Quantized vortex dynamics and interaction patterns in superconductivity based on the reduced dynamical law

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. By deriving several non-autonomous first integrals of the ODEs, we obtain qualitatively dynamical properties of a cluster of quantized vortices, including global existence, finite time collision, equilibrium solution and invariant solution manifolds. 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 initial setups, we can obtain analytical solutions for the nonlinear ODEs.

1. Introduction. In this paper, we study analytically and numerically stability and interaction patterns of the following system of ordinary differential equations (ODEs) describing the dynamics of N ≥ 2 quantized vortices in superconductivity based on the reduced dynamical law [21,12,15,26,27] x j (t) = 2m j N k=1,k =j with initial data x j (0) = x 0 j = (x 0 j , y 0 j ) T ∈ R 2 , 1 ≤ j ≤ N.

QUANTIZED 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], the dynamics 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 interaction patterns of the reduced dynamical law (1.1) with (1.2), which will generate important insights about quantized vortex dynamics and interaction patterns in superconductivity and is much simpler than to 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 collision when they have opposite winding numbers. For N = 3, we prove orbital stability when they have the same 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 and several 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 they have opposite winding numbers. In section 3, we prove orbital stability when they have the same winding number and find different collision patterns when they have different winding numbers for the dynamics of N = 3 vortices. Analytical solutions of the ODEs (1.1) are presented under several initial 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 establish dynamical properties of the system of ODEs (1.1) with the initial data (1.2) for describing the dynamics -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)| > 0 for 0 ≤ t < T c and d jl (T c ) = 0, then we say that they will be finite time collision or annihilation (cf. Fig. 2.1a); otherwise, i.e. d jl (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 there exists a finite time 0 < T c < +∞ such that min 1≤j<l≤N d jl (t) > 0 for 0 ≤ t < T c , lim t→T − c x j (t) = x 0 ∈ R 2 for j, k ∈ I with x 0 a fixed point and min j∈I d jl (t) > 0 for 0 ≤ t ≤ T c and l ∈ J := {m | 1 ≤ m ≤ N, m / ∈ I}, then we say that all vortices in the set I will form a (finite time) collision cluster among the N vortices (cf. Fig. 2.1b). Define it is easy to see that 0 < T max ≤ +∞ by noting (1.2). If T max < +∞, a finite time 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. T max = +∞, there is no collision among all the N quantized vortices (or the ODEs (1.1) with (1.2) is global well-posed in time).
2.1. Invariant solution manifolds. Let α > 0 be a positive constant, 0 ≤ θ 0 < 2π be a constant, x 0 ∈ R 2 be a given point and Q(θ) be the rotational matrix defined as Then it is easy to see that the ODEs (1.1) with (1.2) is translational and rotational invariant with the proof omitted here for brevity.
where e ∈ R 2 is a given unit vector. In fact, S e (x 0 ) is a line in 2D passing the point x 0 and parallel to the unit vector e. For X = (x 1 , . . . , (2.1) Noting the symmetric structure in (1.1) and (2.1), we can assume with the initial data by noting (2.1) The ODEs (2.3) with (2.4) is locally well-posed. Thus X(t) ∈ S e (x 0 ) for 0 ≤ t < T max .

2.2.
Non-autonomous first integrals. Let N + and N − be the number of vortices with winding number m = 1 and m = −1, respectively, then we have In addition, it is easy to get be the solution of the ODEs (1.1) with (1.2), then H 1 (X, t), H 2 (X, t) and H 3 (X, t) are non-autonomous first integrals of (1.1), i.e.
Proof. Differentiating the left equation in (2.8) (with X = X(t)) with respect to t, we have Using summation by parts and noting (2.7) and (1.1), we obtain (2.13) Plugging (2.13) into (2.12), we get which immediately implies the left equation in (2.10) by noting the initial condition (1.2). Similarly, differentiating (2.9) (with X = X(t)) with respect to t, we get Similar to (2.13), noting (1.1) and (2.7), we get which immediately implies the right equation in (2.10) by noting the initial condition (1.2). Therefore H 1 (X, t), H 2 (X, t) and H 3 (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 0 = +1 or −1 be fixed. When the N quantized vortices have the same winder number, e.g. m 0 , we have there is no finite time collision among the N quantized vortices. In addition, 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 which immediately implies that 2 ≤ M < N . Denote the non-empty sets I = {1, . . . , M } and J = {M + 1, . . . , N }, and define Then we have which yields Differentiating (2.20) with respect to t, we obtaiṅ .
Similar to (2.12) via (2.13) (with details omitted here for brevity), we geṫ This is a contradiction, and thus T max = +∞, i.e. there is no finite time collision among the N quantized vortices.
Then it is easy to see that d min (t) and D min (t) are continuous and piecewise smooth functions. In addition, for 1 ≤ j < l ≤ N , noting (1.1), we havė When the N vortices are initially collinear, we have Theorem 2.2. Suppose the N vortices have the same winding number, i.e. m j = m 0 for 1 ≤ j ≤ N in (1.1), and the initial data X 0 in (1.2) is collinear, then d min (t) and D min (t) are monotonically increasing functions.
Proof. Since X 0 is collinear, there exist x 0 ∈ R 2 and a unit vector e ∈ R 2 such that X 0 ∈ S e (x 0 ), by Lemma 2.2, we know that , without loss of generality, we assume that there exists 1 ≤ i 0 ≤ N − 1 (otherwise by re-ordering) such that Plugging j = i 0 and l = i 0 + 1 into (2.23) and noting (2.25), (2.24) and (2.22), we gavė Here we used dmin(t) ≥ |j − l| for 1 ≤ j < l ≤ N by noting (2.25). Thus D min (t) (and d min (t)) is a monotonically increasing function over [t 0 , t 1 ). Therefore, D min (t) (and d min (t)) is a monotonically increasing 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 ≥ 0.
Similarly, when 2 ≤ N ≤ 4 and X 0 ∈ R 2×N * , we have Proof. Taking 0 ≤ t 0 < t 1 such that d min (t) is smooth on [t 0 , t 1 ), without loss of generality, we assume that d min (t) = d 12 (t) for t 0 ≤ t < t 1 (otherwise by re-ordering). Taking j = 1 and l = 2 in which implies that D min (t) and d min (t) are monotonically increasing functions over t ∈ [t 0 , t 1 ]. When N = 4, without loss of generality, we can assume then we getḊ which implies that D min (t) and d min (t) are monotonically increasing functions over t ∈ [t 0 , t 1 ].
Remark 2.1. When N ≥ 5 and the initial data X 0 ∈ R 2×N * is not collinear, d min (t) might not be a monotonically increasing function, especially when 0 ≤ t ≪ 1. Based on our extensive numerical results, for any given X 0 ∈ R 2×N * , there exits a constant T 0 > 0 depending on X 0 such that d min (t) is a monotonically increasing function when t ≥ T 0 . Rigorous mathematical justification is ongoing.
(iii) If M 0 > 0 and 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 are omitted here for brevity.
(iv) When M = N , for any given X 0 ∈ R 2×N * , we get H 0 1 > 0. Noting (2.27) and which immediately implies that the N vortices cannot be a collision cluster when t ∈ [0, T max ] for any given X 0 . When 2 ≤ M < N and N ≥ 3, without loss of generality, we assume I = {1, . . . , M } and denote J = {M + 1, . . . , N }. Thus M 1 = 1≤j<l≤M m j m l ≥ 0. We will proceed the proof by the method of contradiction. Assume that the M vortices x 1 , . . . , .
This is a contradiction and thus the set of vortices {x j (t) | j ∈ I} cannot be a collision cluster among the N vortices for 0 ≤ t ≤ T max .

ZHIGUO XU, WEIZHU BAO AND SHAOYUN SHI
which immediately implies that ρ 2 (t) is a monotonically decreasing function and lim t→+∞ ρ 2 (t) = 0. Thus we have Thus the vortex x 2 (t) moves towardsx 0 along the line S e (x 0 ). Based on the results in Theorem 2.1, we know that at least two vortices must move to infinity when t → +∞. Thus we have Thus the other two vortices x 1 (t) and x 3 (t) repel with each other and move outwards to far field along the line S e (x 0 ) when t → +∞.

4.1.
For the interaction of two clusters. Here we take N = 2n with n ≥ 2.
Proof. The proof is analogue to that of Proposition 4.1 and thus it is omitted here for brevity. Taking m j = m 0 and m n+j = −m 0 for 1 ≤ j ≤ n and the initial data X 0 in (1.2) as (4.14), then the solution of the ODEs (1.1) with (4.14) can be given as (4.15), where which implies that the N = 2n vortices will be a (finite time) collision cluster.

4.2.
For the interaction of two clusters and a single vortex. Here we take N = 2n + 1 with n ≥ 2.
Proof. Due to symmetry, we get x N (t) ≡ 0 for t ≥ 0. The rest of the proof is analogue to that of Proposition 4.1 and thus it is omitted here for brevity.
Proof. Due to symmetry, we get x N (t) ≡ 0 for t ≥ 0, and the rest of the proof is analogue to that of Proposition 4.1 and 4.3, thus it is omitted here for brevity.  .22), where (i) when n = 2, then x 1 (t), x 2 (t) and x 5 (t) be a collision cluster among the 5 vortices and they will collide at the origin (0, 0) T in finite time; (ii) when n = 3, then with α 5 and β 5 being two positive constants satisfying Specifically, when n ≫ 1, we have α 5 ≈ 2n − 6, and β 5 ≈ 6n − 6.
3) When n ≥ 4, the proof is analogue to that of Proposition 4.1 and thus it is omitted here for brevity.
where C 3 = 3a 2 1 +a 2 2 (a 2 1 +a 2 2 ) 2 > 0. Denote T c < T c , then we have which implies that only the three vortices x 1 (t), x 2 (t) and x 5 (t) be a collision cluster among 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 x n+1 (t), . . . , x 2n (t) and x N (t) cannot be a collision cluster among the N vortices since they have the same winding number; and similarly, only the n + 1 vortices with x 1 (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 1 := 1≤j<l≤n m j m l + n j=1 m j m N = 1 2 [(n − 1) 2 − n − 1] = 1 2 n(n − 3) ≥ 0. Thus, in order to have a finite time collision, there exist 1 ≤ j 0 ≤ n and 1 ≤ l 0 ≤ n such that the vortex dipole x j0 (t) and x n+l0 (t) will collide at t = T c , i.e. ρ 1 (T c ) = ρ 2 (T c ) = 0. Therefore, the N = 2n + 1 vortices will be a (finite time) collision cluster.

5.
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 patterns of quantized vortices in superconductivity. By deriving several non-autonomous first integrals of the ODEs system, we proved global well-posedness of the N vortices when they have the same winding number and demonstrated that finite time collision might happen when they have different winding numbers. When N = 3, we established rigorously orbital stability when they have the same winding number 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 the analytical solutions of the ODEs system. The analytical and numerical results demonstrated the rich dynamics and interaction patterns of N vortices in superconductivity.