2, then x1(t), x2(t) and x5(t) be a collision cluster among the 5vortices and they will collide at the origin (0, 0)T in finite time;

(ii) when n = 3, then

ρ1(t) ∼(

a1a2

a1 + a2

)2

, ρ2(t) ∼ 12t, t� 1; (4.25)

(iii) when n ≥ 4, then

ρ1(t) ∼ α5t, ρ2(t) ∼ β5t, t� 1,


0 < α5 < β5, α5 + β5 = 8n− 12, β5 − α5 = 4nβn/25 + α

n/25

βn/25 − αn/25

.

Specifically, when n� 1, we have α5 ≈ 2n− 6, and β5 ≈ 6n− 6.

Proof. Similar to the proof of Proposition 4.1 and 4.3, the solution of the ODEs(1.1) with (4.21) can be given as (4.22), where

ρ1(t) + ρ2(t) = 8n− 12, ρ2(t)− ρ1(t) = 4n+8nρ

n/21 (t)

ρn/22 (t)− ρn/21 (t)

, t > 0,

which implies

ρ1(t) = 2n− 6− 4nρn/21 (t)

ρn/22 (t)− ρn/21 (t)

,

ρ2(t) = 6n− 6 +4nρ

n/21 (t)

ρn/22 (t)− ρn/21 (t)

, t > 0.

(4.26)


1) When n = 2, (4.26) reduces to

ρ1(t) = −2− 8ρ1(t)

ρ2(t)− ρ1(t), ρ2(t) = 6 +

8ρ1(t)

ρ2(t)− ρ1(t), t > 0. (4.27)

Solving (4.27) with the initial data (4.13), we get

ρ1(t) = 2t+ C1 −√

8t2 + 8C1t+ C22 ,

ρ2(t) = 2t+ C1 +√

8t2 + 8C1t+ C22 , t ≥ 0.

Thus there exists a Tc := 12

[−C1 +

√C2

1 + 2a21a

22

]> 0, such that

ρ1(Tc) = 0, ρ2(Tc) > 0, ρ1(t) > 0, ρ2(t) > 0, t ∈ [0, Tc),

which immediately implies that x1(t), x2(t) and x5(t) be a collision cluster amongthe 5 vortices and they will collide at the origin (0, 0)T when t→ T−c .

2) When n = 3, (4.26) reduces to

ρ1(t) = − 12ρ3/21 (t)

ρ3/22 (t)− ρ3/2

1 (t), ρ2(t) =

12ρ3/22 (t)

ρ3/22 (t)− ρ3/2

1 (t), t > 0, (4.28)

which immediately implies

d

dt

[1√ρ1(t)

+1√ρ2(t)

]= 0 =⇒ 1√

ρ1(t)+

1√ρ2(t)

≡ a1 + a2

a1a2, t ≥ 0. (4.29)

Since 0 < a1 < a2, then ρ1(t) and ρ2(t) are monotonically decreasing and increasingfunctions, respectively. From (4.29), we know that 0 < ρ1(t) < ρ2(t) for t ≥ 0 andthus Tmax = +∞, i.e. there is no finite time collision. Noting that M0 > 0, byTheorem 2.4, we have

limt→+∞

ρ2(t) = +∞. (4.30)

Combining (4.30), (4.29) and (4.28), we obtain (4.25) immediately.3) When n ≥ 4, the proof is analogue to that of Proposition 4.1 and thus it is

omitted here for brevity.In addition, Figure 4.3 depicts the solution ρ1(t) and ρ2(t) of (4.26) obtained

numerically with ρ01 = 1 and ρ0

2 = 4 for different n ≥ 2.

Proposition 4.7. Taking mN = −m0, mj = m0 for 1 ≤ j ≤ 2n = N − 1 and theinitial data X0 in (1.2) as (4.23), then the solution of the ODEs (1.1) with (4.23)can be given as (4.24), where

(i) when n = 2, only the three vortices x1(t), x2(t) and x5(t) be a collision clusteramong the 5 vortices and they will collide at the origin (0, 0)T in finite time;

(ii) when n = 3, then

ρ1(t) ∼(

a1a2

a2 − a1

)2

, ρ2(t) ∼ 12t, t� 1;

(iii) when n ≥ 4, then

ρ1(t) ∼ α6t, ρ2(t) ∼ β6t, t� 1,


0 < α6 < β6, α6 + β6 = 8n− 12, β6 − α6 = 4nβn/26 − αn/26

βn/26 + α

n/26

.


0.2 0.4 0.6 0.8 10

1

2

3

4

5

t

ρ 1(t) 0 0.1

1

n=2

n=3

n=4

n=5

0.2 0.4 0.6 0.8 10

4

10

20

30

t

ρ 2(t) 0 0.02 0.04 0.06

4

5

6

n=2

n=3

n=4

n=5



Specifically, when n� 1, we have α6 ≈ 2n− 6, and β6 ≈ 6n− 6.

Proof. The proof is analogue to that of Proposition 4.6 and thus it is omitted herefor brevity.

Proposition 4.8. Taking mN = −m0, mj = m0 and mn+j = −m0 for 1 ≤ j ≤ nand the initial data X0 in (1.2) as (4.23), then the solution of the ODEs (1.1) with(4.23) can be given as (4.24), where

(i) when n = 2, only the three vortices x1(t), x2(t) and x5(t) be a collision clusteramong the 5 vortices and they will collide at the origin (0, 0)T in finite time;

(ii) When n ≥ 3, all the N = 2n+ 1 vortices be a collision cluster and they willcollide at the origin (0, 0)T when t→ Tc := 1

4 [a21 + a2

2].

Proof. Similar to the proof of Proposition 4.1 and 4.3, the solution of the ODEs(1.1) with (4.23) can be given as (4.24), where

ρ1(t)+ ρ2(t) = −4, ρ2(t)− ρ1(t) = 8−4n+8nρ

n/21 (t)

ρn/22 (t) + ρ

n/21 (t)

, t > 0, (4.31)

which implies

ρ1(t) = 2n− 6− 4nρn/21 (t)

ρn/22 (t) + ρ

n/21 (t)

,

ρ2(t) = 2− 2n+4nρ

n/21 (t)

ρn/22 (t) + ρ

n/21 (t)

, t > 0.

(4.32)

Solving (4.31) with initial data (4.13), we get

ρ1(t) + ρ2(t) = −4t+ a21 + a2

2, t ≥ 0,

which implies that a finite time collision must happen and 0 < Tmax ≤ Tc :=14 (a2

1 + a22).

1) When n = 2, from (4.32), we obtain

ρ1(t) = −10ρ1(t) + 2ρ2(t)

ρ1(t) + ρ2(t)< 0, ρ2(t) =

6ρ1(t)− 2ρ2(t)

ρ1(t) + ρ2(t), t > 0. (4.33)


0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5ρ 1(t

)

t

n= 2

n= 3

n= 4

n= 5

0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

ρ 2(t)

t

n= 2

n= 3

n= 4

n= 5



Solving (4.33) with the initial data (4.13), we have

ρ1(t) = (−2t+ C1)(2C3(−2t+ C1)− 1),

ρ2(t) = (−2t+ C1)(3− 2C3(−2t+ C1)), t ≥ 0.

where C3 =3a21+a22

(a21+a22)2> 0. Denote

0 < Tmax =2C1C3 − 1

4C3=a2

1(a21 + a2

2)

6a21 + 2a2

2

=2a2

1

3a21 + a2

2

Tc < Tc,

then we have

ρ1(Tmax) = 0, ρ2(Tmax) > 0, ρ1(t) > 0, ρ2(t) > 0, t ∈ [0, Tmax),

which implies that only the three vortices x1(t), x2(t) and x5(t) be a collision clusteramong the 5 vortices and they will collide at the origin (0, 0)T when t→ T−max.

2) When n ≥ 3, by Theorem 2.4, only the n+ 1 vortices with xn+1(t), . . . ,x2n(t)and xN (t) cannot be a collision cluster among the N vortices since they have thesame winding number; and similarly, only the n + 1 vortices with x1(t), . . . ,xn(t)and xN (t) cannot be a collision cluster among the N vortices since their collectivewinding number defined as M1 :=

∑1≤j<l≤nmjml +

∑nj=1mjmN = 1

2 [(n − 1)2 −n − 1] = 1

2n(n − 3) ≥ 0. Thus, in order to have a finite time collision, there exist1 ≤ j0 ≤ n and 1 ≤ l0 ≤ n such that the vortex dipole xj0(t) and xn+l0(t) willcollide at t = Tc, i.e. ρ1(Tc) = ρ2(Tc) = 0. Therefore, the N = 2n+ 1 vortices willbe a (finite time) collision cluster.

In addition, Figure 4.4 depicts the solution ρ1(t) and ρ2(t) of (4.32) obtainednumerically with ρ1(0) = 1 and ρ2(0) = 4 for different n ≥ 2.

5. Conclusion. Based on the reduced dynamical law of a system of ordinary dif-ferential equations (ODEs) for the dynamics of N vortex centers, we have obtainedstability and interaction patterns of quantized vortices in superconductivity. Byderiving several non-autonomous first integrals of the ODEs system, we provedglobal well-posedness of the N vortices when they have the same winding numberand demonstrated that finite time collision might happen when they have differentwinding numbers. When N = 3, we established rigorously orbital stability whenthey have the same winding number and classified their collision patterns when


they have different winding numbers. Finally, under several special initial setupsincluding interaction of two clusters, we obtained explicitly the analytical solutionsof the ODEs system. The analytical and numerical results demonstrated the richdynamics and interaction patterns of N vortices in superconductivity.

Acknowledgments. This work was supported partially by the Academic ResearchFund of Ministry of Education of Singapore grant No. R-146-000-223-112 (W.B.),the National Natural Science Foundation of China grant No. 11771177, 11501242(S.S. and Z.X.), Program for Changbaishan Scholars of Jilin Province and Programfor JLU Science, Technology Innovative Research Team No. 2017TD-20 (S.S), Sci-ence Foundation of Jilin, China grant No. 20170520055JH (Z.X.).

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Received December 2016; revised October 2017.

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