Periodic Vortex Streets and Complex Monodromy

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

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2014
Автори:	Hemery, A.D., Veselov, P.V.
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Опубліковано:	Інститут математики НАН України 2014
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Цитувати:	Periodic Vortex Streets and Complex Monodromy / A.D. Hemery, P.V. Veselov // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 39 назв. — англ.

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spelling	Hemery, A.D. Veselov, P.V. 2019-02-09T11:15:54Z 2019-02-09T11:15:54Z 2014 Periodic Vortex Streets and Complex Monodromy / A.D. Hemery, P.V. Veselov // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 76B47; 34M05; 81R12 DOI: http://dx.doi.org/10.3842/SIGMA.2014.114 https://nasplib.isofts.kiev.ua/handle/123456789/146403 The explicit constructions of periodic and doubly periodic vortex relative equilibria using the theory of monodromy-free Schrödinger operators are described. Several concrete examples with the qualitative analysis of the corresponding travelling vortex streets are given. We are very grateful to John Gibbons and Boris Khesin for helpful and encouraging discussions. We also thank all the referees for the critical comments and constructive suggestions. This work was mainly done in spring 2012 when the first author (ADH) was a PhD student at Loughborough University. The work of APV was partly supported by the EPSRC (grant EP/J00488X/1). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Periodic Vortex Streets and Complex Monodromy Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Periodic Vortex Streets and Complex Monodromy
spellingShingle	Periodic Vortex Streets and Complex Monodromy Hemery, A.D. Veselov, P.V.
title_short	Periodic Vortex Streets and Complex Monodromy
title_full	Periodic Vortex Streets and Complex Monodromy
title_fullStr	Periodic Vortex Streets and Complex Monodromy
title_full_unstemmed	Periodic Vortex Streets and Complex Monodromy
title_sort	periodic vortex streets and complex monodromy
author	Hemery, A.D. Veselov, P.V.
author_facet	Hemery, A.D. Veselov, P.V.
publishDate	2014
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	The explicit constructions of periodic and doubly periodic vortex relative equilibria using the theory of monodromy-free Schrödinger operators are described. Several concrete examples with the qualitative analysis of the corresponding travelling vortex streets are given.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/146403
citation_txt	Periodic Vortex Streets and Complex Monodromy / A.D. Hemery, P.V. Veselov // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 39 назв. — англ.
work_keys_str_mv	AT hemeryad periodicvortexstreetsandcomplexmonodromy AT veselovpv periodicvortexstreetsandcomplexmonodromy
first_indexed	2025-11-24T16:08:40Z
last_indexed	2025-11-24T16:08:40Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 114, 18 pages Periodic Vortex Streets and Complex Monodromy Adrian D. HEMERY † and Alexander P. VESELOV ‡§ † Charterhouse School, Godalming, Surrey, GU7 2DX, UK E-mail: adh@charterhouse.org.uk ‡ Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK E-mail: A.P.Veselov@lboro.ac.uk § Moscow State University, Russia Received August 28, 2014, in final form December 10, 2014; Published online December 23, 2014 http://dx.doi.org/10.3842/SIGMA.2014.114 Abstract. The explicit constructions of periodic and doubly periodic vortex relative equi- libria using the theory of monodromy-free Schrödinger operators are described. Several con- crete examples with the qualitative analysis of the corresponding travelling vortex streets are given. Key words: vortex; equilibria; monodromy; integrability 2010 Mathematics Subject Classification: 76B47; 34M05; 81R12 1 Introduction The study of vortex dynamics is a classical subject going back to Helmholtz [35]. If we iden- tify the plane with the set of complex numbers C then the dynamics of N point vortices z1(t), . . . , zN (t) with circulations (or, vorticities) Γ1, . . . ,ΓN is determined by the system dz̄j dt = 1 2πi N∑ k 6=j Γk zj − zk , j = 1, . . . , N. In the periodic setting we have the equations dz̄j dt = 1 2πi N∑ k 6=j Γk cot(zj − zk), j = 1, . . . , N, 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 N∑ k 6=j Γk cot(zj − zk)− v̄ = 0, j = 1, . . . , N, (1.1) where v = dzj 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, Γ1+Γ2 = 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. mailto:adh@charterhouse.org.uk mailto:A.P.Veselov@lboro.ac.uk http://dx.doi.org/10.3842/SIGMA.2014.114 2 A.D. Hemery and A.P. Veselov Figure 1. Classical von Kármán vortex street moving to the left. 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. Let k = (k1, . . . , kn), k1 < k2 < · · · < kn be a set of distinct natural numbers, and φ = (φ1, . . . , φn), φj ∈ C/πZ be a set of n complex numbers considered modulo π. Let Wk,φ(z) = W ( χk1,φ1(z), . . . , χkn,φn(z) ) (1.2) be the Wronskian with χkj ,φj (z) = sin(kjz + φj) and Wk,φ,κ(z) = W ( χk1,φ1(z), . . . , χkn,φn(z), eiκz ) , (1.3) where κ ∈ C is another complex parameter. Consider the combined configuration Σk,φ,κ of complex roots of Wk,φ,κ(z) = 0 and Wk,φ(z) = 0. Prescribe their circulations as follows: a zero zi of Wk,φ,κ(z) = 0 has the circulation Γi = mi equal to the multiplicity of zi while for a zero zj of Wk,φ(z) the circulation Γj = −mj is negative multiplicity of zj (for common zeros of Wk,φ,κ(z) and Wk,φ(z) the circulation is the difference of the corresponding multiplicities). Theorem 1.1. The configuration Σk,φ,κ is a periodic relative vortex equilibrium moving with constant velocity v = − κ̄ 2π for any non-critical κ /∈ {k1, . . . , kn}. In the frame moving with the vortices the complex potential of the flow is W = 1 2πi logψ(κ, z), where ψ(κ, z) = Wk,φ,κ(z) Wk,φ(z) is a trigonometric Baker–Akhiezer function for the corresponding monodromy-free Schrödinger operator L = −D2 + ∑ mj(mj − 1) sin2(z − zj) , D = d dz . (1.4) For critical value κ = kj we have an equilibrium vortex configuration with complex potential W = 1 2πi logψj(z), ψj = Wk(j),φ(j)(z) Wk,φ(z) , where k(j), φ(j) are the sets k, φ without kj and φj 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 Periodic Vortex Streets and Complex Monodromy 3 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 k1 = 1, φ1 = 0, then ψ(κ, z) = (iκ− cot z)eiκz. 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 e2iz = κ+ 1 κ− 1 if κ 6= 1. Assuming that κ is real and positive, we have 2 cases: κ > 1 (fast), κ < 1 (slow) when z = 1 2i log κ+ 1 κ− 1 and z = 1 2i log κ+ 1 1− κ + π 2 respectively. In the slow case we have the von Kármán street shown in Fig. 1, the fast case is shown in Fig. 2. Figure 2. Fast von Kármán vortex street, known to be unstable. The complex potential of the flow in the moving frame is W = 1 2πi logψ(κ, z), the instanta- neous complex potential of the flow in the fixed frame is W = 1 2πi logψ(κ, z)e−iκz = 1 2πi log(iκ− cot z). 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 monodromy- free 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]. Theorem 1.2. Let ψ(z) = N∏ i=1 σmi(z − zi)eBz (1.5) 4 A.D. Hemery and A.P. Veselov �3 �2 �1 1 2 3 �0.3 �0.2 �0.1 0.1 0.2 0.3 �3 �2 �1 1 2 3 �0.4 �0.2 0.2 Figure 3. Vortex streets Σk,0,κ with k = (7, 8), κ = 4 (left) and κ = 7.4 (right) with the graphs of the asymptotic curves (6.3), (6.4). with mi ∈ Z, m1 + · · ·+mN = 0 and B ∈ C be a solution of the Schrödinger equation −ψ′′ + ( N∑ i=1 mi(mi − 1)℘(z − zi) ) ψ = Eψ. (1.6) Then W (z) = 1 2πi logψ(z) is the complex potential of a doubly periodic relative vortex equilibrium in the moving frame. Conversely, let the set z1, . . . , zN with integer circulations m1, . . . ,mN with zero sum be a dou- bly 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 monodromy- free 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 algebro- geometric) 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]. 2 Monodromy-free Schrödinger operators and Stieltjes relations Consider the Schrödinger operator L = −D2 + u(z) 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 −ϕ′′ + u(z)ϕ = Eϕ (2.1) has all solutions meromorphic for all values of E. Near a pole (which can be assumed for simplicity to be z = 0) the potential can be represented as Laurent series u = ∞∑ i=−2 ciz i. Following the classical Frobenius approach one can look for the Periodic Vortex Streets and Complex Monodromy 5 solutions of the form ϕ = z−µ ( 1 + ∞∑ i=1 ξiz i ) . 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: c−2 = m(m+ 1), m ∈ Z+. (2.2) 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 condi- tions when this does not happen. Lemma 2.1 ([10]). The logarithmic terms are absent for all λ if and only if in addition to (2.2) all the first m+ 1 odd coefficients at the Laurent expansion of the potential are vanishing: c2k−1 = 0, k = 0, 1, . . . ,m. Let ψ(z) be a solution of the corresponding Schrödinger equation( −D2 + u(z) ) ψ(z) = λψ(z) and f(z) = D logψ(z). Then the potential u(z) can be expressed as u(z) = f ′ + f2 + λ. 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 ′ + f2 + const is monodromy-free if and only if at any pole z0 with Resz=z0 f = m the following generalised Stieltjes relations are satisfied: Resz=z0 f 2 = Resz=z0 f 4 = · · · = Resz=z0 f 2\|m\| = 0. The proof is simple: substituting f = ±m z − z0 + ∑ k=0 αk(z − z0)k with m ∈ Z+ into u = f ′ + f2 that c−2 = m(m ± 1) we can check that the trivial monodromy conditions c2k−1 = 0, k = 0, 1, . . . ,m − 1 are equivalent to the vanishing of the coefficients α2k = 0, k = 0, 1, . . . ,m− 1. The last relation c2m−1 = 0 is then fulfilled automatically, see [33]. In particular, we always have the original Stieltjes relation: at every pole zi of f Resz=zi f 2 = 0, which Stieltjes used to give electrostatic interpretation of the zeroes of some classical polyno- mials [25]. We are going to use the same idea to produce some new relative vortex equilibria. 6 A.D. Hemery and A.P. Veselov 3 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 L0 = −D2. The corresponding potentials have the form u(z) = −2D2 logW ( χk1,φ1(z), . . . , χkn,φn(z) ) , (3.1) where k1 < k2 < · · · < kn are distinct natural numbers, φi ∈ C/πZ are arbitrary complex numbers modulo π, χl,φ(z) = sin(lz + φ), l ∈ N. The Schrödinger equation( −D2 + u(z) ) ψ = κ2ψ with u(z) given by (3.1) has solution ψ(κ, z) = Wn(κ, z) Wn(z) , where Wn(z) = Wk,φ(z), Wn(κ, z) = Wk,φ,κ(z) are the Wronskians (1.2), (1.3). Note that Wn(κ, z) = Pn(κ, z)eiκz, where Pn(κ, z) = Wn(z)inκn + n−1∑ j=0 Aj(z)κ j is a polynomial in κ with coefficients being trigonometric polynomials in z, so the ratio ψ(κ, z) = Wn(κ, z)/Wn(z) is (a version of) the corresponding trigonometric Baker–Akhiezer function [18]. Let Wn(z) = C M∏ i=1 sinmi(z − ai), Wn(κ, z) = C ′ N∏ i=1 sinni(z − bi)eiκz with some constants C and C ′ be the corresponding factorisations with possible multiplicities. Then the log-derivative f = D logψ(κ, z) = D logWn(κ, z)−D logWn(z) has the form f = M∑ i=1 mi cot(z − bi)− N∑ j=1 nj cot(z − aj) + iκ = M+N∑ i=1 Γi cot(z − zi) + iκ, where z1, . . . , zM+N = a1, . . . , aM , b1, . . . , bN and Γi = mi for i = 1, . . . ,M and ΓM+j = −nj for j = 1, . . . , N . It may happen that ai = bj , so we have a cancelation with Γ = mi − nj in that case. We claim that the set of vortices with position at z1, . . . , zM+N with the corresponding circulations Γ1, . . . ,ΓM+N described above is a periodic relative vortex equilibrium configuration moving with velocity v = κ̄/2π. Indeed, by Stieltjes relations Resz=zj f 2 = 2Γj ∑ k 6=j Γk cot(zj − zk) + 2iΓjκ = 0 for all j = 1, . . . ,M +N . Comparing with (1.1) we see that dz̄j dt = 1 2πi ∑ k 6=j Γk cot(zj − zk) = − κ 2π , so we have the vortex configuration moving with constant speed v = −κ̄/2π. Periodic Vortex Streets and Complex Monodromy 7 Note that in the frame moving with the vortices the corresponding flow can be written as dz̄ dt = 1 2πi  M∑ i=1 mi cot(z − bi)− N∑ j=1 nj cot(z − aj) + iκ  = d dz 1 2πi logψ, so by definition W = 1 2πi logψ(κ, z) is the complex potential of the flow [1, 21]1. The instanta- neous complex potential of the flow in the fixed frame is W∗ = W − κ 2π z = 1 2πi log ( ψe−iκz ) = 1 2πi log Pn(κ, z) Wn(z) . At the critical level κ = kj the Wronskian Wk,φ,κ(z) reduces to Wk,φ,kj (z) = Wk(j),φ(j)(z), so we have ψ = Wk(j),φ(j)(z) Wk,φ(z) , where k(j) = (k1, . . . , k̂j , . . . , kn), where k̂j means that kj 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]. The corresponding vortex equilibrium Σ(n,m, l) depends on integer parameters n > m ≥ 1 and even l > 0. It corresponds to k = (1, 2, . . . , n−m,n−m+ 2, n−m+ 4, . . . , n+m− 2, n+m+ l) (or, more explicitly kj = j for j = 1, . . . , n−m, kn−m+j = n−m+ 2j for j = 1, . . . ,m− 1 and kn = n+m+ l) and φ = 0. One can show that in this case Wk(n),φ(n)(z) = C1 cos m(m−1) 2 z sin n(n−1) 2 z, Wk,φ(z) = C2 cos m(m+1) 2 z sin n(n+1) 2 z l∏ j=1 sin(z − zj), where z1, . . . , zl ∈ (0, π) are certain simple real zeros located symmetrically with respect to π/2 (see [11]). This means that the function W = 1 2πi logψ(z) with ψ(z) = Wk,φ(z)/Wk(n),φ(n)(z) = C cosm z sinn z l∏ j=1 sin(z − zj) is the complex potential of the collinear vortex equilibrium configuration Σ(n,m, l) consisting of z = 0 with circulation n, z = π/2 with circulation m and z = z1, . . . , zl with circulation 1. One can produce more equilibriums by changing z → qz and extend this configuration to a moving vortex street by considering ψ(z) = Wk,φ,κ(z)/Wk,φ(z) with non-critical κ. 1There is a sign discrepancy in the definition of the complex potential of the flow between Milne-Thomson [21] and Acheson [1]. We follow Acheson here. 8 A.D. Hemery and A.P. Veselov 4 Whittaker–Hill equation and relative vortex equilibria in the background flow Since the Darboux transformation preserves the class of monodromy-free operators one can use it in a more general situation to produce vortex equilibria in the presence of a background flow. We demonstrate it here in the example of the classical Whittaker–Hill equation −ψ′′ − ( 4αs cos 2x+ 2α2 cos 4x ) ψ = λψ. This equation is special because for natural values of parameter s it has precisely s elementary eigenfunctions of the form ψj(z) = ϕj(z)e α cos 2x, j = 1, . . . , s, where ϕj(z) are some trigonomet- ric polynomials [38]. For example, for s = 3 we have ϕ1 = 1 + √ 1 + 16α2 − 1 4α cos 2x, ϕ2 = sin 2x, ϕ3 = 1− √ 1 + 16α2 4α + cos 2x. Let I = {i1, . . . , in} be a set of distinct natural numbers, WI = W (ψi1 , . . . , ψin) = W (ϕi1 , . . . , ϕin)enα cos 2x be the Wronskian of the corresponding eigenfunctions. Following [17] consider the Darboux transformation of the Whittaker–Hill operator with the potential ũ = − ( 4αs cos 2x+ 2α2 cos 4x ) − 2D2 logWI , and its eigenfunction ψJI(z) = WJ(z)/WI(z), where J = {i1, . . . , in, in+1}, WJ = W (ψi1 , . . . , ψin , ψin+1) = W (ϕi1 , . . . , ϕin , ϕin+1)e(n+1)α cos 2x. The corresponding log-derivative f = D logψJI(z) has the form f = M∑ i=1 mi cot(z − bi)− N∑ j=1 nj cot(z − aj)− 2α sin 2z = M+N∑ i=1 Γi cot(z − zi)− 2α sin 2z, where as before ai and bj are the zeros of the denominator WI and numerator WJ with circula- tions being negative multiplicities and multiplicities respectively. By Proposition 2.2 we have Resz=zi f 2 = 2Γi ∑ j 6=i Γj cot(zi − zj)− 4Γiα sin 2zi = 0, or, ∑ j 6=i Γj cot(zi − zj)− 2α sin 2zi = 0 for all i = 1, . . . ,M +N . These Stieltjes relations can be interpreted as the vortex equilibrium relations in a periodic background flow (see Fig. 4). Theorem 4.1. The function W = 1 2πi logψJI(z) = 1 2πi(logWJ(z) − logWI(z)) is the complex potential of a vortex equilibrium in the background flow with complex potential α 2πi cos 2z. Two examples of vortex equilibria in the case s = 5 are shown in Fig. 5. Periodic Vortex Streets and Complex Monodromy 9 0 1 2 3 �4 �2 0 2 4 Figure 4. The background flow with complex potential W = cos 2z. �1 1 2 3 �1.0 �0.5 0.5 1.0 �3 �2 �1 1 2 3 �0.6 �0.4 �0.2 0.2 0.4 0.6 Figure 5. Vortex equilibria in the Whittaker–Hill case with s = 5, α = 1.5 and J = {4, 5}, I = {4} (left), J = {1, 5}, I = {1} (right). 5 Doubly-periodic vortex equilibria The classical Lamé operator has the form L = −D2 + s(s+ 1)℘(z), 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 ψi(z) by the Lamé functions we have new doubly periodic vortex equilibria. In fact, we have the following very general result. Let σ(z), ζ(z), ℘(z) be classical Weierstrass functions with periods 2ω1, 2ω2, see Whittaker–Watson [39]. Suppose that ψ(z) = N∏ i=1 σmi(z − zi)eBz with integer mi with zero sum, is a solution of the Schrödinger equation −ψ′′ + u(z)ψ = Eψ with the potential u(z) = N∑ i=1 mi(mi − 1)℘(z − zi). 10 A.D. Hemery and A.P. Veselov Then we claim that W (z) = 1 2πi logψ(z) is the complex potential of a relative doubly periodic vortex equilibrium in the moving frame. Indeed, the function f = D logψ = N∑ i=1 miζ(z − zi) +B must satisfy the Riccati equation f ′ + f2 = u(z)− E. Since the residues of the potential are zero, the same must be true for f2, which implies the Stieltjes conditions N∑ j 6=i mjζ(zi − zj) +B = 0, i = 1, . . . , N. (5.1) Now recall the equations for the dynamics of the doubly periodic configuration of vortices. Let z1, . . . , zN with circulations Γ1, . . . ,ΓN be part of such configuration within the parallelogram defined by the periods 2ω1, 2ω2. Then their dynamics is described by the equations dz̄k dt = 1 2πi  N∑ j 6=k Γjζ(zk − zj) + C  , i = 1, . . . , N, where C = a N∑ j=1 Γjzj + b N∑ j=1 Γj z̄j and a, b are solutions of the linear system aωi + bω̄i = ηi, i = 1, 2, with the usual notation ηi = ζ(ωi) (see [27, 30]). If z1, . . . , zN are the zeros of ψ and Γi = mi, then due to Stieltjes relations (5.1) we have dz̄k dt = 1 2πi (C −B) = A, 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 logψ(z), the instantaneous one in the fixed frame is W∗ = W +Az = 1 2πi log ( ψe(C−B)z ) . Conversely, assume that we have a doubly periodic relative vortex equilibrium z1, . . . , zN with integer circulations Γi = mi, so that dz̄k dt = 1 2πi  N∑ j 6=k Γjζ(zk − zj) + C  = A, then we have Stieltjes relations (5.1) with B = C − 2πiA, which guarantee the Riccati relation (since the functions are elliptic) and hence the Schrödinger equation of the required form. This proves Theorem 1.2. Periodic Vortex Streets and Complex Monodromy 11 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 we only mention classical Hermite’s solution of the Lamé equation Lψ = Eψ ψ = s∏ r=1 σ(ar − z) σ(z)σ(ar) eBz, B = s∑ r=1 ζ(ar), which gives doubly periodic vortex streets with z = 0 of circulation −s and zr = ar, r = 1, . . . , s of circulation 1, depending on an arbitrary parameter E (see the details in Whittaker and Watson [39, Section 23.7]). 6 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 k1 = 1, k2 = 2 An example of the corresponding vortex street is shown at Fig. 6. �1 1 2 3 �0.8 �0.6 �0.4 �0.2 0.2 0.4 0.6 Figure 6. Configuration Σk,φ,κ with k = (1, 2), φ = (0, π/2) and κ = 8. The circulations of the red and blue vortices are Γ = 1 and Γ = −1 respectively. Let us assume that φ1 = φ2 = 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. Indeed in this case W (sin z, sin 2z) = −2 sin3 z and W ( sin z, sin 2z, eiκz ) = − sin zeiκz (( 2 + κ2 ) cos 2z − 3iκ sin 2z + ( 4− κ2 )) , so we have the cancelation of sin z and ψ = W (sin z, sin 2z, eiκz) W (sin z, sin 2z) = ( 2 + κ2 ) cos 2z − 3iκ sin 2z + ( 4− κ2 ) 2 sin2 z eiκz. The equation( 2 + κ2 ) cos 2z − 3iκ sin 2z + ( 4− κ2 ) = 0 12 A.D. Hemery and A.P. Veselov can be rewritten as (κ− 1)(κ− 2)e2iz + (κ+ 1)(κ+ 2)e−2iz − 2(κ+ 2)(κ− 2) = 0, which is a quadratic in X = e2iz with roots X1,2 = ( κ2 − 4 ) ± √ 3 ( 4− κ2 ) (κ− 1)(κ− 2) . So, we have a relative equilibrium with the vortices at the following locations (and π-periodically) z0 = 0 with Γ = −2, z1,2 = 1 2i log ( (κ2 − 4)± √ 3(4− κ2) (κ− 1)(κ− 2) ) with Γ = 1 moving with the velocity v = −κ̄/2π for non-critical κ 6= 1, 2. When κ = 0 we have X1,2 = −2 ± √ 3 and an equilibrium configuration shown on the very left of Fig. 7 with red vortices z1,2 = π/2± i 2 log(2 + √ 3). Figs. 7–10 show what happens when the parameter κ grows. �3 �2 �1 1 2 3 �0.6 �0.4 �0.2 0.2 0.4 0.6 �3 �2 �1 1 2 3 �1.0 �0.8 �0.6 �0.4 �0.2 0.2 Figure 7. Σk,0,κ with k = (1, 2) and κ = 0 (equilibrium, L), κ = 0.5 (R). Circulation of the fixed vortex at 0 is −2, while the other vortices have circulation 1. �3 �2 �1 1 2 3 �1.0 �0.5 0.5 1.0 �3 �2 �1 1 2 3 �1.5 �1.0 �0.5 Figure 8. Σk,0,κ with k = (1, 2) and κ = 1 (critical, L), κ = 1.2 (R). Let us look at what happens near the critical values κ = 1 and κ = 2. Let κ = 1 + ε with small ε, then z1,2 ≈ 1 2i log ( (−3 + 2ε)± (3− ε) −ε ) , so the limiting value is z1 = 1 2i log(−1) = 1 2i log eiπ = π 2 while z2 ≈ 1 2i log(6/ε) → −i∞ goes to infinity. At the critical value κ = 1 we have a genuine equilibrium shown at Fig. 8 with 0 and π/2 having circulations −2 and 1 respectively corresponding to ψ = sin 2z/ sin3 z = 2 cos z/ sin2 z. Periodic Vortex Streets and Complex Monodromy 13 �3 �2 �1 1 2 3 �1.4 �1.2 �1.0 �0.8 �0.6 �0.4 �0.2 �3 �2 �1 1 2 3 �1.2 �1.0 �0.8 �0.6 �0.4 �0.2 Figure 9. Σk,0,κ with k = (1, 2) and κ = 1.9 (L), κ = 2.1 (R). �3 �2 �1 1 2 3 �0.5 �0.4 �0.3 �0.2 �0.1 �3 �2 �1 1 2 3 �0.25 �0.20 �0.15 �0.10 �0.05 Figure 10. Σk,0,κ with k = (1, 2) and κ = 3 (L), κ = 6 (R) Similarly, setting κ = 2 + ε, ε > 0 small z1,2 ≈ 1 2i log ( 4ε± (−12ε)1/2 ε ) ≈ 1 2i log ( ± 2i(3/ε)1/2 ) , so in the limit ε → 0 z1,2 go to −i∞ with the real parts approaching ±π/4. If instead we let κ = 2− ε, ε > 0, then z1,2 ≈ 1 2i log ( −4ε± (12ε)1/2 −ε ) = 1 2i log ( 4∓ 2(3/ε)1/2 ) , so this time in the limit ε→ 0 the real parts of z1,2 approach 0 and π/2 (see Fig. 9). At the critical level κ = 2 we have the trivial equilibrium with vortices of circulation −2 at 0 corresponding to ψ = sin z/ sin3 z = sin−2 z with both vortices with circulation 1 gone to infinity. When κ → ∞, then X1,2 → 1 so that red vortices z1,2 → nπ, n ∈ Z approach blue one (see Fig. 10). For κ < 0 the configuration will look the same as the corresponding positive configuration but will move to the left, instead. 6.2 Case n = 2 with k1 = m, k2 = n, φ1 = φ2 = 0 The corresponding configuration of vortices Σm,n,κ is given by the zeros of Wm,n = W (sinmz, sinnz) = ((n−m) sin(m+ n)z − (m+ n) sin(n−m)z)/2, W κ m,n = W (sinmz, sinnz, eiκz) = ( (n+m) ( κ2 −mn ) sin(n−m)z − (n−m) ( κ2 +mn ) sin(n+m)z + iκ ( n2 −m2 ) (cos(n−m)z − cos(n+m)z) ) eiκz/2. (6.1) When κ = 0 we have a genuine equilibrium. In that case W 0 m,n = mnW (cosmz, cosnz), where W (cosmz, cosnz) = n−m 2 sin(n+m)z + n+m 2 sin(n−m)z. 14 A.D. Hemery and A.P. Veselov The number of common zeros of W 0 m,n and Wm,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 Wm,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. �3 �2 �1 1 2 3 �0.15 �0.10 �0.05 0.05 0.10 0.15 Figure 11. Vortex equilibrium configuration Σ10,12,0,0 against the asymptotic curve (6.3). 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 Wm,n we have \| sin(m+ n)z\| = m+ n n−m \| sin(n−m)z\|. (6.2) 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 appro- ximate formula for the curve on which the zeros lie \|y\| = ± 1 m+ n ( log \| sin(n−m)x\|+ log 2(m+ n) n−m ) . (6.3) 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 ( κ2 +mn ) sin(n+m)z ) = (n+m) ( κ(n−m) cos(n−m)z − i ( κ2 −mn ) sin(n−m)z ) . Periodic Vortex Streets and Complex Monodromy 15 Writing the left hand side in terms of exponentials, we get n−m 2 ( (κ− n)(κ−m)ei(n+m)z − (κ+ n)(κ+m)e−i(n+m)z ) = (n+m) ( κ(n−m) cos(n−m)z − i ( κ2 −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 ≈ sinx and similarly for cos z. Taking the modulus of both sides we have n−m 2 (κ+ n)(κ+m)e(n+m)y ≈ (n+m) (( κ2 − n2 )( κ2 −m2 ) sin2(m− n)x+ κ2(n−m)2 )1/2 . Taking logarithms of both sides we arrive at the formula y = 1 2(n+m) log ∣∣∣∣κ− nκ+ n κ−m κ+m ∣∣∣∣+ 1 (n+m) log 2(n+m) n−m + 1 2(n+m) log ∣∣∣∣sin2(n−m)x+ κ2(n−m)2 (κ2 − n2)(κ2 −m2) ∣∣∣∣ , 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 y± = 1 2(n+m) log ∣∣∣∣κ− nκ+ n κ−m κ+m ∣∣∣∣ ± 1 2(n+m) ( 2 log 2(n+m) n−m + log ∣∣∣∣sin2(n−m)x+ κ2(n−m)2 (κ2 − n2)(κ2 −m2) ∣∣∣∣) . (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 = sinnz/Wm,n and ψ = sinmz/Wm,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 Wm,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 sinnx. • 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 sinmx. • 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. 16 A.D. Hemery and A.P. Veselov �3 �2 �1 1 2 3 �0.3 �0.2 �0.1 0.1 0.2 0.3 �3 �2 �1 1 2 3 �0.3 �0.2 �0.1 0.1 0.2 0.3 �3 �2 �1 1 2 3 �0.4 �0.3 �0.2 �0.1 0.1 0.2 0.3 Figure 12. Σ7,8,κ with κ = 0 (equilibrium, L), κ = 2 (M), κ = 6 (R). �3 �2 �1 1 2 3 �0.4 �0.2 0.2 �3 �2 �1 1 2 3 �0.3 �0.2 �0.1 0.1 0.2 0.3 �3 �2 �1 1 2 3 �0.4 �0.2 0.2 Figure 13. Σ7,8,κ with κ = 6.9 (L), κ = 7 (critical, M), κ = 7.2 (R). �3 �2 �1 1 2 3 �0.4 �0.3 �0.2 �0.1 0.1 0.2 0.3 �3 �2 �1 1 2 3 �0.4 �0.3 �0.2 �0.1 0.1 0.2 0.3 �3 �2 �1 1 2 3 �0.3 �0.2 �0.1 0.1 0.2 0.3 Figure 14. Σ7,8,κ with κ = 7.5 (L), κ = 7.6 (M), κ = 8 (critical, R). �3 �2 �1 1 2 3 �0.3 �0.2 �0.1 0.1 0.2 0.3 �3 �2 �1 1 2 3 �0.3 �0.2 �0.1 0.1 0.2 0.3 �3 �2 �1 1 2 3 �0.3 �0.2 �0.1 0.1 0.2 0.3 Figure 15. Σ7,8,κ with κ = 10 (L), κ = 30 (M), κ = 50 (R). 7 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 Σk,φ,κ non-integer kj . Periodic Vortex Streets and Complex Monodromy 17 The geometry of the trigonometric configurations Σk,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). Acknowledgements We are very grateful to John Gibbons and Boris Khesin for helpful and encouraging discus- sions. We also thank all the referees for the critical comments and constructive suggestions. This work was mainly done in spring 2012 when the first author (ADH) was a PhD student at Loughborough University. The work of APV was partly supported by the EPSRC (grant EP/J00488X/1). 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Periodic Vortex Streets and Complex Monodromy

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