Periodic Vortex Streets and Complex Monodromy

The explicit constructions of periodic and doubly periodic vortex relative equilibria using the theory of monodromy-free Schr\"odinger operators are described. Several concrete examples with the qualitative analysis of the corresponding travelling vortex streets are given.


Introduction
The study of vortex dynamics is a classical subject going back to Helmholtz [35]. If we identify the plane with the set of complex numbers C then the dynamics of N point vortices z 1 (t), . . . , z N (t) with circulations (or, vorticities) Γ 1 , . . . , Γ N is determined by the system In the periodic setting we have the equations where we assume for simplicity that the period L = π (see [4,22] and the pioneering paper by Friedman and Polubarinova [14]). In this paper we consider the periodic relative equilibria of the vortices described by the system 1 2πi where v = dz j dt is the common constant velocity of the vortices. A classical example is given by the so-called von Kármán vortex street [20,36,37], corresponding to the case N = 2, Γ 1 +Γ 2 = 0.
More recently the vortex dynamics in periodic domains was studied in [4,22,26,31], but the dynamics in general is known to be non-integrable and even the description of all relative equilibria remains a largely open question. One of the results of this paper is the following class of the explicit periodic relative vortex equilibria, most of which seem to be new.
Theorem 1.1. The configuration Σ k,φ,κ is a periodic relative vortex equilibrium moving with constant velocity v = −κ 2π for any non-critical κ / ∈ {k 1 , . . . , k n }. In the frame moving with the vortices the complex potential of the flow is is a trigonometric Baker-Akhiezer function for the corresponding monodromy-free Schrödinger operator For critical value κ = k j we have an equilibrium vortex configuration with complex potential W = 1 2πi log ψ j (z), where k (j) , φ (j) are the sets k, φ without k j and φ j respectively.
The proof (see Section 3 below) is based on a simple observation that the conditions of relative equilibrium coincide with the Stieltjes relations [33] and thus hold for all periodic trigonometric monodromy-free operators of the form (1.4). Such operators play an important role in the theory of Huygens principle as it was shown by Berest and Loutsenko [7]. They were classified in [6,8] (see Theorem 4.3 in [8]) and all turned out to be iterated Darboux transformations applied to trivial potential u = 0. For rational potentials a similar observation was known already for quite a while [5] (see also [3] and references therein), but in the periodic setting we have not seen this in the literature although it looks quite natural.
Von Kármán vortex streets correspond to the simplest case n = 1. Indeed, let us consider for simplicity k 1 = 1, φ 1 = 0, then We have one zero and one pole modulo π: the pole is z = 0 and the zero is the solution of cot z = iκ, which is equivalent to Assuming that κ is real and positive, we have 2 cases: κ > 1 (fast), κ < 1 (slow) when respectively. In the slow case we have the von Kármán street shown in Fig. 1, the fast case is shown in Fig. 2. The complex potential of the flow in the moving frame is W = 1 2πi log ψ(κ, z), the instantaneous complex potential of the flow in the fixed frame is For the critical value κ = 1 the zero of ψ goes to infinity: ψ = 1 sin z and we have trivial vortex equilibrium, consisting of points lπ, l ∈ Z with circulations −1.
Two examples in the case n = 2 are shown in Fig. 3, the details and more examples with some qualitative analysis are presented in Section 6. All the pictures in the paper were produced using Mathematica. The colour of the points indicates the sign of the circulations (which are generically ±1): red means positive, blue -negative circulations. The axes on all figures are x and y, such that z = x + iy.
Our approach based on the ideas of [33] is very general and can be applied to all monodromyfree operators. In particular, we apply it to the classical Whittaker-Hill operator and its Darboux transformations [17] to construct periodic equilibria in the presence of background flow with complex potential W = A cos 2z.
In the doubly periodic case we have more general result, when only one meromorphic solution is required. Let σ(z) and ℘(z) be classical Weierstrass elliptic functions [39].
(1.5) with m i ∈ Z, m 1 + · · · + m N = 0 and B ∈ C be a solution of the Schrödinger equation Then W (z) = 1 2πi log ψ(z) is the complex potential of a doubly periodic relative vortex equilibrium in the moving frame.
Conversely, let the set z 1 , . . . , z N with integer circulations m 1 , . . . , m N with zero sum be a doubly periodic relative vortex equilibrium. Then the function (1.5) with suitable constant B is a solution of equation (1.6) for some energy E.
The classical theory of Lamé equation going back to Hermite [39] and modern theory of elliptic solitons [2,9,19,24,28,29,32,34], which can be defined as the theory of monodromyfree operators of the form (1.6) (or Picard potentials in terminology of [15]), provides many examples of such solutions and thus new doubly periodic relative equilibria. In particular, the logarithm of the corresponding elliptic Baker-Akhiezer function is a complex potential for a relative doubly periodic vortex equilibrium in the moving frame. We should mention though that in contrast to trigonometric case an effective description of all elliptic finite-gap (or algebrogeometric) operators still remains an open problem, which was first emphasized by S.P. Novikov as part of the effectivisation programme in finite-gap theory.
Note that in all the examples we produce the circulations are integers, which might be useful for applications to liquid helium, where circulations are known to be quantized [13,30].

Monodromy-free Schrödinger operators and Stieltjes relations
Consider the Schrödinger operator in the complex domain z ∈ C with meromorphic potential u(z) having poles only of second order. The operator L is called monodromy-free if the corresponding Schrödinger equation The corresponding µ must satisfy the characteristic equation µ(µ + 1) = c −2 , which means that the equation (2.1) has a meromorphic solution only if the coefficient c −2 at any pole has a very special form: This condition is in fact not sufficient: the corresponding solution ϕ may have a logarithmic term. The following important lemma due to Duistermaat and Grünbaum [10] gives the conditions when this does not happen.
Let ψ(z) be a solution of the corresponding Schrödinger equation and f (z) = D log ψ(z). Then the potential u(z) can be expressed as

Proposition 2.2 ([33]
). Let f be a meromorphic function having the poles of the first order with integer residues. The Schrödinger operator L with the potential u = f + f 2 + const is monodromy-free if and only if at any pole z 0 with Res z=z 0 f = m the following generalised Stieltjes relations are satisfied: The proof is simple: substituting we can check that the trivial monodromy conditions c 2k−1 = 0, k = 0, 1, . . . , m − 1 are equivalent to the vanishing of the coefficients α 2k = 0, k = 0, 1, . . . , m − 1. The last relation c 2m−1 = 0 is then fulfilled automatically, see [33].
In particular, we always have the original Stieltjes relation: at every pole z i of f Res z=z i f 2 = 0, which Stieltjes used to give electrostatic interpretation of the zeroes of some classical polynomials [25]. We are going to use the same idea to produce some new relative vortex equilibria.

Trigonometric monodromy-free operators and periodic vortex streets
As we have already mentioned all π-periodic trigonometric monodromy-free operators of the form (1.4) are known [6,8] to be the result of several Darboux transformations applied to The corresponding potentials have the form where k 1 < k 2 < · · · < k n are distinct natural numbers, φ i ∈ C/πZ are arbitrary complex numbers modulo π, The Schrödinger equation coefficients being trigonometric polynomials in z, so the ratio ψ(κ, z) = W n (κ, z)/W n (z) is (a version of) the corresponding trigonometric Baker-Akhiezer function [18]. Let with some constants C and C be the corresponding factorisations with possible multiplicities.
We claim that the set of vortices with position at z 1 , . . . , z M +N with the corresponding circulations Γ 1 , . . . , Γ M +N described above is a periodic relative vortex equilibrium configuration moving with velocity v =κ/2π. Indeed, by Stieltjes relations for all j = 1, . . . , M + N . Comparing with (1.1) we see that Note that in the frame moving with the vortices the corresponding flow can be written as The instantaneous complex potential of the flow in the fixed frame is At the critical level κ = k j the Wronskian W k,φ,κ (z) reduces to wherek j means that k j is omitted from the list, and similarly for φ (j) . Some of the zeros disappear at infinity and the relative equilibriium becomes a genuine one, in agreement with the general claim by Montaldi, Solière and Tokieda [22] that if the sum of the vorticities is not zero the only relative equilibria are the usual equilibria. This completes the proof of Theorem 1.1. We will discuss many examples of corresponding relative vortex equilibria in Section 6. Here we will mention only new collinear vortex equilibria (when all the vortices are on a real line), related to Baker-Akhiezer configurations found by M. Feigin and D. Johnston [11].
One can produce more equilibriums by changing z → qz and extend this configuration to a moving vortex street by considering ψ(z) = W k,φ,κ (z)/W k,φ (z) with non-critical κ. This equation is special because for natural values of parameter s it has precisely s elementary eigenfunctions of the form ψ j (z) = ϕ j (z)e α cos 2x , j = 1, . . . , s, where ϕ j (z) are some trigonometric polynomials [38]. For example, for s = 3 we have Let I = {i 1 , . . . , i n } be a set of distinct natural numbers, be the Wronskian of the corresponding eigenfunctions. Following [17] consider the Darboux transformation of the Whittaker-Hill operator with the potential u = − 4αs cos 2x + 2α 2 cos 4x − 2D 2 log W I , The corresponding log-derivative f = D log ψ JI (z) has the form where as before a i and b j are the zeros of the denominator W I and numerator W J with circulations being negative multiplicities and multiplicities respectively.

By Proposition 2.2 we have
Res

Doubly-periodic vortex equilibria
The classical Lamé operator has the form where ℘(z) is the Weierstrass elliptic function and s is an integer. For the Lamé operator there are explicit formulae for the 2s + 1 eigenfunctions, known as Lamé functions, going back to Hermite [39]. Replacing in the previous section ψ i (z) by the Lamé functions we have new doubly periodic vortex equilibria.
Suppose that Then we claim that W (z) = 1 2πi log ψ(z) is the complex potential of a relative doubly periodic vortex equilibrium in the moving frame. Indeed, the function must satisfy the Riccati equation Since the residues of the potential are zero, the same must be true for f 2 , which implies the Stieltjes conditions and a, b are solutions of the linear system aω i + bω i = η i , i = 1, 2, with the usual notation η i = ζ(ω i ) (see [27,30]). If z 1 , . . . , z N are the zeros of ψ and Γ i = m i , then due to Stieltjes relations (5.1) we have which shows that the configuration of zeros of ψ is indeed a relative vortex equilibrium moving with velocity v =Ā. The complex potential of the flow in the moving frame is W = 1 2πi log ψ(z), the instantaneous one in the fixed frame is Conversely, assume that we have a doubly periodic relative vortex equilibrium z 1 , . . . , z N with integer circulations Γ i = m i , so that Of course, in that form the claim becomes almost a tautology since both conditions are equivalent to the Stieltjes relations (5.1), which it is not clear how to solve. Fortunately we have several concrete examples coming from the theory of elliptic solitons [2,9,15,19,24,28,29,32], which will be discussed elsewhere. Here

Examples: pictures and analysis
We restrict ourselves with the analysis of the vortex configurations given by Theorem 1.1. As we will see in spite of the simple explicit formulae there are many natural questions to answer already in this case.
As we have seen above, in the simplest case n = 1 we have the original von Kármán vortex streets [20,36,37].
6.1 Case n = 2 with k 1 = 1, k 2 = 2 An example of the corresponding vortex street is shown at Fig. 6. Let us assume that φ 1 = φ 2 = 0. As we will see in that case because of the cancelation we have only three vortices per period: one with circulation −2 (located at 0) and two with circulation +1.
6.2 Case n = 2 with k 1 = m, k 2 = n, φ 1 = φ 2 = 0 The corresponding configuration of vortices Σ m,n,κ is given by the zeros of The number of common zeros of W 0 m,n and W m,n depends on arithmetic of m and n. If d is the greatest common divisor of m and n then W 0 m,n and W m,n have d common zeros at z = lπ/d, l = 0, . . . , d − 1 of multiplicities 1 and 3 respectively. Thus we have per period m + n − 3d vortices of circulation −1, d vortices of circulation −2 and m + n − d vortices of circulation 1 (see the example with m = 10, n = 12 below). The same is true for generic values of κ. Since the picture suggests that the vortices lie on some curve let us try to find its shape. Our arguments here are similar to the analysis of the Wronskians of Hermite polynomials in [12].
For complex zeros of both W 0 m,n and W m,n we have | sin(m + n)z| = m + n n − m | sin(n − m)z|.
Let us assume that m, n are large compared with the difference m − n. Then in the upper half-plane z = x + iy, y > 0 the negative exponential term in sin(m + n)z, with modulus e (m+n)y /2, will dominate. Taking the modulus of both sides and assuming that y is small, so that sin(n − m)z ≈ sin(n − m)x, equation (6.2) becomes 1 2 e (m+n)y ≈ m + n n − m | sin(n − m)x|.
Taking logarithms and combining with the lower half-plane case, we have the following approximate formula for the curve on which the zeros lie 3) Fig. 11 shows a good agreement with this formula already for m = 7, n = 8. When the parameter κ (which is essentially velocity) increases from zero the red vortices lie on their own independent curves. We will now derive a formula for these curves, from equation (6.1). Setting W κ m,n = 0, then collecting terms with argument (n + m)z onto the left and those with argument (n − m)z onto the right, we have (n − m) κ(n + m) cos(n + m)z − i κ 2 + mn sin(n + m)z Writing the left hand side in terms of exponentials, we get n − m 2 (κ − n)(κ − m)e i(n+m)z − (κ + n)(κ + m)e −i(n+m)z = (n + m) κ(n − m) cos(n − m)z − i κ 2 − mn sin(n − m)z .
Since (n + m) is assumed to be large, in the upper half-plane the negative exponential term will dominate. Set z = x + iy and assume that y is small enough that sin z ≈ sin x and similarly for cos z. Taking the modulus of both sides we have Taking logarithms of both sides we arrive at the formula and by a similar calculation in the lower half-plane (where we keep the positive exponential term instead) we have a formula for the lower line of vortices. Combining the two gives us . (6.4) Formula (6.4) works well away from the critical values κ = m or κ = n, when we have the equilibria corresponding to ψ = sin nz/W m,n and ψ = sin mz/W m,n respectively (see Figs. 13 and 14). Conjecturally the formulae (6.3), (6.4) define the asymptotic curves for the zeros in the limit of large m and n with fixed difference n − m (cf. [12], where the Wronskians of Hermite polynomials were studied).
The configurations corresponding to m = 7 and n = 8 are displayed in the sequence of pictures shown in Figs. 12-15 for increasing values of the parameter κ. We give a qualitative description of what is happening at each stage: • κ = 0. The zeros of W m,n and W κ m,n are interlaced. • 0 < κ < m. The zeros of W κ m,n move downward whilst maintaining a similar overall form until κ approaches m, when they flatten out.
• κ = m. The first critical value. The bottom line of vortices tends to −i∞. The top line of vortices sit on the real axis at the zeros of sin nx.
• m < κ < n. The bottom line returns and a vortex is exchanged between the top and bottom lines.
• κ = n. The second critical case. Again, the bottom line of vortices tends to −i∞ and the top line of vortices sit on the real axis, only this time at the zeros of sin mx.
• κ > n. The red vortices move upwards and tend towards the blue vortices as κ → ∞.
Recall that in all these examples the velocity is horizontal since κ is real.

Concluding remarks
An explicit description of all monodromy-free operators is known only in a few cases: rational class of potentials decaying or with quadratic growth at infinity [23] and trigonometric class described in Section 3. Already in the sextic rational case the situation is far from clear, see [16] for the latest results in this direction. The same is true about monodromy-free perturbations of Whittaker-Hill operator [17] and already mentioned elliptic case. The link with vortex dynamics adds one more reason to the importance of these problems. It would be interesting to analyse from this point of view a class of the monodromy-free potentials in terms of the Painlevé-IV transcendents described in [33]. Another interesting question is to study the quasi-periodic case by allowing in the construction of Σ k,φ,κ non-integer k j .
The geometry of the trigonometric configurations Σ k,φ,κ is also worthy of studying further, in particular, the asymptotic shape of the corresponding vortex streets. It would be nice also to see if the shape of the corresponding Young diagram of k plays any role here, similarly to the case of Hermite polynomials in [12].
Finally, from the point of view of possible applications the stability of the new equilibria is crucial and is to be investigated (see Lamb [20] for the conditions of stability for the original von Kármán streets).