A Finite Genus Solution of the Veselov s Discrete Neumann System

Transcription

1 Commun. Theor. Phys Vol. 58, No. 4, October 5, 202 A Finite Genus Solution of the Veselov s Discrete Neumann System CAO Ce-Wen and XU Xiao-Xue Æ Department of Mathematics, Zhengzhou University, Zhengzhou 45000, China Received May 9, 202 Abstract The Veselov s discrete Neumann system is derived through nonlinearization of a discrete spectral problem. Based on the commutative relation between the Lax matrix and the Darboux matrix with finite genus potentials, a special solution is calculated with the help of the Baker Akhiezer Kriechever function. PACS numbers: Ik, Jr, Jb Key words: Veselov s discrete Neumann system, Baker Akhiezer Kriechever function, finite genus solution Introduction Prominent progress has been made in recent years in the study of lattice integrable systems, including the problem of integrable discretization of given soliton models. [ 0] In this paper we are going to discuss the Neumann system with 2 xxq j + uq j = α j q j, j N, q, q = const., u = q, q Aq, q q x, q x, 2 which is a well-known classical integrable system appeared in the nineteenth century, describing the harmonic oscillation of a particle constrained on the sphere. Three essentially different integrable discrete versions of Eq. are found by Veselov, Ragnisco, and Adler, respectively. [0 4] Our purpose is to study the Veselov s discrete Neumann system, which reads [0 3] q j + q j = β α j /2 Bqq j, j N, which defines the generating function q 2 j with q, q = const., 3 Bq = βi A /2 q, q + q βi A q, q. 4 In Sec. 2, a self-contained exposition of the Liouville theory of the Neumann system is given. Special attention is paid to the proof of functional independence of the integrals, which is essential for the Liouvile integrability. In Sec. 3, the discrete equation 3 is constructed through nonlinearization of a discrete spectral problem. In Sec. 4, the finite genus solution of Eq. 3 is calculated with the help of the Baker Akhiezer Kriechever function based on the basic commutative relation between the Lax matrix and the Darboux matrix with finite genus potentials. 2 The Liouville Theory of the Neumann System Let R 2N, dp dq be the phase space; A = diagα,...,α N with distinct non-zero diagonal entries; ξ, η = N j= ξ jη j. Consider the Lax matrix Lλ; p, q as 0 ε j Qλ p, q Q λ p, p Lλ; p, q = + =, λ α j= j Q λ q, q Q λ p, q pj q j p 2 j ε j =, Q λ ξ, η = λi A ξ, η, 6 p j q j Fλ; p, q = detlλ = Q λ q, q + Q λ p, pq λ q, q Q 2 λ p, q. 7 A series of integrals are given as the coefficients of the expansion Fλ = F j λ j, 8 j=0 F 0 = q, q, F = Aq, q + p, p q, q p, q 2, 9 Supported by the National Natural Science Foundation of China under Grant No cwcao@zzu.edu.cn c 20 Chinese Physical Society and IOP Publishing Ltd

2 470 Communications in Theoretical Physics Vol. 58 F s = A s q, q + j+k=s ; j, k 0 A j p, p A k q, q A j p, q A k p, q. 0 More intrinsic is the new generating function Hλ = j=0 H jλ j defined by λfλ = [λhλ] 2, which gives rise to the integrals {H j } recursively as H 0 = q, q, H = F /2H 0, H s = F s H j H k. 2 2H 0 j+k=s; j,k Consider the Hamiltonian system of Fλ d pj Fλ/ qj = = Wλ, α j dt λ q j Fλ/ p j pj q j, j N, 3 Wλ, µ = 2 Lλ. 4 λ µ Resorting to this it is easy to prove the basic Lax equation along the Fλ-flow d dt λ Lµ = [Wλ, µ, Lµ], λ, µ C. 5 As a corollary we obtain d/dt λ L 2 = [W, L 2 ]. By L 2 = I detl we have dfµ/dt λ = 0. Since the Poisson bracket is a derivative along the flow, {f, Fλ} = df/dt λ, we obtain Lemma The generating function has a factorization {Fλ, Fµ} = {Fλ, Hµ} = {Hλ, Hµ} = 0, λ, µ C, 6 {F j, F k } = {F j, H k } = {H j, H k } = 0, j, k = 0,, 2,... 7 Fλ = N q, q αλ j= λ λ j = H 2 0 Rλ α 2 λ, 8 where αλ = N j= λ α j. The polynomial Rλ has degree 2N and defines an algebraic curve with genus g = N. For any non-branch λ C, there are two points on R: pλ = λ, ξ = Rλ, R : ξ 2 + Rλ = 0, 9 τpλ = λ, ξ = Rλ, where τ : R R is the map of changing the sheets of R. The elliptic variables {ν j } are defined by By Eq 5 we have Putting µ = ν k, we obtain L 2 λ = Q λ q, q = q, q nλ αλ, N nλ = λ ν j. 20 j= d L 2 µ = 4 L 2 λl µ L λl 2 µ. dt λ λ µ 2 Rν k k= dν k = 2H 0 dt λ αλ nλ λ ν k n ν k, 2a ν g s k 2 dν k = 2H 0 Rν k dt λ αλ λg s, s g, 2b where the interpolation formula of polynomials is used to derive Eq. 2b. Consider the basis {ω s} of holomorphic differentials on R and the quasi-abel Jacobi variables {φ s } defined as ω s = λg s dλ 2 Rλ, s g ; 22 φ s = k= pνk ω s, s g. 23

3 No. 4 Communications in Theoretical Physics 47 After rewriting Eq. 2b with the help of {φ s }, we obtain Lemma 2 The Fλ- and F j -flow are straightened out by the quasi-abel Jacobi variables as dφ s = {φ s dt, Fλ} = 2H 0 λ αλ λg s, s g ; 24 dφ s = {φ dt s, F j } = 2H 0 A j s, s g; j = 0,, 2,..., 25 j where A 0 = ; A k = 0 as k < 0; and A k as k > 0 are defined by Π N j= α jλ = A k λ k. Lemma 3 F 0, F,...,F N are functionally independent in M = {p, q R 2N : q 0}. Proof First, by Eq. 25, we have {φ s, F 0} = 0 s g and A A 2 A N 2 φ,..., φ N A A N 3 t,..., t N = 2H A It needs only to prove the linear independence of df 0, df,..., df N in the cotangent space T p,q R2N at any point p, q M. Suppose c 0 df 0 + c df + + c N df N = 0. Then we have k=0 c {φ s, F } + + c N {φ s, F N } = 0, s g. By Eq. 26 we obtain c = = c N = 0. Thus c 0 df 0 = 0, which implies c 0 = 0 since at any point p, q of M we have df = d q, q = 2q j dq j 0. j= Consider the Hamiltonian system H H = 2 q, q Aq, q + p, p q, q p, q 2, 27a pj x = p, q αj p, p + H /H 0 pj, 27b q j H 0 q, q p, q j N, which is reduced into the Neumann system after cancelling p j. Since {H, F k } = 0, k, finally we obtain Proposition The Neumann system is completely integrable, which can be put in the Hamiltonian form 27b with integrals F 0, F,...,F N, involutive in pairs and functionally independent in the open dense subset M = {p, q R 2N : q 0} of the phase space. The partial fraction expansion of Fλ yields the well-known Uhlenbeck Devaney integrals {E s }, which are linearly equivalent to the integrals F 0, F,..., F N whose coefficients give rise to the Vandermonde determinant: Fλ = F j = s= E s λ α s ; E s = q 2 s + k N; k s q j p s q k p k q s 2 α s α k, 28 α j s E s, 0 j N. 29 s= This proves the involutivity and the functional independence of E,...,E N in the subset M. 3 Construction of the Lattice Neumann System The Hamiltonian form 27b of the Neumann system suggests a discrete spectral problem a λ + β + ab/c χ = D β λχ, D β λ; a, b, c =, 30 λ β c b

4 472 Communications in Theoretical Physics Vol. 58 with detd β λ = λ β. Define a map by N replicas of Eq. 30 as S : R 2N R 2N, p, q p, q, pj = q j αj β Dβ α j ; a, b, c pj q j, j N. 3 Our purpose is to choose a suitable constraint between a, b, c and p, q so that the map S is integrable and shares the same integrals {F k } as the Neumann system. It turns out that this leads exactly to the Veselov s discretization 3. Lemma 4 Let P β t; p, q = t 2 L 2 β + 2tL β L 2 β. Then Lλ; p, qd β λ; a, b, c D β λ; a, b, clλ; p, q 0 a = P β b/c; p, q c b 0 c c 2 c 2 q, q Proof Equation 3 implies ε j D β α j = D β α j ε j. Thus [ ] 0 LD β D β L =, D β ε j D β λ D β λε j λ λ α j= j = q, q c 2 ca b p, q p, q c 0 c q, q By Eq. 3 we have p, q + p, q = a q, q b q, q /c, c 2 q, q = c 2 + Q β cp + bq, cp + bq = c 2 P β b/c. The proof is completed by substituting these expressions in Eq. 33. The quadratic polynomial P β t has two roots as t ± β; p, q = By Eq. 32, under the constraints L 2 β L β ± Fβ. 34 c = q, q = H 0, b = ct + β; p, q, 35 we have LD = DL. Hence det L = det L, i.e. F = F, which gives rise to the invariance under the action of the map S: Fλ : p, q = Fλ : p, q, λ C; 36 F j p, q = F j p, q, H j p, q = H j p, q, j = 0,, 2, Further, by Eq. 3 we have q j = αj β c p j + a q j, 38 c 2 q, q = c 2 P β a/c; p, q. 39 Thus a/c is a root of the quadratic polynomial. We choose an implicit constraint as By Eqs. 3 and 38 we have q j = αj β cp j + bq j, a = ct β; p, q. 40 q j = αj β cp j + aq j. After cancelling p j we obtain q j + q j = αj β b + aq j. 4 This is exactly the lattice Neumann system 3 with Bq = b + a = 2c Fβ L β It is easy to derive expression 4 from Eq. 3. Proposition 2 The Veselov s discrete Neumann system 3 can be put in the form of an integrable map S : R 2N R 2N, given by Eq. 3 under the constraints 35, 40, which shares the same integrals F 0 = q, q, F,...,F N as the continuous Neumann system. 4 The Finite Genus Solution A discrete flow pm, qm = S m, q 0 is defined by the iteration of the map S. For any given constant c, at the level {H 0 = c}, define the finite genus potentials as a m = ct β; pm +, qm +, 43 b m = ct + β; pm, qm. 44 Let D m β λ = D β λ; a m, b m, c, L m λ = Lλ; pm, qm. Then by lemma 4, we have the commutative relation L m+ λd m β λ = Dβ m λl mλ. 45 Define a discrete spectral problem with finite genus potentials as χm +, λ = D β λ β m λχm, λ. 46 To simplify the calculation, consider a modified version as hm +, λ = D β m λhm, λ, 47

5 No. 4 Communications in Theoretical Physics 473 with hm, λ = λ β m/2 χm, λ, whose fundamental solution matrix Mm, λ satisfies Mm +, λ = D β m λmm, λ, M0, λ = I, 48a Mm, λ = D β m λdβ m 2 λ Dβ 0 λ, 48b detmm, λ = λ β m, 48c L m λmm, λ = Mm, λl 0 λ. 48d By Eq. 48b, each entry of Mm, λ is polynomial of λ. According to Eq. 45, the solution space E λ of the linear equation 47 is invariant under the action of the linear operator L m λ, whose eigenvalues are determined as detρi L m λ = ρ 2 + Fλ = 0, 49 ρ ± λ = ±ρ λ = ± Fλ = ±H 0 Rλ αλ. 50 For given non-branch λ C, ρ + λ and ρ λ are values of a meromorphic function on the Riemann surface R at the points pλ and τ pλ, respectively. The corresponding eigenvectors h ± m, λ satisfy h ± m +, λ = D β m λh ± m, λ, 5a h ± m, λ = h ± m, λ h 2 ± m, λ c ± = Mm, λ λ, 5b L m λ ρ ± λ h ±m, λ = 0. 5c Let m = 0 in Eq. 5c, we solve c ± λ = L 0 λ ± ρ λ L 2 0 λ = L2 0 λ L 0 λ ρ λ, 52 with c + λ c λ = L2 0 λ/l 2 0 λ. c + λ and c λ are also values of a meromorphic function on R at the points pλ and τ pλ, respectively. Thus by Eq. 5b, two meromorphic functions h m, p and h 2 m, p are defined with the values, respectively, as h s m, pλ = h s + m, λ, h s m, τpλ = h s m, λ, s =, Lemma 5 As λ, h ± m, λ = c ±ζ m [ + Oζ], 54a h 2 ± m, λ = ±ζ m [ + Oζ], 54b where ζ = i/ λ is the local coordinate of R at infinity, with λ = ζ 2. Proof As λ, by induction we have Oλ k c λ k [ + Oλ ] M2k, λ = c λ k [ + Oλ ] Oλ k, λ k [ + Oλ ] Oλ k M2k, λ = Oλ k λ k [ + Oλ. 55 ] From Eqs. 50 and 52 we obtain The proof is completed by substituting these estimations in Eq. 5b. Lemma 6 Formula of Dubrovin Novikov s type h 2 + ρ λ = c ζ[ + Oζ 2 ], 56 c ± λ = ±c ζ [ + Oζ]. 57 m, λ h2 m, λ = λ βm Proof Resorting to the commutative relation 48d, we have h + h h + h2 c + h 2 + h h 2 λ = Mm, λ c λ c + λ + h2 c λ m,λ = = L 2 detmm, λ L 2 0 λ M T m, λ = g j= 0 λ[l mλ + ρ λ I]Mm, λ iσ 2 M T m, λ L 2 m λ L m λ + ρ λ L mλ ρ λ L 2 mλ λ ν j m λ ν j L 2 0 λmm, λ[l 0λ + ρ λ I] iσ 2 M T m, λ, where σ 2 is the usual Pauli matrix. Thus h 2 + h2 = λ β m L 2 mλ/l 2 0 λ. This completes the proof. By Eq. 54b, h 2 m, p has a pole with order m at. With the help of Eq. 58, we obtain Lemma 7 The meromorphic function h 2 m, p has the divisor as [pν j m pν j 0] + m[pβ ]. 59 j= Put the basis of holomorphic differentials on R in the vector form ω = ω,...,ω g T, where ω s is given by

6 474 Communications in Theoretical Physics Vol. 58 Eq. 22. The normalized basis ω = C ω defines the periodic vectors δj = ω, Bj = ω, j g, 60 a j b j which span a lattice T in C g. By a dipole technique due to Toda, [5] we have pνjm pβ ω + m ω 0, mod T. 6 j= pν j0 Define the Abel Jacobi variable on the Jacobi variety JR = C g /T as pνjm φm = ω. 62 j= Proposition 3 The discrete flow S m is straightened out by the Abel Jacobi variable as φm φ0 + mω β, mod T, 63 Ω β = pβ ω. 64 The meromorphic function h 2 m, p can be represented by its divisor up to a constant factor as h 2 θ[ Ap + φm + K] m, p = const. θ[ Ap + φ0 + K] p exp m ω[pβ, ], 65 where A : DivR JR, Ap = p ω, is the Abel map; K is the Riemann constant vector; ω[p, q] is the dipole, which is an Abel differential of the third kind, having only two simple poles at p and q with residues + and, respectively. The constant factor can be calculated and cancelled by resorting the asymptotic behavior at infinity given by Eq. 54b. Thus we obtain Lemma 8 The Baker Akhiezer Kriechever function h 2 m, p has an explicit expression as h 2 m, p = θ[ p ω + mω β + K0] θ[k0] θ[ p ω + K0] θ[mω β + K0] p m, exp ω[pβ, ] 66 r p ζ 2 r = lim ζ exp ω[pβ, ], 67 ζ 0 Km = A + φm + K = mω β + K0. 68 By putting λ = α j in Eq. 5a, it is easy to see that the function q j m = α j β m/2 h 2 m, pα j 69 solves the lattice Neumann Eq. 3. Finally we obtain Proposition 4 The Veselov s discrete Neuman system 3 has an explicit solution as q j m = θ[ω α j + mω β + K0] θ[k0] θ[ω αj + K0] θ[mω β + K0] pαj m, r αj β exp ω[pβ, ] 70 where Ω αj K + ω. = pα ω; Ω j β = ω, and K0 = φ0 + pβ References [] V.E. Adler, A.I. Bobenko, and Yu. B. Suris, Commun. Math. Phys [2] J. Atkinson, J. Hietarinta, and F. Nijhoff, J. Phys. A: Math. Theor [3] C. Cao and X. Xu, J. Phys. A: Math. Theor [4] C. Cao and G. Zhang, J. Phys. A: Math. Theor [5] C. Cao and G. Zhang, Chin. Phys. Lett [6] B. Grammaticos, Y. Kosmann-Schwarzbach, and T. Tamizhmani, Discrete Integrable Systems, Springer, Berlin [7] J. Hietarinta and D.J. Zhang, J. Phys. A: Math. Theor [8] F. Nijhoff, J. Atkinson, and J. Hietarinta, J. Phys. A: Math. Theor [9] F. Nijhoff and H. Capel, Acta Appl. Math [0] Yu. B. Suris, The Problem of Integrable Discretization: Hamiltonian Proach, Birkhäuser, Basel [] A.P. Veselov, Funct. Anal. Appl [2] A.P. Veselov, Russ. Math. Surv [3] J. Moser and A. P. Veselov, Commun. Math. Phys [4] O. Ragnisco, Phys. Lett. A [5] M. Toda, Theory of Nonlinear Lattices, Springer, Berlin 98.