Quantized vortex dynamics and interaction patterns in superconductivity based on the reduced dynamical law
aa r X i v : . [ m a t h . C A ] D ec Manuscript submitted to doi:10.3934/xx.xx.xx.xxAIMS’ JournalsVolume X , Number , XX pp. X–XX
QUANTIZED VORTEX DYNAMICS AND INTERACTION PATTERNS INSUPERCONDUCTIVITY BASED ON THE REDUCED DYNAMICAL LAW
Zhiguo Xu
School of Mathematics, Jilin University, Changchun 130012, P. R. China
Weizhu Bao
Department of Mathematics, National University of Singapore, Singapore 119076
Shaoyun Shi
School of Mathematics, Jilin University, Changchun 130012, P. R. China
Abstract.
We study analytically and numerically stability and interaction patterns of quantizedvortex lattices governed by the reduced dynamical law – a system of ordinary differential equations(ODEs) – in superconductivity. By deriving several non-autonomous first integrals of the ODEs,we obtain qualitatively dynamical properties of a cluster of quantized vortices, including globalexistence, finite time collision, equilibrium solution and invariant solution manifolds. For a vortexlattice with 3 vortices, we establish orbital stability when they have the same winding numberand find different collision patterns when they have different winding numbers. In addition, underseveral special initial setups, we can obtain analytical solutions for the nonlinear ODEs. Introduction.
In this paper, we study analytically and numerically stability and interactionpatterns of the following system of ordinary differential equations (ODEs) describing the dynamics of N ≥ x j ( t ) = 2 m j N X k =1 ,k = j m k x j ( t ) − x k ( t ) | x j ( t ) − x k ( t ) | , ≤ j ≤ N, t > , (1.1)with initial data x j (0) = x j = ( x j , y j ) T ∈ R , ≤ j ≤ N. (1.2)Here t is time, x j ( t ) = ( x j ( t ) , y j ( t )) T ∈ R is the center of the j -th (1 ≤ j ≤ N ) quantized vortexat time t , m j = +1 or − j -th (1 ≤ j ≤ N )quantized vortex. We always assume that the initial data satisfies X := ( x , . . . , x N ) ∈ R × N ∗ := { X = ( x , . . . , x N ) ∈ R × N | x j = x l ∈ R for 1 ≤ j < l ≤ N } and denote its mass center as¯ x := N P Nj =1 x j . Throughout this paper, we assume that N ≥ N quantized vortices – particle-like or topological defects – in the Ginzburg-Landau equation[20, 12, 15] ∂ t ψ ( x , t ) = ∇ ψ ( x , t ) + 1 ε (1 − | ψ ( x , t ) | ) ψ ( x , t ) , x ∈ R , t > , (1.3)with initial condition ψ ( x ,
0) = ψ ( x ) = Π Nj =1 φ m j ( x − x j ) , x ∈ R , (1.4) Mathematics Subject Classification.
Primary: 34C60, 34D05; Secondary: 34A33, 34D30, 65L07.
Key words and phrases.
Quantized vortex, reduced dynamical law, superconductivity, interaction pattern, non-autonomous first integral, winding number, orbital stability, finite time collision, collision cluster. for superconductivity when either ε = 1 and d := min ≤ j
1) with ( r, θ ) the polar coordinates in 2D and f ( r ) satisfying[21, 12, 15, 26, 27]1 r ddr (cid:18) r df ( r ) dr (cid:19) − r f ( r ) + 1 ε (1 − f ( r )) f ( r ) = 0 , < r < + ∞ ,f (0) = 0 , lim r → + ∞ f ( r ) = 1 . Here φ m ( x ) is a typical quantized vortex in 2D, which is zero of the order parameter at the vortex cen-ter located at the origin and has localized phase singularity with integer m topological charge usuallycalled also as winding number or index or circulation. In fact, quantized vortices have been widelyobserved in superconductor [11, 15, 4], liquid helium [19], Bose-Einstein condensates [24, 2, 13]; andthey are key signatures of superconductivity and superfluidity. The study of quantized vortices andtheir dynamics is one of the most important and fundamental problems in superconductivity andsuperfluidity [21, 3, 18, 25, 6, 8, 9, 10, 14, 16, 5, 22, 23].Based on the reduced dynamical law, i.e. (1.1), for the quantized vortex dynamics in super-conductivity, when two quantized vortices have the same winding number (i.e. vortex pair), theyundergo a repulsive interaction; and respectively, when they have opposite winding numbers (i.e.vortex dipole or vortex-antivortex), they undergo an attractive interaction [21, 26, 27]. For N ≥ X ∈ R × N ∗ , it is 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 the analytical solutionof (1.1) with (1.2): when m = m (i.e. vortex pair), the two vortices move outwards by repellingeach other along the line passing through their initial locations x = x and they never collide atfinite time; and when m = − m (i.e. vortex dipole or vortex-antivortex), the two vortices movetowards each other along the line passing through their initial locations x = x and they will collideat (cid:0) x + x (cid:1) in finite time [21, 26, 27]. For analytical solutions of the ODEs (1.1) with severalspecial initial setups in (1.2), we refer to [26, 27] and references therein. In addition, define the masscenter of the N vortices as ¯ x ( t ) := 1 N N X j =1 x j ( t ) , t ≥ , (1.5)then it was proven that the mass center is conserved under the dynamics of (1.1) with (1.2) [26, 27]¯ x ( t ) ≡ ¯ x (0) = ¯ x , t ≥ . (1.6)Introduce W ( X ) = − X ≤ j = k ≤ N m j m k ln | x j − x k | = − ln Y ≤ j = k ≤ N | x j − x k | m j m k , X ∈ R × N ∗ , (1.7)then (1.1) can be reformulated as ˙ X ( t ) = −∇ X W ( X ) , t > , (1.8)which implies that W ( X ( t )) ≤ W ( X ( t )) ≤ W ( X (0)) = W ( X ) , ≤ t ≤ t . (1.9)In addition, let z j ( t ) := x j ( t ) + iy j ( t ) ∈ C for 1 ≤ j ≤ N , then (1.1) can be reformulated as˙ z j ( t ) = 2 m j N X k =1 ,k = j m k z j ( t ) − z k ( t ) | z j ( t ) − z k ( t ) | = 2 m j N X k =1 ,k = j m k z j ( t ) − z k ( t ) , ≤ j ≤ N, t > , (1.10)where ¯ z denotes the complex conjugate of z ∈ C . UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 3
For rigorous mathematical justification of the derivation of the above reduced dynamical law(1.1) with (1.2) for superconductivity, we refer to [12, 15] and references therein, and respectively,for numerical comparison of quantized vortex center dynamics under the Ginzburg-Landau equation(1.3) with (1.4) and its corresponding reduced dynamical law (1.1) with (1.2), we refer to [26, 27]and references therein. Based on the mathematical and numerical results [12, 15, 26, 27], thedynamics of the N quantized vortex centers under the reduced dynamical law agrees qualitatively(and quantitatively when they are well-separated) with that under the Ginzburg-Landau equation.The main aim of this paper is to study analytically and numerically the dynamics and interactionpatterns of the reduced dynamical law (1.1) with (1.2), which will generate important insights aboutquantized vortex dynamics and interaction patterns in superconductivity and is much simpler thanto solve the Ginzburg-Landau equation (1.3) with (1.4). We establish global existence of the ODEs(1.1) when the N quantized vortices have the same winding number and possible finite time collisionwhen they have opposite winding numbers. For N = 3, we prove orbital stability when they have thesame winding number and find different collision patterns when they have different winding numbers.Analytical solutions 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 solution manifolds andseveral non-autonomous first integrals of the ODEs (1.1) and establish its global existence when the N quantized vortices have the same winding number and possible finite time collision when theyhave opposite winding numbers. In section 3, we prove orbital stability when they have the samewinding number and find different collision patterns when they have different winding numbers forthe dynamics of N = 3 vortices. Analytical solutions of the ODEs (1.1) are presented under severalinitial setups with symmetry in section 4. Finally, some conclusions are drawn in section 5.2. Dynamical properties of a cluster with N quantized vortices. In this section, we establishdynamical properties of the system of ODEs (1.1) with the initial data (1.2) for describing thedynamics – reduced dynamical law – of a cluster with N quantized vortices in superconductivity.For any two vortices x j ( t ) and x l ( t ) (1 ≤ j < l ≤ N ), if there exists a finite time 0 < T c < + ∞ such that d jl ( t ) := | x j ( t ) − x l ( t ) | > ≤ t < T c and d jl ( T c ) = 0, then we say that they willbe finite time collision or annihilation (cf. Fig. 2.1a); otherwise, i.e. d jl ( t ) > t ≥
0, thenwe say that they will not collide. When N ≥ I ⊆ { , , . . . , N } be a set with at least 2elements, if there exists a finite time 0 < T c < + ∞ such that min ≤ j
Let α > ≤ θ < π be a constant, x ∈ R be a given point and Q ( θ ) be the rotational matrix defined as Q ( θ ) = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) , ≤ θ < π. Then it is easy to see that the ODEs (1.1) with (1.2) is translational and rotational invariant withthe proof omitted here for brevity.
Lemma 2.1.
Let X ( t ) = ( x ( t ) , x ( t ) , . . . , x N ( t )) ∈ R × N ∗ be the solution of the ODEs (1.1) with (1.2) , then we have ZHIGUO XU, WEIZHU BAO AND SHAOYUN SHI −1 0 1−101 y x − x x x (a) Figure 2.1.
Illustrations of a finite time collision of a vortex dipole in a vortexcluster with 3 vortices (a) and a (finite time) collision cluster with 3 vortices ina vortex cluster with 5 vortices (b). Here and in the following figures, ‘+’ and‘ − ’ denote the initial vortex centers with winding numbers m = +1 and m = − (i) If x j → x j + x for ≤ j ≤ N in (1.2) , then x j ( t ) → x j ( t ) + x for ≤ j ≤ N .(ii) If x j → α x j for ≤ j ≤ N in (1.2) , then x j ( t ) → α x j ( t/α ) for ≤ j ≤ N .(iii) If x j → Q ( θ ) x j for ≤ j ≤ N in (1.2) , then x j ( t ) → Q ( θ ) x j ( t ) for ≤ j ≤ N . Denote S e ( x ) := (cid:8) X = ( x , . . . , x N ) ∈ R × N ∗ | | ( x j − x ) · e | = | x j − x | × | e | , ≤ j ≤ N (cid:9) , where e ∈ R is a given unit vector. In fact, S e ( x ) is a line in 2D passing the point x and parallelto the unit vector e . For X = ( x , . . . , x N ) ∈ R × N ∗ , if there exist x ∈ R and a unit vector e ∈ R such that X ∈ S e ( x ), then we say that X is collinear. Lemma 2.2.
If the initial data X ∈ R × N ∗ in (1.2) is collinear, i.e. there exist x ∈ R and aunit vector e ∈ R such that X ∈ S e ( x ) , then the solution X ( t ) of (1.1) - (1.2) is collinear, i.e. X ( t ) ∈ S e ( x ) for ≤ t < T max .Proof. From X ∈ S e ( x ), there exist a j ∈ R (1 ≤ j ≤ N ) satisfying a j = a l for 1 ≤ j < l ≤ N such that x j = x + a j e , ≤ j ≤ N. (2.1)Noting the symmetric structure in (1.1) and (2.1), we can assume x j ( t ) = x + a j ( t ) e , ≤ j ≤ N, t ≥ . (2.2)Plugging (2.2) into (1.1), we have˙ a j ( t ) = 2 m j N X k =1 ,k = j m k a j ( t ) − a k ( t ) | a j ( t ) − a k ( t ) | , ≤ j ≤ N, t > , (2.3)with the initial data by noting (2.1) a j (0) = a j , ≤ j ≤ N. (2.4)The ODEs (2.3) with (2.4) is locally well-posed. Thus X ( t ) ∈ S e ( x ) for 0 ≤ t < T max . UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 5
Let e ∈ R be a unit vector, denote θ jN := j − πN and x (0) j = Q ( θ jN + θ ) e for 1 ≤ j ≤ N anddefine S N e ( x , θ ) := n X r = ( x + r x (0)1 , . . . , x + r x (0) N ) ∈ R × N ∗ | r > o ,S N e ( x ) := [ ≤ θ < π S N e ( x , θ ) . Lemma 2.3.
Assume the N vortices have the same winding number, i.e. m = . . . = m N = ± . Ifthere exists a unit vector e ∈ R , x ∈ R and ≤ θ < π such that the initial data X ∈ S N e ( x , θ ) in (1.2) , then the solution X ( t ) of (1.1) satisfies X ( t ) ∈ S N e ( x , θ ) for t ≥ .Proof. Since X ∈ S N e ( x , θ ), there exists a r > x j = x + r Q ( θ jN + θ ) e , ≤ j ≤ N. (2.5)Noting the symmetric structure in (1.1) and (2.5), we can assume x j ( t ) = x + r ( t ) Q ( θ jN + θ ) e , ≤ j ≤ N, t ≥ . (2.6)Plugging (2.6) into (1.1), noting m = . . . = m N and (2.5), we have [26, 27]˙ r ( t ) = N − r ( t ) , t > , r (0) = r , which implies r ( t ) = p r + 2( N − t for t ≥
0. Thus X ( t ) ∈ S N e ( x , θ ) for t ≥ θ ∈ R , x ∈ R and a unit vector e ∈ R , S e ( x ) is aninvariant solution manifold of the ODEs (1.1) with (1.2). In addition, when m = . . . = m N , then S N e ( x , θ ) is also an invariant solution manifold of the ODEs (1.1) with (1.2). Specifically, when X ∈ S N e ( , θ ) and m = . . . = m N , then the ODEs (1.1) with (1.2) admits the self-similar solution X ( t ) = p r + 2( N − t X with r = | x | = . . . = | x N | for t ≥
0. For more self-similar solutions ofthe ODEs (1.1) with special initial setups, we refer to [26, 27] and references therein.2.2.
Non-autonomous first integrals.
Let N + and N − be the number of vortices with windingnumber m = 1 and m = −
1, respectively, then we have0 ≤ N + ≤ N, ≤ N − ≤ N, N + + N − = N. In addition, it is easy to get M = X ≤ j Lemma 2.4. Let X ( t ) = ( x ( t ) , x ( t ) , . . . , x N ( t )) ∈ R × N ∗ be the solution of the ODEs (1.1) with (1.2) , then H ( X , t ) , H ( X , t ) and H ( X , t ) are non-autonomous first integrals of (1.1) , i.e. H ( X ( t ) , t ) ≡ H := X ≤ j From (2.8) and (2.9), it is easy to see that H ( X ( t ) , t ) = 12( N − 1) [ H ( X ( t ) , t ) + H ( X ( t ) , t )] , t ≥ . (2.19)Differentiating (2.19) with respect to t , noticing (2.15) and (2.18), we have dH ( X ( t ) , t ) dt = 12( N − (cid:20) dH ( X ( t ) , t ) dt + dH ( X ( t ) , t ) dt (cid:21) = 0 , t ≥ , which immediately implies the right equation in (2.10) by noting the initial condition (1.2). Therefore H ( X , t ), H ( X , t ) and H ( X , t ) are three non-autonomous first integrals of the ODEs (1.1).2.3. Global existence in the case with the same winding number. Let m = +1 or − N quantized vortices have the same winder number, e.g. m , we have Theorem 2.1. Suppose the N vortices have the same winding number, i.e. m j = m for ≤ j ≤ N in (1.1) , then T max = + ∞ , i.e. there is no finite time collision among the N quantized vortices. Inaddition, at least two vortices move to infinity as t → + ∞ .Proof. The proof will be proceeded by the method of contradiction. Assume 0 < T max < + ∞ , i.e.there exist M (2 ≤ M ≤ N ) vortices (without loss of generality, we assume here that they are x , . . . , x M ) that collide at a fixed point x ∈ R and the rest N − M vortices are all away from thispoint. Taking t = T max in the left equation in (2.10), noting (2.7), (2.8) and | N + − N − | = N , weget 0 < H = H ( X ( T max ) , T max ) = − N M T max + X ≤ j 0, then there exists a 0 < T < T max such that0 ≤ D I ( t ) < ε, T ≤ t ≤ T max . UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 9 Differentiating (2.20) with respect to t , we obtain˙ D I ( t ) = 2 X ≤ j 0, combining (2.8) and (2.10), we get P Nj =1 | x j ( t ) | = lim t → + ∞ H +4 M t → + ∞ when t → + ∞ . Hence there exists an 1 ≤ i ≤ N such that | x i ( t ) | → + ∞ when t → + ∞ . Due to the conservation of mass center, i.e. ¯ x ( t ) := N P Nj =1 x j ( t ) ≡ ¯ x (0), there exists atleast another 1 ≤ j = i ≤ N such that | x j ( t ) | → + ∞ when t → + ∞ . Thus there exist at leasttwo vortices move to infinity when t → + ∞ .Define d min ( t ) = min ≤ j Suppose the N vortices have the same winding number, i.e. m j = m for ≤ j ≤ N in (1.1) , and the initial data X in (1.2) is collinear, then d min ( t ) and D min ( t ) are monotonicallyincreasing functions.Proof. Since X is collinear, there exist x ∈ R and a unit vector e ∈ R such that X ∈ S e ( x ),by Lemma 2.2, we know that X ( t ) ∈ S e ( x ) for t ≥ 0. Thus there exist a j ( t ) (1 ≤ j ≤ N ) satisfying a j ( t ) = a l ( t ) for 1 ≤ j < l ≤ N such that x j ( t ) = x + a j ( t ) e , t ≥ , ≤ j ≤ N. (2.24)Taking 0 ≤ t < t such that d min ( t ) is smooth on [ t , t ), without loss of generality, we assume thatthere exists 1 ≤ i ≤ N − a ( t ) < a ( t ) < . . . < a i ( t ) < a i +1 ( t ) < . . . < a N ( t ) , d min ( t ) = d i ,i +1 ( t ) , t ≤ t < t . (2.25)Plugging j = i and l = i + 1 into (2.23) and noting (2.25), (2.24) and (2.22), we gave˙ D min ( t ) = ˙ D i ,i +1 ( t ) = 4 d min ( t ) N X k =1 ,k = i ,i +1 (cid:18) d i k ( t ) − d i +1 ,k ( t ) (cid:19) = 4 " − i − X k =1 d ( t ) d i k ( t ) d i +1 ,k ( t ) − N X k = i +2 d ( t ) d i k ( t ) d i +1 ,k ( t ) ≥ " − i − X k =1 i − k )( i + 1 − k ) − N X k = i +2 k − i )( k − i − = 4 (cid:18) i + 1 N − i (cid:19) > , t ≤ t < t . Here we used d jl ( t ) d min ( t ) ≥ | j − l | for 1 ≤ j < l ≤ N by noting (2.25). Thus D min ( t ) (and d min ( t )) is amonotonically increasing function over [ t , t ). Therefore, D min ( t ) (and d min ( t )) is a monotonicallyincreasing function over its every piecewise smooth interval. Due to that it is a continuous function,thus D min ( t ) (and d min ( t )) is a monotonically increasing function for t ≥ ≤ N ≤ X ∈ R × N ∗ , we have Theorem 2.3. Suppose ≤ N ≤ and the N vortices have the same winding number, i.e. m j = m for ≤ j ≤ N in (1.1) , then d min ( t ) and D min ( t ) are monotonically increasing functions.Proof. Taking 0 ≤ t < t such that d min ( t ) is smooth on [ t , t ), without loss of generality, weassume that d min ( t ) = d ( t ) for t ≤ t < t (otherwise by re-ordering). Taking j = 1 and l = 2 in(2.23), we get for t ≤ t < t ˙ D min ( t ) = ˙ D ( t ) = 4 x ( t ) − x ( t )) · N X k =1 ,k =1 , (cid:18) x ( t ) − x k ( t ) d k ( t ) − x ( t ) − x k ( t ) d k ( t ) (cid:19) . When N = 2 or 3, noting 0 < d ( t ) ≤ d jl ( t ) for 1 ≤ j = l ≤ N , we get˙ D min ( t ) > − N X k =1 ,k =1 , (cid:18) d ( t ) d k ( t ) + d ( t ) d k ( t ) (cid:19) ≥ − N − ≥ , t ≤ t < t , which implies that D min ( t ) and d min ( t ) are monotonically increasing functions over t ∈ [ t , t ]. When N = 4, without loss of generality, we can assume d ( t ) ≤ d ( t ) ≤ d ( t ) , d ( t ) ≤ d ( t ) ≤ d ( t ) , t ≤ t < t , UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 11 then we get ˙ D min ( t ) > (cid:20) − (cid:18) d ( t ) d ( t ) + d ( t ) d ( t ) (cid:19)(cid:21) ≥ , t ≤ t < t , which implies that D min ( t ) and d min ( t ) are monotonically increasing functions over t ∈ [ t , t ]. Remark 2.1. When N ≥ X ∈ R × N ∗ is not collinear, d min ( t ) might not bea monotonically increasing function, especially when 0 ≤ t ≪ 1. Based on our extensive numericalresults, for any given X ∈ R × N ∗ , there exits a constant T > X such that d min ( t )is a monotonically increasing function when t ≥ T . Rigorous mathematical justification is ongoing.2.4. Finite time collision in the case with opposite winding numbers. When the N vorticeshave opposite winding numbers, we have Theorem 2.4. Suppose the N vortices have opposite winding numbers, i.e. | N + − N − | < N , wehave(i) If M < , finite time collision happens, i.e. < T max < + ∞ , and there exists a collisioncluster among the N vortices. In addition, T max ≤ T a := − H NM .(ii) If M = 0 , then the solution of (1.1) is bounded, i.e. | x j ( t ) | ≤ q H = vuut N X j =1 | x j | , t ≥ , ≤ j ≤ N. (2.26) (iii) If M > and there is no finite time collision, i.e. T max = + ∞ , then at least two vorticesmove to infinity as t → + ∞ .(iv) Let I ⊆ { , , . . . , N } be a set with M ( ≤ M ≤ N ) elements. If the collective windingnumber of I defined as M := P j,l ∈ I,j = l m j m l ≥ , then the set of vortices { x j ( t ) | j ∈ I } cannotbe a collision cluster among the N vortices for ≤ t ≤ T max .Proof. (i) Combining (2.8) and (2.10), we get0 ≤ X ≤ j 0, when t → T a = − H NM , we have 4 N M t + H → 0. Thus finite time collision happensat t = T max ≤ T a < + ∞ .(ii) If M = 0, combining (2.8) and (2.10), we get0 ≤ | x j ( t ) | ≤ N X j =1 | x j ( t ) | ≡ H = N X j =1 | x j | , t ≥ , which immediately implies (2.26).(iii) If M > T max = + ∞ , then there exists no finite time collision cluster among the N vortices. The proof can be proceeded similarly as the last part in Theorem 2.1 and details areomitted here for brevity.(iv) When M = N , for any given X ∈ R × N ∗ , we get H > 0. Noting (2.27) and M = M ≥ X ≤ j 3, without loss of generality, we assume I = { , . . . , M } anddenote J = { M + 1 , . . . , N } . Thus M = P ≤ j 0. We will proceed the proof by the method of contradiction. Assume that the M vortices x , . . . , x M collide at x ∈ R when t → T c satisfying 0 < T c ≤ T max , i.e. x j ( t ) → x when t → T − c for 1 ≤ j ≤ M and | x j ( t ) − x | > t → T − c for M + 1 ≤ j ≤ N . Denote d := min j ∈ J min ≤ t ≤ T c | x j ( t ) − x | and we have d > 0. Sincelim t → T c D I ( t ) = 0, there exists 0 < T < T c , such that D I ( t ) < d and d I,J ( t ) > d for t ∈ [ T , T c ).Choose T ∈ [ T , T c ), such that0 < T c − T < d M ( M − N − M ) . Since D I ( t ) is a continuous function, there exists T ∈ [ T , T c ], such that D I ( T ) = max t ∈ [ T ,T c ] D I ( t ) > 0. Similar to (2.21), we have D I ( T ) = D I ( T ) − D I ( T c ) = − Z T c T ddt D I ( t ) dt = − Z T c T X ≤ j If the N vortices be a collision cluster at < T max < + ∞ under a given initialdata X ∈ R × N ∗ , then we have M < , H = N H , H = ( N − H . (2.28) Proof. Due to the conservation of mass center and X ∈ R × N ∗ , we get¯ x ( t ) ≡ ¯ x = ⇒ lim t → T − max x j ( t ) = x j ( T max ) = ¯ x , ≤ j ≤ N. (2.29)Plugging (2.29) into (2.8) and (2.10), we get H ( X ( T max ) , T max ) = X ≤ j If the ODEs (1.1) admits an equilibrium solution, then N ≥ is a square of aninteger, i.e. N = ( N + − N − ) and ≤ N + = 12 (cid:16) N ± √ N (cid:17) < N, ≤ N − = N − N + < N. (2.32) UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 13 Proof. Assume X ( t ) ≡ X ∈ R × N ∗ be an equilibrium solution of (1.1), noting (2.8) and (2.10), weget H ( X ( t ) , t ) = H − N M t ≡ H , t ≥ . (2.33)Thus M = 0. Noting (2.7), we have4 ≤ N = ( N + − N − ) = (2 N + − N ) = (2 N − − N ) . (2.34)Thus N ≥ Remark 2.2. When N = 4, an equilibrium solution of (1.1) was constructed in [26, 27] by taking m = − x = (0 , T and m = m = m = 1, x j located in the vertices of a right trianglecentered at the origin. Here we want to remark that any equilibrium solution of (1.1) is dynamicallyunstable.3. Interaction patterns of a cluster with quantized vortices. In this section, we assume N = 3 in (1.1) and (1.2).3.1. Structural/obital stability in the case with the same winding number. Assume that m = m = m and by Theorem 2.1, we know the ODEs (1.1) with (1.2) is globally well-posed, i.e. T max = + ∞ . Lemma 3.1. If the initial data X ∈ R × ∗ in (1.2) with N = 3 is collinear, then one vortex movesto the mass center ¯ x and the other two vortices repel with each other and move outwards to far fieldalong the line when t → + ∞ .Proof. Since X ∈ R × ∗ is collinear, there exist x ∈ R and a unit vector e ∈ R such that x j = x + a j e , j = 1 , , . Without loss of generality, we assume that a < a < a , a < , a > , a + a + a = 0 . Based on the results in Lemma 2.2 and Theorem 2.1, we know that there exist a j ( t ) ( j = 1 , , x j ( t ) = x + a j ( t ) e , j = 1 , , , (3.1)satisfying a ( t ) < a ( t ) < a ( t ) , a ( t ) + a ( t ) + a ( t ) ≡ , t ≥ . (3.2)Plugging (3.1) into (1.1) with N = 3 and m = m = m , noting (3.2), we get˙ a ( t ) = − a ( t ) − a ( t ) − a ( t ) − a ( t ) = 6 a ( t )[ a ( t ) − a ( t )][ a ( t ) − a ( t )] < , ˙ a ( t ) = 2 a ( t ) − a ( t ) − a ( t ) − a ( t ) = − a ( t )[ a ( t ) − a ( t )][ a ( t ) − a ( t )] , t > , ˙ a ( t ) = 2 a ( t ) − a ( t ) + 2 a ( t ) − a ( t ) = 6 a ( t )[ a ( t ) − a ( t )][ a ( t ) − a ( t )] > , with the initial data a j (0) = a j , j = 1 , , . (3.3)Thus a ( t ) is a monotonically decreasing function and a ( t ) is a monotonically increasing functionfor t ≥ 0. Let ρ ( t ) = a ( t ) ≥ 0, then we have˙ ρ ( t ) = − ρ ( t )[ a ( t ) − a ( t )][ a ( t ) − a ( t )] < , t > , which immediately implies that ρ ( t ) is a monotonically decreasing function and lim t → + ∞ ρ ( t ) = 0.Thus we have lim t → + ∞ a ( t ) = 0 = ⇒ lim t → + ∞ x ( t ) = ¯ x = 13 X j =1 x j . Thus the vortex x ( t ) moves towards ¯ x along the line S e (¯ x ). Based on the results in Theorem 2.1,we know that at least two vortices must move to infinity when t → + ∞ . Thus we have a ( t ) → −∞ , a ( t ) → + ∞ when t → + ∞ . Thus the other two vortices x ( t ) and x ( t ) repel with each other and move outwards to far fieldalong the line S e ( x ) when t → + ∞ . Theorem 3.1. Assume the initial data X ∈ R × ∗ in (1.2) with N = 3 is not collinear, then thereexists a unit vector e ∈ R such that lim t → + ∞ d S ( t ) := inf X ∈ S e (¯ x ) k X ( t ) − X k = 0 . (3.4) Proof. Without loss of generality, as shown in Fig. 3.1a, we assume ¯ x = and d := d (0) ≤ d := d (0) ≤ d := d (0). Thus 0 < θ := θ (0) ≤ θ := θ (0) ≤ θ := θ (0) < π satisfying θ + θ + θ = π (cf. Fig. 3.1a). From (1.1) with N = 3, we get˙ d ( t ) = 4 d ( t ) + 2 cos( θ ( t )) d ( t ) + 2 cos( θ ( t )) d ( t ) , ˙ θ ( t ) = B ( t ) (cid:2) d ( t ) + d ( t ) − d ( t ) (cid:3) , (3.5)˙ d ( t ) = 4 d ( t ) + 2 cos( θ ( t )) d ( t ) + 2 cos( θ ( t )) d ( t ) , ˙ θ ( t ) = B ( t ) (cid:2) d ( t ) + d ( t ) − d ( t ) (cid:3) , (3.6)˙ d ( t ) = 4 d ( t ) + 2 cos( θ ( t )) d ( t ) + 2 cos( θ ( t )) d ( t ) , ˙ θ ( t ) = B ( t ) (cid:2) d ( t ) + d ( t ) − d ( t ) (cid:3) , (3.7)where B ( t ) := 4 A ( t ) / ( d ( t ) d ( t ) d ( t )) with A ( t ) denoting the area of the triangle with vertices x ( t ), x ( t ) and x ( t ). Denote ρ ( t ) = d ( t ) , ρ ( t ) = d ( t ) , ρ ( t ) = d ( t ) , t ≥ . From (3.5)-(3.7) and noting the initial data, we get π ≤ θ ( t ) < π and 0 < θ ( t ) ≤ π are monotoni-cally decreasing and increasing functions, respectively, and ρ ( t ) ≤ ρ ( t ) , < θ ( t ) ≤ π ≤ θ ( t ) < π, t ≥ 0; lim t → + ∞ θ ( t ) = lim t → + ∞ θ ( t ) = π . (3.8)Combining this with θ ( t )+ θ ( t )+ θ ( t ) ≡ π for t ≥ 0, we get lim t → + ∞ θ ( t ) = π , which immediatelyimplies (3.4).For θ ∈ R and X = ( x , . . . , x N ) ∈ S N e ( , θ ), define Q ( θ ) X := ( Q ( θ ) x , . . . , Q ( θ ) x N ). Definition 3.1. For the self-similar solution ˜ X ( t ) = p r + 2( N − t ˜ X with ˜ X = (˜ x , . . . , ˜ x N ) ∈ S N e ( , θ ) and r = | ˜ x | of the ODEs (1.1) with m = . . . = m N , if for any ε > 0, there exists δ > X in (1.2) satisfies k X − ˜ X k < δ , the solution X ( t ) of the ODEs(1.1) with (1.2) satisfies sup t ≥ inf r> , θ ∈ [0 , π ) (cid:13)(cid:13)(cid:13) X ( t ) − ¯ x − rQ ( θ ) ˜ X ( t ) (cid:13)(cid:13)(cid:13) < ε, then the self-similar solution ˜ X ( t ) is called as orbitally stable. Theorem 3.2. For any θ ∈ R and ˜ X = (˜ x , ˜ x , ˜ x ) ∈ S e ( , θ ) , the solution ˜ X ( t ) = p t + r ˜ X with r = | ˜ x | of the ODEs (1.1) with N = 3 and m = m = m is orbitally stable. UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 15 θ ( t ) θ ( t ) θ ( t ) x ( t ) x ( t ) x ( t ) d ( t ) d ( t ) d ( t ) (a) θ ( t ) x ( t ) x ( t ) x ( t ) θ ( t ) θ ( t ) φ ( t ) φ ( t ) r ( t ) r ( t ) r ( t ) O(b) θ ( t ) θ ( t ) θ ( t ) x ( t ) x ( t ) x ( t ) d ( t ) d ( t ) d ( t ) (c) Figure 3.1. Interaction of 3 vortices with the same winding number (a and b) andopposite winding numbers (c). Proof. By using Lemma 2.1, without loss of generality, we can assume that θ = 0, r = 1 and¯ x = . In addition, as shown in Fig. 3.1b, we assume x j ( t ) = r j ( t )(cos( θ j ( t )) , sin( θ j ( t )) T , j = 1 , , , (3.9)satisfying θ := θ (0) < θ := θ (0) < θ := θ (0) < π and k X − ˜ X k ≤ δ with 0 < δ ≤ sufficiently small and to be determined later. In fact, from k X − ˜ X k = X j =1 ( r j (0) − + X j =1 r j (0) sin ( ϕ j ) , (3.10)with ϕ j = θ j − ( j − π for j = 1 , , 3, we can get | r (0) − | + | r (0) − | + | r (0) − | ≤ δ < , | θ | + (cid:12)(cid:12)(cid:12)(cid:12) θ − π (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) θ − π (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ < π . (3.11)Plugging (3.9) into (1.6), we get r ( t ) cos( θ ( t )) + r ( t ) cos( θ ( t )) + r ( t ) cos( θ ( t )) ≡ ,r ( t ) sin( θ ( t )) + r ( t ) sin( θ ( t )) + r ( t ) sin( θ ( t )) ≡ , t ≥ . Solving the above equations, we obtain r ( t ) = − r ( t ) sin( θ ( t ) − θ ( t ))sin( θ ( t ) − θ ( t )) = − r ( t ) sin ( φ ( t ) + φ ( t ))sin ( φ ( t )) , (3.12) r ( t ) = r ( t ) sin( θ ( t ) − θ ( t ))sin( θ ( t ) − θ ( t )) = r ( t ) sin ( φ ( t ))sin ( φ ( t )) , t ≥ , (3.13)where (cf. Fig. 3.1b) φ ( t ) = θ ( t ) − θ ( t ) , φ ( t ) = θ ( t ) − θ ( t ) , t ≥ . (3.14)By Lemma 2.4, we have r ( t ) + r ( t ) + r ( t ) = 12 t + H , t ≥ , (3.15)with H = r (0) + r (0) + r (0). Substituting (3.12) and (3.13) into (3.15), we can get r ( t ) = (12 t + H ) / sin ( φ ( t )) D / ( t ) , D ( t ) := sin ( φ ( t )) + sin ( φ ( t )) + sin ( φ ( t ) + φ ( t )) . (3.16)Plugging (3.9) into (1.1) with N = 3, noting (3.12)-(3.14) and (3.16), we have˙Φ( t ) = 212 t + H F ( φ ( t ) , φ ( t )) = 212 t + H F (Φ( t )) , t > , (3.17)where Φ( t ) := ( φ ( t ) , φ ( t )) T and F (Φ) = ( f (Φ) , f (Φ)) T is defined as f (Φ) = sin ( φ )sin ( φ ) sin ( φ + φ ) (cid:20) sin ( φ + φ ) D (Φ) + sin ( φ ) D (Φ) − sin ( φ ) + sin ( φ + φ ) D (Φ) (cid:21) ,f (Φ) = sin ( φ )sin ( φ ) sin ( φ + φ ) (cid:20) sin ( φ + φ ) D (Φ) + sin ( φ ) D (Φ) − sin ( φ ) + sin ( φ + φ ) D (Φ) (cid:21) , with D (Φ) = 1 D (Φ) (cid:0) sin ( φ ) + sin ( φ + φ ) + 2 sin ( φ ) sin ( φ + φ ) cos ( φ ) (cid:1) ,D (Φ) = 1 D (Φ) (cid:0) sin ( φ ) + sin ( φ ) − φ ) sin ( φ ) cos ( φ + φ ) (cid:1) ,D (Φ) = 1 D (Φ) (cid:0) sin ( φ ) + sin ( φ + φ ) + 2 sin ( φ ) sin ( φ + φ ) cos ( φ ) (cid:1) ,P (Φ) = 1 D (Φ) (cid:20) sin ( φ ) − sin ( φ + φ ) cos (cid:18) φ − π (cid:19) + sin ( φ ) cos (cid:18) φ + φ − π (cid:19)(cid:21) ;and ˙ θ ( t ) = 212 t + H g (Φ( t )) , t > , (3.18)with g (Φ) = g ( φ , φ ) = sin ( φ ) sin ( φ + φ ) ( D (Φ) − D (Φ))sin ( φ ) D (Φ) D (Φ) . Let s = 14 ln (cid:18) t + H H (cid:19) , Ψ( s ) = Φ( t ) − (2 π/ , π/ T , s ≥ , (3.19)then (3.17) can be re-written as˙Ψ( s ) = F (cid:0) Ψ( s ) + (2 π/ , π/ T (cid:1) = − s ) + G (Ψ( s )) , s > , (3.20)where G (Ψ) = F (cid:0) Ψ + (2 π/ , π/ T (cid:1) + 2Ψ . UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 17 It is easy to verify that Ψ( s ) ≡ is an equilibrium solution of (3.20). By the variation-of-constantformula, we have Ψ( s ) = e − s Ψ(0) + Z s e − s − τ ) G (Ψ( τ )) dτ, s ≥ . (3.21)By using the Taylor expansion, there exist constants K j > j = 1 , , 3) and 0 < δ < k G (Ψ) k ≤ k Ψ k , k G (Ψ) k ≤ K k Ψ k , | g (Φ) | = | g (Ψ + (2 π/ , π/ T ) | ≤ K k Ψ k , (3.22) | − P (Φ) | = (cid:12)(cid:12) − P (Ψ + (2 π/ , π/ T ) (cid:12)(cid:12) ≤ K k Ψ k , when k Ψ k < δ . (3.23)For any 0 < δ ≤ δ , when k Ψ(0) k ≤ δ ( ⇔ k Φ(0) − (2 π/ , π/ T k ≤ δ ) and let S > k Ψ( s ) k ≤ δ for 0 ≤ s ≤ S , noting (3.21) and (3.22) and using the triangle inequality, we have k Ψ( s ) k ≤ e − s k Ψ(0) k + Z s e − s − τ ) k Ψ( τ ) k dτ, ≤ s ≤ S, which is equivalent to e s k Ψ( s ) k = k Ψ k + Z s e τ k Ψ( τ ) k dτ, ≤ s ≤ S. Using the Gronwall’s inequality, we get k Ψ( s ) k ≤ k Ψ(0) k e − s ≤ k Ψ(0) k ≤ δ , ≤ s ≤ S. (3.24)From (3.24) and using the standard extension theorem for ODEs, we can obtain k Ψ( s ) k ≤ k Ψ(0) k e − s , ≤ s < + ∞ . (3.25)Combining (3.25) and (3.21), using the triangle inequality, we obtain k Ψ( s ) k ≤ e − s k Ψ(0) k + e − s Z s e τ k G (Ψ( τ )) k dτ ≤ e − s k Ψ(0) k + e − s Z s e τ K k Ψ( τ ) k dτ ≤ e − s k Ψ(0) k + e − s Z s e τ K k Ψ(0) k e − τ dτ ≤ (cid:2) k Ψ(0) k + K k Ψ(0) k (cid:3) e − s ≤ (1 + K ) δ e − s , ≤ s < + ∞ , which immediately implies k Φ( t ) − (2 π/ , π/ T k < (1 + K ) δ s H t + H , ≤ t < + ∞ . Noting (3.19) and (3.22), we have | g (Φ) | = | g (Ψ + ( π/ , π/ T ) | ≤ K k Ψ k ≤ K (1 + K ) δ s H t + H , ≤ t < + ∞ . This implies that the ODE (3.18) is globally solvable, and the solution can be written as θ ( t ) = θ (0) + Z t s + H g ( φ ( s ) , φ ( s )) ds, ≤ t ≤ + ∞ . Denote θ ∞ = lim t →∞ θ ( t ) and θ ( t ) = θ ( t ) − θ ∞ , then we haveinf r> k X ( t ) − rQ ( θ ( t )) ˜ X ( t ) k = inf r> (cid:8) t + H + 3 r − r d ( t ) (cid:9) = 12 t + H − d ( t )= 12 t + H − P (Φ( t ))) , t ≥ , (3.26) where d ( t ) := r ( t ) + r ( t ) cos( φ ( t ) − π/ 3) + r ( t ) cos( φ ( t ) + φ ( t ) − π/ . Noting (3.23), we have | − P (Φ( t )) | = (cid:12)(cid:12) − P (Ψ( t ) + (2 π/ , π/ T ) (cid:12)(cid:12) ≤ K k Ψ( t ) k ≤ K (1 + K ) δ H t + H , t ≥ . (3.27)For any ε > 0, taking δ = ε K (1+ K ) H and 0 < δ = min (cid:8) , δ , δ (cid:9) , when k X − ˜ X k < δ ,noting (3.27) and (3.26), we getsup t ≥ inf r> k X ( t ) − rQ ( θ ( t )) ˜ X ( t ) k ≤ K (1 + K ) H δ < ε, (3.28)which completes the proof by taking δ = δ in the above proof.3.2. Collision patterns in the case with opposite winding numbers. Without loss of gen-erality, we assume m = m = +1 and m = − N = 3. Then we have M = [( N + − N − ) − N ] = (1 − 3) = − < 0, thus finite time collision must happen. Theorem 3.3. For any given initial data X ∈ R × ∗ in (1.2) with N = 3 , we have(i) If | x − x | = | x − x | , then the three vortices be a collision cluster and they will collide at ¯ x when t → T − max = H with H = P ≤ j 3) such that x j ( t ) = x + a j ( t ) e , j = 1 , , , satisfying a ( t ) + a ( t ) + a ( t ) ≡ a + a + a and a ( t ) < a ( t ) for 0 ≤ t < T max and˙ a ( t ) = − a ( t ) − a ( t ) + 2 a ( t ) − a ( t ) = 2[ a ( t ) − a ( t )][ a ( t ) − a ( t )][ a ( t ) − a ( t )] , ˙ a ( t ) = − a ( t ) − a ( t ) − a ( t ) − a ( t ) = 2[3 a ( t ) − ( a + a + a )][ a ( t ) − a ( t )][ a ( t ) − a ( t )] , t > , ˙ a ( t ) = 2 a ( t ) − a ( t ) − a ( t ) − a ( t ) = 2[ a ( t ) − a ( t )][ a ( t ) − a ( t )][ a ( t ) − a ( t )] , with the initial data (3.3).If | a | = | a | , i.e. a = − a > 0, then the above ODEs with (3.3) admits the unique solution as a ( t ) = − q ( a ) − t, a ( t ) ≡ , a ( t ) = q ( a ) − t, ≤ t < T max := 12 ( a ) , which immediately implies that the three vortices be a collision cluster and they collide at ¯ x = x when t → T − max = H with H = P ≤ j 0. If 0 = a < a < a , then we can show that a ( t ) < a ( t ) < a ( t )for 0 ≤ t < T max and a ( t ), a ( t ) and a ( t ) are monotonically decreasing, increasing and increasingfunctions over t ∈ [0 , T max ), respectively. Thus only x and x form a collision cluster among the UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 19 a < a < a , then we can show that a ( t ) < a ( t ) < a ( t )for 0 ≤ t < T max and a ( t ) and a ( t ) are monotonically increasing and decreasing functions over t ∈ [0 , T max ), respectively. In addition we have a ( T max ) ≤ a ( T max ) < a ( T max ) = a + a + a − a ( T max ) − a ( T max ) > 0, therefore, again only x and x form a collision cluster among the 3vortices.(ii) If the initial data X ∈ R × ∗ in (1.2) with N = 3 is not collinear, i.e. the initial locationsof the 3 vortices form a triangle. Without loss of generality, as shown in Fig. 3.1c, we assume¯ x = and d := d (0) ≤ d := d (0). Thus 0 < θ := θ (0) ≤ θ := θ (0) < π satisfying θ + θ + θ = π (cf. Fig. 3.1b). From (1.1) with N = 3, we get˙ d ( t ) = − d ( t ) + 2 cos( θ ( t )) d ( t ) − θ ( t )) d ( t ) , ˙ θ ( t ) = − B ( t ) (cid:2) d ( t ) + d ( t ) (cid:3) < , (3.29)˙ d ( t ) = 4 d ( t ) − θ ( t )) d ( t ) − θ ( t )) d ( t ) , ˙ θ ( t ) = B ( t ) (cid:2) d ( t ) + d ( t ) + 2 d ( t ) (cid:3) > , (3.30)˙ d ( t ) = − d ( t ) − θ ( t )) d ( t ) + 2 cos( θ ( t )) d ( t ) , ˙ θ ( t ) = − B ( t ) (cid:2) d ( t ) + d ( t ) (cid:3) < . (3.31)If d = d , then 0 < θ = θ < π (cf. Fig. 3.1c), this together with (3.29)-(3.31) yields d ( t ) = d ( t ) , < θ ( t ) = θ ( t ) < π , ≤ t < T max ; lim t → T − max θ ( t ) = lim t → T − max θ ( t ) = 0 , which immediately implies that the three vortices are forming a collision cluster. By using Theorem2.4, we get T max = H / < d < d , then 0 < θ < θ < π (cf. Fig. 3.1b). From (3.29) and (3.31), we have˙ d ( t ) − ˙ d ( t ) = [4 − θ ( t ))]( d ( t ) − d ( t )) d ( t ) d ( t ) + 2[cos( θ ( t )) − cos( θ ( t ))] d ( t ) > , ˙ θ ( t ) − ˙ θ ( t ) = B ( t ) (cid:2) d ( t ) − d ( t ) (cid:3) > , t > . Then we have d ( t ) ≥ d ( t ) + d − d , θ ( t ) ≥ θ ( t ) + θ − θ , ≤ t ≤ T max . This, together with that θ ( t ) and θ ( t ) are monotonically decreasing functions, 0 < θ ( t ) = π − θ ( t ) − θ ( t ) < π is a monotonically increasing functions and finite time collision must happen (cf.Fig. 3.1c), we get that lim t → T − max d ( t ) = 0 and lim t → T − max θ ( t ) = 0. Thus only the two vortices x ( t ) and x ( t ) form a collision cluster among the 3 vortices. By using Theorem 2.4, we get thecollision time 0 < T max < H .4. Analytical solutions under special initial setups. Let 0 ≤ θ < π be a constant, n ≥ < a < a be two constants, C := (cid:0) a + a (cid:1) , C := (cid:0) a − a (cid:1) , and m = +1 or − 1. Denote θ jn = 2( j − πn + θ , α jn = 2( j − πn + πn + θ , ≤ j ≤ n. For the interaction of two clusters. Here we take N = 2 n with n ≥ Proposition 4.1. Taking m j = m for ≤ j ≤ N = 2 n and the initial data X in (1.2) as x j = a (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , x n + j = a (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , ≤ j ≤ n, (4.1) then the solution of the ODEs (1.1) with (4.1) can be given as x j ( t ) = √ ρ ( t ) (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , x n + j ( t ) = p ρ ( t ) (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , ≤ j ≤ n, t ≥ , (4.2) where when n = 2 , ρ ( t ) = C + 6 t − q C + 8 C t + 24 t , ρ ( t ) = C + 6 t + q C + 8 C t + 24 t , t ≥ 0; (4.3) and when n ≥ , ρ ( t ) ∼ α t, ρ ( t ) ∼ β t, t ≫ , (4.4) with α and β being two positive constants satisfying < α < β , α + β = 8 n − , β − α = 4 n β n/ + α n/ β n/ − α n/ . (4.5) Specifically, when n ≫ , we have α ≈ n − , β ≈ n − . (4.6) Proof. Noting the symmetry of the ODEs (1.1) with the initial data (4.1), we can take the solutionansatz (4.2). Substituting (4.2) into (1.1) and (1.2), we obtain˙ ρ ( t ) = 4 n X j =2 n ( θ n ) · (cid:0) n ( θ n ) − n ( θ jn ) (cid:1)(cid:12)(cid:12)(cid:12) n ( θ n ) − n ( θ jn ) (cid:12)(cid:12)(cid:12) + 4 n X j =1 n ( θ n ) · (cid:16) ρ ( t ) n ( θ n ) − p ρ ( t ) ρ ( t ) n ( θ jn ) (cid:17)(cid:12)(cid:12)(cid:12)p ρ ( t ) n ( θ n ) − p ρ ( t ) n ( θ jn ) (cid:12)(cid:12)(cid:12) = 2 n − n X j =1 ρ ( t ) − p ρ ( t ) ρ ( t ) cos( θ n − θ jn ) ρ ( t ) + ρ ( t ) − p ρ ( t ) ρ ( t ) cos( θ n − θ jn ) , t > , (4.7)˙ ρ ( t ) = 4 n X j =1 n ( θ n ) · (cid:16) ρ ( t ) n ( θ n ) − p ρ ( t ) ρ ( t ) n ( θ jn ) (cid:17)(cid:12)(cid:12)(cid:12)p ρ ( t ) n ( θ n ) − p ρ ( t ) n ( θ jn ) (cid:12)(cid:12)(cid:12) + 4 n X j =2 n ( θ n ) · (cid:0) n ( θ n ) − n ( θ jn ) (cid:1)(cid:12)(cid:12)(cid:12) n ( θ n ) − n ( θ jn ) (cid:12)(cid:12)(cid:12) = 2 n − n X l =1 ρ ( t ) − p ρ ( t ) ρ ( t ) cos( θ n − θ jn ) ρ ( t ) + ρ ( t ) − p ρ ( t ) ρ ( t ) cos( θ n − θ jn ) , t > , (4.8)where n ( θ ) = (cos( θ ) , sin( θ )) T , θ ∈ R . (4.9)Summing (4.7) and (4.8), we have˙ ρ ( t ) + ˙ ρ ( t ) = 8 n − , t > , (4.10)Subtracting (4.7) from (4.8), we get˙ ρ ( t ) − ˙ ρ ( t ) = 4 n ρ n/ ( t ) + ρ n/ ( t ) ρ n/ ( t ) − ρ n/ ( t ) = 4 n + 8 nρ n/ ( t ) ρ n/ ( t ) − ρ n/ ( t ) , t > . (4.11)Here we use the equality n X j =1 x − x + 1 − x cos( θ n − θ jn ) = n x n + 1 x n − , < x ∈ R . Combining (4.10) and (4.11), we obtain˙ ρ ( t ) = 2 n − − nρ n/ ( t ) ρ n/ ( t ) − ρ n/ ( t ) , ˙ ρ ( t ) = 6 n − nρ n/ ( t ) ρ n/ ( t ) − ρ n/ ( t ) , t ≥ , (4.12)with the initial data ρ (0) = ρ := a < ρ (0) = ρ := a . (4.13) UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 21 ρ ( t ) t n =2 n =3 n =4 n =5 ρ ( t ) t n =2 n =3 n =4 n =5 Figure 4.1. Time evolution of ρ ( t ) (left) and ρ ( t ) (right) of (4.12) with ρ = 1and ρ = 4 for different n ≥ 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 winding number, i.e. T max = + ∞ by usingTheorem 2.1, we get ρ ( t ) < ρ ( t ) for t ≥ ρ ( t ) > , ˙ ρ ( t ) − ˙ ρ ( t ) > , t ≥ . Therefore, we conclude that ρ ( t ) and ρ ( t ) − ρ ( t ) are monotonically increasing functions when t ≥ t → + ∞ ρ ( t ) = + ∞ by noting Theorem 2.4. From (4.12), we can conclude that there existtwo positive constants 0 < α < β such that (4.4) is valid. Plugging (4.4) into (4.10), we get (4.5)immediately. When n ≫ 1, (4.5) yields α + β = 8 n − , β − α ≈ n, which immediately implies (4.6). In addition, Figure 4.1 depicts the solution ρ ( t ) and ρ ( t ) of(4.12) obtained numerically with ρ (0) = 1 and ρ (0) = 4 for different n ≥ Proposition 4.2. Taking m j = m for ≤ j ≤ N = 2 n and the initial data X in (1.2) as x j = a (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , x n + j = a (cid:0) cos( α jn ) , sin( α jn ) (cid:1) T , ≤ j ≤ n, (4.14) then the solution of the ODEs (1.1) with (4.14) can be given as x j ( t ) = p ρ ( t ) (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , x n + j ( t ) = p ρ ( t ) (cid:0) cos( α jn ) , sin( α jn ) (cid:1) T , ≤ j ≤ n, t ≥ , (4.15) where when n = 2 , ρ ( t ) = C + 6 t − C (cid:18) tC (cid:19) / , ρ ( t ) = C + 6 t + C (cid:18) tC (cid:19) / , t ≥ and when n ≥ , ρ ( t ) ∼ α t, ρ ( t ) ∼ β t, t ≫ , with α and β being two positive constants satisfying < α < β , α + β = 8 n − , β − α = 4 n β n/ − α n/ β n/ + α n/ . Specifically, when n ≫ , α ≈ n − and β ≈ n − .Proof. The proof is analogue to that of Proposition 4.1 and thus it is omitted here for brevity. Proposition 4.3. Taking m j = m and m n + j = − m for ≤ j ≤ n and the initial data X in (1.2) as (4.14) , then the solution of the ODEs (1.1) with (4.14) can be given as (4.15) , where ρ ( t ) > , ρ ( t ) > , ≤ t < T c := 14 ( a + a ) , lim t → T − c ρ ( t ) = lim t → T − c ρ ( t ) = 0 , which implies that the N = 2 n 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 theinitial data (4.14), we can take the solution ansatz (4.15). In addition, plugging (4.15) into (1.1)and (1.2), we get˙ ρ ( t ) = 2 n − − n X l =1 ρ ( t ) − p ρ ( t ) ρ ( t ) cos( θ n − α ln ) ρ ( t ) + ρ ( t ) − p ρ ( t ) ρ ( t ) cos( θ n − α ln ) , (4.16)˙ ρ ( t ) = 2 n − − n X l =1 ρ ( t ) − p ρ ( t ) ρ ( t ) cos( α n − θ ln ) ρ ( t ) + ρ ( t ) − p ρ ( t ) ρ ( t ) cos( α n − θ ln ) , (4.17)with the initial data (4.13).Summing (4.16) and (4.17), we obtain˙ ρ ( t ) + ˙ ρ ( t ) = 4 n − − n X l =1 n − − n = − , t ≥ . (4.18)Subtracting (4.16) from (4.17), we get˙ ρ ( t ) − ˙ ρ ( t ) = − n ρ n/ ( t ) − ρ n/ ( t ) ρ n/ ( t ) + ρ n/ ( t ) = − n + 8 nρ n/ ( t ) ρ n/ ( t ) + ρ n/ ( t ) , t > . (4.19)Here we use the equality n X j =1 x − x + 1 − x cos( θ n − θ jn + πn ) = n x n − x n + 1 , < x ∈ R . Combining (4.18) and (4.19), we obtain˙ ρ ( t ) = 2 n − − nρ n/ ( t ) ρ n/ ( t ) + ρ n/ ( t ) , ˙ ρ ( t ) = − n − nρ n/ ( t ) ρ n/ ( t ) + ρ n/ ( t ) , t ≥ , (4.20)with the initial data (4.13).Solving (4.18) by noting (4.13), we get ρ ( t ) + ρ ( t ) = − t + a + a , ≤ t < T c := 14 ( a + a ) . Noticing N + = N − = n = N , thus M = − N = − n < N = 2 n vortices. Thus there exist 1 ≤ j ≤ n and 1 ≤ l ≤ n such that the vortex dipole x j and x n + l will collide at t = T c , i.e. ρ ( T c ) = ρ ( T c ) = 0. Therefore,the N = 2 n vortices will be a (finite time) collision cluster. In addition, Figure 4.2 depicts thesolution ρ ( t ) and ρ ( t ) of (4.20) obtained numerically with ρ (0) = 1 and ρ (0) = 4 for different n ≥ Remark 4.1. When a = a , i.e. ρ = ρ , we can get ρ ( t ) = ρ ( t ) = − t + a , which also implies the N = 2 n vortices will be a (finite time) collision cluster. UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 23 ρ ( t ) t n = 2 n = 3 n = 4 n = 5 ρ ( t ) t n = 2 n = 3 n = 4 n = 5 Figure 4.2. Time evolution of ρ ( t ) (left) and ρ ( t ) (right) of (4.20) with ρ = 1and ρ = 4 for different n ≥ For the interaction of two clusters and a single vortex. Here we take N = 2 n + 1 with n ≥ Proposition 4.4. Taking m j = m for ≤ j ≤ N = 2 n + 1 and the initial data X in (1.2) as x N = , x j = a (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , x n + j = a (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , ≤ j ≤ n, (4.21) then the solution of the ODEs (1.1) with (4.21) can be given as x N ( t ) ≡ , x j ( t ) = √ ρ ( t ) (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , x n + j ( t ) = p ρ ( t ) (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , ≤ j ≤ n, (4.22) where when n = 2 , ρ ( t ) = C + 10 t − q C + 8 C t + 40 t , ρ ( t ) = C + 10 t + q C + 8 C t + 40 t , t ≥ and when n ≥ , ρ ( t ) ∼ α t, ρ ( t ) ∼ β t, t ≫ , with α and β being two positive constants satisfying < α < β , α + β = 8 n + 4 , β − α = 4 n β n/ + α n/ β n/ − α n/ . Specifically, when n ≫ , we have α ≈ n + 2 , and β ≈ n + 2 . Proof. Due to symmetry, we get x N ( t ) ≡ for t ≥ 0. The rest of the proof is analogue to that ofProposition 4.1 and thus it is omitted here for brevity. Proposition 4.5. Taking m j = m for ≤ j ≤ N = 2 n + 1 and the initial data X in (1.2) as x N = , x j = a (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , x n + j = a (cid:0) cos( α jn ) , sin( α jn ) (cid:1) T , ≤ j ≤ n, (4.23) then the solution of the ODEs (1.1) with (4.23) can be given as x N ( t ) ≡ , x j ( t ) = p ρ ( t ) (cid:0) cos( θ jn ) , sin( θ jn ) (cid:1) T , x n + j ( t ) = p ρ ( t ) (cid:0) cos( α jn ) , sin( α jn ) (cid:1) T , ≤ j ≤ n, (4.24) where when n = 2 , ρ ( t ) = C + 10 t − C (cid:18) tC (cid:19) / , ρ ( t ) = C + 10 t + C (cid:18) tC (cid:19) / , t ≥ and when n ≥ , ρ ( t ) ∼ α t, ρ ( t ) ∼ β t, t ≫ , with α and β being two positive constants satisfying < α < β , α + β = 8 n + 4 , β − α = 4 n β n/ − α n/ β n/ + α n/ . Specifically, when n ≫ , α ≈ n + 2 and β ≈ n + 2 .Proof. Due to symmetry, we get x N ( t ) ≡ for t ≥ 0, and the rest of the proof is analogue to thatof Proposition 4.1 and 4.3, thus it is omitted here for brevity. Proposition 4.6. Taking m N = − m , m j = m for ≤ j ≤ n = N − and the initial data X 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 x ( t ) , x ( t ) and x ( t ) be a collision cluster among the vortices and theywill collide at the origin (0 , T in finite time;(ii) when n = 3 , then ρ ( t ) ∼ (cid:18) a a a + a (cid:19) , ρ ( t ) ∼ t, t ≫ 1; (4.25) (iii) when n ≥ , then ρ ( t ) ∼ α t, ρ ( t ) ∼ β t, t ≫ , with α and β being two positive constants satisfying < α < β , α + β = 8 n − , β − α = 4 n β n/ + α n/ β n/ − α n/ . Specifically, when n ≫ , we have α ≈ n − , and β ≈ n − . 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˙ ρ ( t ) + ˙ ρ ( t ) = 8 n − , ˙ ρ ( t ) − ˙ ρ ( t ) = 4 n + 8 nρ n/ ( t ) ρ n/ ( t ) − ρ n/ ( t ) , t > , which implies˙ ρ ( t ) = 2 n − − nρ n/ ( t ) ρ n/ ( t ) − ρ n/ ( t ) , ˙ ρ ( t ) = 6 n − nρ n/ ( t ) ρ n/ ( t ) − ρ n/ ( t ) , t > . (4.26)1) When n = 2, (4.26) reduces to˙ ρ ( t ) = − − ρ ( t ) ρ ( t ) − ρ ( t ) , ˙ ρ ( t ) = 6 + 8 ρ ( t ) ρ ( t ) − ρ ( t ) , t > . (4.27)Solving (4.27) with the initial data (4.13), we get ρ ( t ) = 2 t + C − q t + 8 C t + C , ρ ( t ) = 2 t + C + q t + 8 C t + C , t ≥ . Thus there exists a T c := h − C + p C + 2 a a i > 0, such that ρ ( T c ) = 0 , ρ ( T c ) > , ρ ( t ) > , ρ ( t ) > , t ∈ [0 , T c ) , which immediately implies that x ( t ), x ( t ) and x ( t ) be a collision cluster among the 5 vorticesand they will collide at the origin (0 , T when t → T − c . UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 25 t ρ ( t ) n =2 n =3 n =4 n =5 t ρ ( t ) n =2 n =3 n =4 n =5 Figure 4.3. Time evolution of ρ ( t ) (left) and ρ ( t ) (right) of (4.26) with ρ = 1and ρ = 4 for different n ≥ n = 3, (4.26) reduces to˙ ρ ( t ) = − ρ / ( t ) ρ / ( t ) − ρ / ( t ) , ˙ ρ ( t ) = 12 ρ / ( t ) ρ / ( t ) − ρ / ( t ) , t > , (4.28)which immediately implies ddt " p ρ ( t ) + 1 p ρ ( t ) = 0 = ⇒ p ρ ( t ) + 1 p ρ ( t ) ≡ a + a a a , t ≥ . (4.29)Since 0 < a < a , then ρ ( t ) and ρ ( t ) are monotonically decreasing and increasing functions,respectively. From (4.29), we know that 0 < ρ ( t ) < ρ ( t ) for t ≥ T max = + ∞ , i.e. thereis no finite time collision. Noting that M > 0, by Theorem 2.4, we havelim t → + ∞ ρ ( 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 forbrevity.In addition, Figure 4.3 depicts the solution ρ ( t ) and ρ ( t ) of (4.26) obtained numerically with ρ = 1 and ρ = 4 for different n ≥ Proposition 4.7. Taking m N = − m , m j = m for ≤ j ≤ n = N − and the initial data X 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 x ( t ) , x ( t ) and x ( t ) be a collision cluster among the vortices and they will collide at the origin (0 , T in finite time;(ii) when n = 3 , then ρ ( t ) ∼ (cid:18) a a a − a (cid:19) , ρ ( t ) ∼ t, t ≫ (iii) when n ≥ , then ρ ( t ) ∼ α t, ρ ( t ) ∼ β t, t ≫ , with α and β being two positive constants satisfying < α < β , α + β = 8 n − , β − α = 4 n β n/ − α n/ β n/ + α n/ . Specifically, when n ≫ , we have α ≈ n − , and β ≈ n − .Proof. The proof is analogue to that of Proposition 4.6 and thus it is omitted here for brevity. Proposition 4.8. Taking m N = − m , m j = m and m n + j = − m for ≤ j ≤ n and the initialdata X 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 x ( t ) , x ( t ) and x ( t ) be a collision cluster among the vortices and they will collide at the origin (0 , T in finite time;(ii) When n ≥ , all the N = 2 n + 1 vortices be a collision cluster and they will collide at theorigin (0 , T when t → T c := [ a + a ] .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˙ ρ ( t ) + ˙ ρ ( t ) = − , ˙ ρ ( t ) − ˙ ρ ( t ) = 8 − n + 8 nρ n/ ( t ) ρ n/ ( t ) + ρ n/ ( t ) , t > , (4.31)which implies˙ ρ ( t ) = 2 n − − nρ n/ ( t ) ρ n/ ( t ) + ρ n/ ( t ) , ˙ ρ ( t ) = 2 − n + 4 nρ n/ ( t ) ρ n/ ( t ) + ρ n/ ( t ) , t > . (4.32)Solving (4.31) with initial data (4.13), we get ρ ( t ) + ρ ( t ) = − t + a + a , t ≥ , which implies that a finite time collision must happen and 0 < T max ≤ T c := ( a + a ).1) When n = 2, from (4.32), we obtain˙ ρ ( t ) = − ρ ( t ) + 2 ρ ( t ) ρ ( t ) + ρ ( t ) < , ˙ ρ ( t ) = 6 ρ ( t ) − ρ ( t ) ρ ( t ) + ρ ( t ) , t > . (4.33)Solving (4.33) with the initial data (4.13), we have ρ ( t ) = ( − t + C )(2 C ( − t + C ) − , ρ ( t ) = ( − t + C )(3 − C ( − t + C )) , t ≥ . where C = a + a ( a + a ) > 0. Denote0 < T max = 2 C C − C = a ( a + a )6 a + 2 a = 2 a a + a T c < T c , then we have ρ ( T max ) = 0 , ρ ( T max ) > , ρ ( t ) > , ρ ( t ) > , t ∈ [0 , T max ) , which implies that only the three vortices x ( t ) , x ( t ) and x ( t ) be a collision cluster among the 5vortices and they will collide at the origin (0 , T when t → T − max .2) When n ≥ 3, by Theorem 2.4, only the n +1 vortices with x n +1 ( t ) , . . . , x n ( t ) and x N ( t ) cannotbe a collision cluster among the N vortices since they have the same winding number; and similarly,only the n + 1 vortices with x ( t ) , . . . , x n ( t ) and x N ( t ) cannot be a collision cluster among the N vortices since their collective winding number defined as M := P ≤ j 1] = n ( n − ≥ 0. Thus, in order to have a finite time collision, there exist1 ≤ j ≤ n and 1 ≤ l ≤ n such that the vortex dipole x j ( t ) and x n + l ( t ) will collide at t = T c , i.e. ρ ( T c ) = ρ ( T c ) = 0. Therefore, the N = 2 n + 1 vortices will be a (finite time) collision cluster.In addition, Figure 4.4 depicts the solution ρ ( t ) and ρ ( t ) of (4.32) obtained numerically with ρ (0) = 1 and ρ (0) = 4 for different n ≥ UANTIZED VORTEX DYNAMICS IN SUPERCONDUCTIVITY 27 ρ ( t ) t n = 2 n = 3 n = 4 n = 5 ρ ( t ) t n = 2 n = 3 n = 4 n = 5 Figure 4.4. Time evolution of ρ ( t ) (left) and ρ ( t ) (right) of (4.32) with ρ = 1and ρ = 4 for different n ≥ Conclusion. Based on the reduced dynamical law of a system of ordinary differential equations(ODEs) for the dynamics of N vortex centers, we have obtained stability and interaction patternsof quantized vortices in superconductivity. By deriving several non-autonomous first integrals of theODEs system, we proved global well-posedness of the N vortices when they have the same windingnumber and demonstrated that finite time collision might happen when they have different windingnumbers. When N = 3, we established rigorously orbital stability when they have the same windingnumber and classified their collision patterns when they have different winding numbers. Finally,under several special initial setups including interaction of two clusters, we obtained explicitly theanalytical solutions of the ODEs system. The analytical and numerical results demonstrated therich dynamics and interaction patterns of N vortices in superconductivity. Acknowledgments. This work was supported partially by the Academic Research Fund of Ministryof Education of Singapore grant No. R-146-000-223-112 (W.B.) and by the National Natural ScienceFoundation of China grant No. 11371166, 11501242 (S.S. and Z.X.). REFERENCES [1] W. Bao, Numerical methods for the nonlinear Schr¨odinger equation with nonzero far-field conditions , MethodsAppl. Anal., (2004), 367-387.[2] W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation , Kinet. Relat.Mod., (2013), 1-135.[3] W. Bao and Q. Tang, Numerical study of quantized vortex interaction in the nonlinear Schroedinger equationon bounded domains , Multiscale Model. Simul., (2014), 411-439.[4] W. Bao and Q. Tang, Numerical study of quantized vortex interaction in the Ginzburg-Landau equation onbounded domains , Commun. Comput. Phys., (2013), 819-850.[5] W. Bao, R. Zeng and Y. Zhang, Quantized vortex stability and interaction in the nonlinear wave equation , Phys.D, (2008), 2391-2410.[6] P. Bauman, C. Chen, D. Phillips and P. Sternberg, Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems , European J. Appl. Math., (1995), 115-126.[7] F. Bethuel, H. Brezis and F. H´elein, “Ginzburg-Landau Vortices”, Birkh¨auser, Boston, 1994.[8] S. J. Chapman and G. Richardson, Motion of vortices in type II superconductors , SIAM J. Appl. Math., (1995), 1275-1296.[9] J. E. Colliander and R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau-Schr¨odinger equation , Internat.Math. Res. Notices, (1998), 333-358.[10] Q. Du, Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity , Comput.Math. Appl., (1994), 119-133.[11] W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity , Phys. D, (1994), 383-404. [12] R. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices , Arch. Rat. Mech., (1998), 99-125.[13] A. Klein, D. Jaksch, Y. Zhang and W. Bao, Dynamics of vortices in weakly interacting Bose-Einstein condensates ,Phys. Rev. A, (2007), 043602.[14] O. Lange and B. Schroers, Unstable manifolds and Schr¨odinger dynamics of Ginzburg-Landau vortices , Nonlin-earity, (2002), 1471-1488.[15] F. Lin, Some dynamical properties of Ginzburg-Landau vortices , Comm. Pure Appl. Math., (1996), 323-360.[16] F. Lin, Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submani-folds , Comm. Pure Appl. Math., (1998), 385-441.[17] F. Lin and J. Xin, On the dynamical law of the Ginzburg-Landau vortices on the plane , Comm. Pure Appl.Math., (1999), 1189-1212.[18] P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation , J. Funct. Anal., (1995),334-344.[19] P. K. Newton and G. Chamoun, Vortex lattice theory: a particle interaction perspective , SIAM Rev., (2009),501-542.[20] J. Neu, Vortices in complex scalar fields , Phys. D, (1990), 385-406.[21] J. Neu, Vortex dynamics of the nonlinear wave equation , Phys. D, (1990), 407-420.[22] Y. Ovchinnikov and I. Sigal, Long-time behavior of Ginzburg-Landau vortices , Nonlinearity, (1998), 1295-1309.[23] Y. Ovchinnikov and I. Sigal, Asymptotic behavior of solutions of Ginzburg-Landau and relate equations , Rev.Math. Phys., (2000), 287-299.[24] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation , Clarendon Press, Oxford, 2003.[25] E. Sandier, The symmetry of minimizing harmonic maps from a two-dimernsional domain to the sphere , Ann.Inst. H. Poincar´e Anal. Non Lin´eaire, (1993), 549-559.[26] Y. Zhang, W. Bao and Q. Du, The dynamics and interaction of quantized vortices in the Ginzburg-Landau-Schr¨odinger equation , SIAM J. Appl. Math., (2007), 1740-1775.[27] Y. Zhang, W. Bao, and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schr¨odingerequation , European J. Appl. Math., (2007), 607-630. E-mail address : [email protected] E-mail address : E-mail address ::