Genus-zero Whitham hierarchies in conformal-map dynamics

Physics Letters B 641 (2006) 466–473 www.elsevier.com/locate/physletb Genus-zero Whitham hierarchies in conformal-map dynamics ✩ Luis Martínez-Alonso a,∗ , Elena Medina b a Departamento de Física Teórica II, Universidad Complutense, E-28040 Madrid, Spain b Departamento de Matemáticas, Universidad de Cádiz, E-11510 Puerto Real, Cádiz, Spain Received 17 February 2006; accepted 24 August 2006 Available online 25 September 2006 Editor: L. Alvarez-Gaumé Abstract A scheme for solving quasiclassical string equations is developed to prove that genus-zero Whitham hierarchies describe the deformations of planar domains determined by rational conformal maps. This property is applied in normal matrix models to show that deformations of simplyconnected supports of eigenvalues under changes of coupling constants are governed by genus-zero Whitham hierarchies.  2006 Elsevier B.V. All rights reserved. MSC: 58B20 Keywords: Whitham hierarchy; Conformal maps; Normal matrix models 1. Introduction Conformal mapping methods have been effectively applied in the analysis of interfacial free-boundary problems involving planar domains [1]. They have provided many exact solutions [2–4] which stimulated the research on possible underlying integrable structures. Thus, Wiegmann and Zabrodin discovered [5,6] that deformations of simply-connected domains with respect to changes of their exterior harmonic moments, treated as independent variables, are described by the dispersionless Toda hierarchy. They also formulated an algebro-geometric analysis [7,8] of the deformations of multiply-connected domains in terms of Whitham equations for Abelian differentials. Recent research has shown [9,10] that many exact solutions of Laplacian growth models correspond to a special type of domains called algebraic or quadrature domains. In the simply-connected case the complement of a quadrature domain D is the image of the exterior of the unit disk under a conformal map given by a rational function z(w) = r w + N0  u0,n n=0 wn + Ns k   s=1 n=1 us,n , (w − as )n (1) where the coefficient r is a positive number and the k poles as = 0 lie inside the unit circle. In this work we prove that the deformations of rational conformal maps under changes of the parameters (r, u0,n , us,m , as ) such that z(ās−1 ) are kept constant, turn out to be described by a solution of the genus-zero Whitham hierarchy W(n) with n = 2k + 2 punctures [11]. It should be noted that according to a recent general result by Takasaki [12] the Whitham hierarchy W(n) is the quasiclassical limit of the n-component KP hierarchy. ✩ Partially supported by MEC project FIS2005-00319. * Corresponding author. E-mail address: [email protected] (L. Martínez-Alonso). 0370-2693/$ – see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.08.082 L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 467 Our analysis is based on solving a system of quasiclassical string equations which leads to the characterization of the conformal map (1) as a function z(w, t) of the Whitham times t. For k = 0 our result agrees with [5] (see also [13]) since W(2) is the dispersionless Toda hierarchy. However, for k  1 the analysis of [5] does not apply because the Schwarz function of the boundary of D has poles outside D, consequently there are infinite exterior harmonic moments of D different from zero and, since z(w) depends on a finite number of parameters only, these harmonic moments are not independent variables. Deformations of quadrature domains naturally arise in the analysis of partition functions of N × N normal matrix models [14]  1 † ZN = e h̄ W (M,M ) dM dM † , (2) with quasiharmonic potentials W (z, z̄) := −zz̄ + V (z) + V (z) of the form   N N k 0 +1 s −1  t s,n zn t0,n + . −ts,0 log(z − βs ) + V (z) = (z − βs )n s=1 n=1 Integrating over eigenvalues and ignoring normalization factors, the partition function reduces to    1  ZN = |zi − zj |2 e h̄ j W (zj ,z̄j ) d2 zj . i>j (3) n=1 (4) j In the large N limit (N → ∞, h̄N fixed) the eigenvalues densely occupy a bounded quadrature domain D in the complex plane (the support of eigenvalues). As a consequence of our analysis we prove that for simply-connected supports of eigenvalues the corresponding rational conformal map z = z(w, t) as a function of the coupling constants t of the partition function represents a solution of the Whitham hierarchy W(2k + 2). 2. String equations in Whitham hierarchies The elements of the phase space for a genus-zero Whitham hierarchy W(M + 1) are characterized by M + 1 punctures qα (α = 0, . . . , M), where q0 := ∞, of the extended complex p-plane and an associated set of local coordinates of the form z0 = p + ∞  c0,n n=1 pn ∞ , zi =  di di,n (p − qi )n , + p − qi i = 1, . . . , M. (5) n=0 In what follows Greek and Latin suffixes will label indices of the sets {0, . . . , M} and {1, . . . , M}, respectively. We will henceforth suppose that there exist positively oriented closed curves Γµ in the complex planes of the variables zµ such that each function zµ (p) −1 (Γ ) on the exterior of Γ (we will assume that the circle γ determines a conformal map of the right-exterior of a circle γµ := zµ µ µ 0 encircles all the γi ) (see Fig. 1). The flows of the Whitham hierarchy can be formulated as the following infinite system of quasiclassical Lax equations ∂zα = {Ωµn , zα }, ∂tµ,n (6) associated to the series of time parameters {t0,n : n  1; ti,n : i = 1, . . . , M, n  0}. Here the Poisson bracket is defined as {F, G} := ∂p F ∂x G − ∂x F ∂p G, x := t01 and the Hamiltonian functions are  n , (n  1), Ωi0 := − log(p − qi ), Ωµn := zµ (7) (µ,+) Fig. 1. Right-exteriors of γµ and Γµ . 468 L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 n ∞ where (·)(i,+) and (·)(0,+) stand for the projectors on the subspaces generated by {(p − qi )−n }∞ n=1 and {p }n=0 in the corresponding spaces of Laurent series. These hierarchies [12] are the dispersionless limits of the multi-component KP hierarchies. In particular, for M = 0 and M = 1 they represent the dispersionless versions of the KP and Toda hierarchies, respectively. In our analysis we will use an extended Lax formalism with Orlov functions mα (zα , t) = ∞  ntα,n zαn−1 + n=1 tα,0  vαn , + zα zαn (8) n2 which verify the same Lax equations (6) as the variables zα , and such that {zα , mα } = 1. The parameter t0,0 in (8) is defined by t0,0 := − M  (9) ti,0 . i=1 The Whitham hierarchy can be formulated as the following system of equations dzα ∧ dmα = dω, (10) ∀α, where ω is the one-form  Ωµn dtµ,n . ω := (11) µ,n To see how to get from the system (10) to the Whitham hierarchy, note that by identifying the coefficients of dp ∧ dtµn and dx ∧ dtµn in (10) we obtain ∂Ωµn ∂zα ∂mα ∂mα ∂zα − = , ∂p ∂tµn ∂p ∂tµn ∂p ∂Ωµn ∂zα ∂mα ∂mα ∂zα − = ∂x ∂tµn ∂x ∂tµn ∂x (12) and, in particular, since Ω01 = p, for (µ, n) = (0, 1), the system (12) implies {zα , mα } = 1. Thus, using this fact and solving (12) for ∂tµn zα and ∂tµn mα , the Lax equations for (zα , mα ) follow. A natural form of characterizing solutions of Whitham hierarchies is provided by systems of string equations Pi (zi , mi ) = P0 (z0 , m0 ), Qi (zi , mi ) = Q0 (z0 , m0 ), i = 1, 2, . . . , M, (13) where {Pα , Qα }M α=0 satisfy {Pα (p, x), Qα (p, x)} = 1. Given a solution (zα (p, t), mα (p, t)) of a system (13), if we denote   Pα (p, t) := Pα zα (p, t), mα (p, t) , Qα (p, t) := Qα zα (p, t), mα (p, t) , it is clear that dPα ∧ dQα = dPβ ∧ dQβ , (14) ∀α, β. On the other hand {Pα (p, x), Qα (p, x)} = {zα , mα } = 1, so that solutions of a system of string equations verify dPα ∧ dQα = dzβ ∧ dmβ , (15) ∀α, β. Theorem. Let (zα (p, t), mα (p, t)) be a solution of (13) which admits expansions of the form (5), (8), (9) and such that the coefficients of the two-forms (15) are meromorphic functions of the complex variable p with finite poles at {q1 , . . . , qM } only. Then (zα (p, t), mα (p, t)) is a solution of the Whitham hierarchy. Proof. In view of the hypothesis of the theorem the coefficients of the two-forms (15) with respect to the basis {dp ∧ dtα,n , dtα,n ∧ dtβ,m } are determined by their principal parts at qµ (µ = 0, . . . , M), so that by taking (15) into account we may write dzα ∧ dmα = M  (dzµ ∧ dmµ )(µ,+) , ∀α. µ=0 Moreover the terms in these decompositions can be found by using the expansions (8) of the functions mµ as follows  ∞    ∞    dvµn  1 dvµn dt µ,0 n−1 n dtµ,n + nzµ + dtµ,n + log zµ dtµ,0 − zµ dzµ ∧ dmµ = dzµ ∧ , =d n n−1 zµ zµ n − 1 zµ n=1 n2 n=1 n2 469 L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 so that  (dzµ ∧ dmµ )(µ,+) = d Thus we find dzα ∧ dmα = dω = d ∞   n dt zµ (µ,+) µ,n  − (1 − δµ0 ) log(p − qµ ) dtµ,0 = d n=1  Ωµn dtµ,n ,  Ωµn dtµ,n . n ∀α µ,n and, consequently, this proves that the functions (zα (p, t), mα (p, t)) determine a solution of the Whitham hierarchy. ✷ 3. Integrable dynamics of quadrature domains Let us consider a rational conformal map z(w) of the form (1) with k poles as inside the unit circle and define N0 Ns k     r ūs,n w n n −1 = + . ū0,n w + z̃(w) := z w̄ w (1 − w ās )n (16) s=1 n=1 n=0 In order to establish the connection between the deformations of the conformal map z(w) and the genus-zero Whitham hierarchies we introduce the change of variable p = Rw := rw + u0,0 , where r and u0,0 are the first coefficients of z(w) in (1). As a function of the new variable p the conformal map is normalized at infinity z(p) = p + O(1/p), p → ∞. Moreover, if we define q0 := ∞, qs := Rbs , qs+k := Ras , q2k+1 := u0,0 , it is clear that z(p) and z̃(p) become rational functions of p with poles at (q0 , qs , q2k+1 ) and (q0 , qs+k , q2k+1 ) respectively. We are going to prove that deformations of z(w) with respect to the coefficients u := (r, u0,n , us,m , as ), such that βs = z(w)|bs are kept constant, are described by the Whitham hierarchy W(2k + 2). To this end we introduce Whitham variables (zα , mα ) on the 2k + 2 punctures qα of the p-plane ⎧ z0 = z, m0 = z̃ (near q0 = ∞), ⎪ ⎪ ⎪ ⎨ zs = 1 , ms = −(z − βs )2 z̃ (near qs ), z−βs (17) 1 ¯s )2 z (near qs+k ), ⎪ z , m = (z̃ − β = s+k s+k ⎪ ¯ z̃−βs ⎪ ⎩ z2k+1 = z̃, m2k+1 = −z (near q2k+1 ). It is clear that these variables are rational functions of p with possible poles at the punctures qα only. Moreover, they satisfy the system of string equations 1 2 zs + βs = ms+k zs+k = −m2k+1 = z0 , (18) −ms zs2 = 1 + β¯s = z2k+1 = m0 . zs+k Obviously the functions zα are of the form (5). On the other hand due to (1) and (16) it follows that the functions mα defined in (17) verify expansions of the form (8) m0 = z̃ = N 0 +1 nt0,n z0n−1 + n=1 ms = −zs−2 z̃ = N s −1 nts,n zsn−1 + n=1 −2 ms+k = zs+k z= t0,0 + ···, z0 N s −1 n=1 w → ∞, ts,0 + ···, zs n−1 nts+k,n zs+k + w → bs , ts+k,0 + ···, zs+k w → as , 470 L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 m2k+1 = −z = N 0 +1 n−1 nt2k+1,n z2k+1 + n=1 t2k+1,0 + ···, z2k+1 (19) w → 0, where the time parameters t := (tα,n ) are rational functions in u (20) tαn = Qα,n (u). The functions Qα,n (u) satisfy certain constraints which can be characterized by considering the map  C : f → Cf, Cf (w) := f w̄ −1 . (21) Observe that in terms of the variable p (22) Cf (p) = f (Ip), where Ip = r 2 /(p − u0,0 ) + u0,0 is the inversion with respect to the circle |p − u0,0 |2 = r 2 . From (17) it is clear that z2k+1 = Cz0 , m2k+1 = −Cm0 , zs+k = Czs , (23) ms+k = −Cms , which implies Q2k+1,n (u) = −Q0,n (u), (24) Qs+k,n = −Qs,n (u). Furthermore, we can prove that  Qα,0 (u) = 0. (25) α Indeed from (17) we deduce m0 dz0 = ms dzs = z̃ dz, m2k+1 dz2k+1 = ms+k dzs+k = −z dz̃. Hence 2πi  Qα,0 = α  α mα dzα = Γα  z̃ ∂p z dp = 0, α γ α where we have taken into account that z̃ ∂p z is a rational function of p with poles at the punctures qα only, and the fact that  γα ∼ 0 in C \ {q1 , . . . , q2k+1 }. α Notice that due to (24) the constraint (25) can be rewritten as  Im Q0,0 + Im Qs,0 (u) = 0. (26) s Under appropriate conditions one can determine u as a function of (t, βs , β̄s ). To this end we consider the system tα,n = Qα,n (u), βs = z(bs , u), (27) where the time parameters t := (tα,n ), α = 0, . . . , 2k + 1; n = 0, . . . , Ñα ; Ñ0 = Ñ2k+1 = N0 + 1, Ñs = Ñs+k = Ns − 1, (28) are assumed to satisfy t2k+1,n = −t¯0,n , ts+k,n = −t¯s,n ,  tα,0 = 0. (29) α Firstly, we observe that u constitutes a set of  Ns + k + 3, 2 N0 + (30) s real variables given by r and the real and imaginary parts of (u0,n , us,m , ai ). On the other hand, in view of (24) and (25)  we may ignore the equations corresponding to α = s + k, 2k + 1 in (27). In this way the system (27) reduces to 2(N0 + 2 + s Ns + k) real equations, but due to (26) one of them is a consequence of the others. Therefore, we are lead to a system of equal number of equations and unknowns which under appropriate conditions will determine u, and consequently (zα , mα ) as functions of (t, βs ). L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 471 In this way we have determined a rational solution (zα (p), mα (p)) of the system of string equations (18) which depends on (t, β1 , . . . , βk ) and satisfies the asymptotic conditions (5) and (8). Therefore, from the above theorem we conclude that this solution evolves with respect to t according to the Whitham hierarchy W(2k + 2). It is interesting to notice the following identity involving the times tα0 and the area T of the domain D. Let Γ be the positively oriented boundary of D and take small positively oriented closed curves Γs′ around the points βs of the z-plane, then we have that     1   1   1 1 1 1 t0,0 = (31) z̃ dz + z̃ dz = zs−2 z̃ dzs + z̃ dz = z̄ dz = − ts,0 + T . 2πi 2πi 2πi 2πi 2πi π s s s Γs′ Γ0 Γ Γs Γ 4. Exact solutions The above analysis can be used to generate solutions of W(2k + 2) of the form (1). Let us consider the case in which only simple poles arise z = rw + u0 + k  s=1 k vs , w − as  v̄s w r z̃ = + ū0 + . w 1 − ās w s=1 z0−1 z00 By identifying the coefficients of and in the expansion of m0 = z̃ as z0 = z → ∞ (w → ∞), we get   k k   v̄s v̄s = t0,1 , r r− = t0,0 , ū0 − ās ās2 (32) s=1 s=1 and identifying the coefficient of zs−1 in the expansion of ms = −(z − βs )2 z̃ as zs → ∞ (w → bs ) yields   k  v̄s vs ′ ās2 = ts,0 . r− ās2 (1 − as ′ ās )2 ′ (33) s =1 Finally, the equations z(w)|w=bs = βs read k  vs ′ ās r + u0 + = βs . ās 1 − ās as ′ ′ (34) s =1 The system (32)–(34) determines (r, u0 , vs , as ) in terms of (t0,0 , t0,1 , ts,0 , βs ). Examples A solution of W(4) is obtained by deforming the conformal map (aircraft wind) z = rw + u0 + v , w−a z̃ = v̄w r + ū0 + . w 1 − āw The solution of the corresponding equations (32)–(34) is given by: a= r 2 − At0,0 , r(β̄ − t0,1 )(1 − A) u0 = v= (r 2 − t 0,0 )(r 2 − t0,0 A)2 , r 3 (β̄ − t0,1 )2 (1 − A)2 r 4 − r 2 t0,0 A + r 2 t 0,1 (β̄ − t0,1 )(1 − A) + |t0,0 |2 A − r 2 t 0,0 , r 2 (β̄ − t0,1 )(1 − A) with r2 = |β̄ − t0,1 |2 (2A3 − 3A2 2|t0,0 |2 A2 , + A) + (t0,0 + t1,0 )(A − 1) + (t0,0 + t 0,0 )A where A = |a|2 is implicitly defined by   2|β̄ − t0,1 |4 A5 + 2(t0,0 + t1,0 ) − 2(t0,0 + t 0,0 ) − 5|β̄ − t0,1 |2 |β̄ − t0,1 |2 A4   + 4 (t0,0 + t 0,0 ) − (t0,0 + t1,0 ) + |β̄ − t0,1 |2 |β̄ − t0,1 |2 A3   + (t0,0 + t1,0 )2 − (t0,0 + t 0,0 )2 + 4|t0,0 |2 − 2(t0,0 + t 0,0 )|β̄ − t0,1 |2 + 2(t0,0 + t1,0 )|β̄ − t0,1 |2 − |β̄ − t0,1 |4 A2 − 2(t0,0 + t1,0 )2 A + (t0,0 + t1,0 )2 = 0. 472 L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 Fig. 2. Solution corresponding to k = 1. Fig. 3. Solution corresponding to k = 2. It can be proved that a reduction of the system (32)–(34) is obtained by setting z = rw + u + l  v̄s vs + , w − as w − ās s=1 l z̃ =  vs w r v̄s w +u+ + , w 1 − ās w 1 − as w (35) s=1 and by assuming u, t0,0 , t0,1 ∈ R, ts+l,0 = t¯s,0 , βs+l = β̄s . In the simplest case l = 1 the reduced system of implicit equations reads u− v1 v̄1 − = t0,1 , a1 ā1 r r− v̄1 v1 − a1 2 ā12 = t0,0 , v̄1 v̄1 v1 r − − 2 2 2 (1 − a1 ā1 )2 ā1 (1 − ā1 ) = t1,0 , v1 ā1 v̄1 ā1 r +u+ + = β1 , ā1 1 − a1 ā1 1 − ā12 (36) and determines a solution of W(6). Figs. 2 and 3 exhibit deformations of domains with respect to changes of the area T such that the Whitham times ts,0 are kept constant. Observe that the evolution of the boundary develops cusp-like singularities [15]. 5. Whitham times and coupling constants of normal matrix models Let us assume that the support of eigenvalues of the normal matrix model (2) is a simply-connected domain D. For example this is the case if k = 1 [8]. Then by applying the saddle point method to the large N limit of (4) it follows that  1 dz′ ∧ dz̄′ z̄ = V ′ (z) + (37) , z ∈ D. 2πi z′ − z D Let us consider now the rational conformal map z(w) associated to D, from (37) it follows that the function z̃(w) can be extended as a meromorphic function of z outside D by  dz′ ∧ dz̄′ 1 . z̃(z) = V ′ (z) + (38) 2πi z′ − z D In fact z̃(z) represents the Schwarz function of the boundary Γ of D. By using (3) and (17) one can rewrite (38) as N −1   N k 0 +1 s   dz′ ∧ dz̄′ 1 n−1 n+1 nz0 t0,n − nzs ts,n + zs ts,0 + . z̃(z) = 2πi z′ − z s=1 n=1 n=1 (39) D By comparing this identity with our definition (19) of the Whitham times and by taking into account (31), we conclude that the Whitham times coincide with the coupling constants of the normal matrix model. Finally, we notice that by using the strategy deployed in the proof of the Theorem of Section 2 and by taking (39) into account, it follows that the deformations with respect to the parameters βs and β̄s are described by the flows N s −2  ∂zα ∂zα ∂zα nts,n = ts,0 + + (Ns − 1)ts,Ns −1 zsNs ∂βs ∂ts,1 ∂ts,n+1 n=1 (s,+)  , zα , (40) L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 473 and N s −2  Ns ∂zα ∂zα ∂zα + + (Ns − 1)ts+k,Ns −1 zs+k = ts+k,0 nts+k,n ∂ts+k,1 ∂ts+k,n+1 ∂ β̄s n=1 (s+k,+)  , zα . (41) References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] M.Z. Bazant, D. Crowdy, Conformal mapping methods for interfacial dynamics, in: S. Yip (Ed.), Handbook of Materials Modeling, vol. 1, Springer, 2005. P.Ya. Polubarinova-Kochina, Dokl. Acad. Nauk SSSR 47 (1945) 254. P.P. Kufarev, Dokl. Acad. Nauk SSSR 75 (1950) 507. S. Richardson, J. Fluid Mech. 56 (1972) 609. P.W. Wiegmann, P.B. Zabrodin, Commun. Math. Phys. 213 (2000) 523. M. 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