Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation

Xiaoxue Xu; Cewen Cao; Guangyao Zhang

doi:10.1080/14029251.2020.1819608

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Volume 27, Issue 4, September 2020, Pages 633 - 646

Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation

Authors

Xiaoxue Xu^*

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China,xiaoxuexu@zzu.edu.cn

Cewen Cao

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China,cwcao@zzu.edu.cn

Guangyao Zhang

School of Science, Huzhou University, Zhejiang, 313000, People’s Republic of China,zgy101003@163.com

^*Corresponding author.

Corresponding Author

Xiaoxue Xu

Received 22 March 2019, Accepted 25 January 2020, Available Online 4 September 2020.

DOI: 10.1080/14029251.2020.1819608 How to use a DOI?
Keywords: lattice Schwarzian Korteweg-de Vries equation; integrable symplectic map; finite genus solution
Abstract: Based on integrable Hamiltonian systems related to the derivative Schwarzian Korteweg-de Vries (SKdV) equation, a novel discrete Lax pair for the lattice SKdV (lSKdV) equation is given by two copies of a Darboux transformation which can be used to derive an integrable symplectic correspondence. Resorting to the discrete version of Liouville-Arnold theorem, finite genus solutions to the lSKdV equation are calculated through Riemann surface method.
Copyright: © 2020 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

Remarkable progress has been made in recent years in the study of discrete soliton equations (see [12] and the references therein). Among the related mathematical theories, the property of multi-dimensional consistency plays an important role in the understanding of discrete integrability. In the 2-dimensional case, it leads to the well-known Adler-Bobenko-Suris (ABS) list [1, 10], which gives a classification of integrable quadrilateral lattice equations. Quite a few works have appeared in the study of the ABS equations, concerning their relations with the usual soliton equations, the Lax pairs, explicit analytic solutions, Bäcklund transformations (BTs), symmetries and conservation laws etc. [3, 6, 7 , 13, 14, 19, 21, 22, 26, 32, 33 ].

The purpose of this paper is to investigate the lSKdV equation, which was first given in [23],

Ξ:=γ12(u−u^)(u˜−u˜^)−γ22(u−u˜)(u^−u˜^)=0,(1.1)

where the usual notation is adopted: u = u(m, n), u˜=u(m+1,n), û = u(m, n+1). Eq. (1.1) is exactly the Q1(0) model (Q1 with δ = 0) in the ABS hierarchy. The approach of Lax representation will be used to confirm the integrability of Eq. (1.1) and to calculate its basic explicit analytic solutions, the finite genus solutions [6, 7].

To produce a purely discrete Lax pair, it is vital to select two appropriate discrete spectral problems. It turns out that a special role is played by the semi-discrete integrable equations, which are also of independent interest, see [18] and references therein, where the well-known Toda, the Volterra and the Ablowitz-Ladik hierarchies are investigated thoroughly. A semi-discrete Lax pair can be constructed with the help of a continuous spectral problem and its Darboux transformation (DT), where the DT is regarded as a discrete spectral problem [4, 18], which usually leads to an integrable symplectic map by using the non-linearization technique [5–7]. Refer to [16], integrable maps are called BTs whose geometrical explanation is given in terms of spectral curves and their Jacobians. And the symplectic correspondences (BTs) compatible with finite gap solutions of KdV have been discussed through DTs for the standard KdV spectral problem [15].

In our case we consider the continuous SKdV equation,

φyφx+14S[φ;x]=0,(1.2)

where S[ϕ;x] denotes the Schwarzian derivative of ϕ [20, 30, 31], i.e.

S[φ;x]=(φxxφx)x−12(φxxφx)2,=φxxxφx−32(φxxφx)2. (1.3)

Technically, it is more convenient to use the derivative version

wy+14(wxx−3wx22w)x=0,(1.4)

with w = ϕ_x. The Eq. (1.4) has a Lax pair given by

∂xχ=(0−wλ−1w−1λ−10)χ,(1.5)

∂yχ=(−wx2wλ−2−wλ−3+14(wxx−3wx22w)λ−1w−1λ−3+14w2(wxx−wx22w)λ−1wx2wλ−2)χ.(1.6)

We note that in [24], the Lax pair for one Schwarzian PDE, which is equivalent to the SKdV hierarchy via expansions on the independent variables and has a fully discrete counterpart (1.1) by considering the independent variables as lattice parameters, has been found. However, we have not been able to blend it with the algebro-geometric technique of nonlinearization employed in the present paper. Fortunately, here each of the linear systems (1.5) and (1.6) can be nonlinearized to produce an integrable Hamiltonian system. Thus we find a Liouville integrable system associated with a spectral problem (see Sec. 2) given by

∂xχ=U(λ,u)χ, U(λ,u)=(uλ0−u),(1.7)

and find that the following DT of Eq. (1.7) is critical:

χ˜=(λ2−γ2)−1/2D(γ)(λ,b)χ, D(γ)(λ,b)=(λγbγb−1λ).(1.8)

The compatibility condition Dx(γ)=U˜D(γ)−D(γ)U gives rise to

bx/b=u+u˜, γb−1=u−u˜.(1.9)

This suggests a constraint b = γ/(u − ũ) and leads to a Lax pair, different from the one in [23], for Eq. (1.1).

Lemma 1.1.

The lSKdV equation (1.1) has a Lax pair

χ˜=(λ2−γ12)−1/2D(γ1)(λ,b′)χ, b′=γ1/(u−u˜),χ^=(λ2−γ22)−1/2D(γ2)(λ,b″)χ, b″=γ2/(u−u^),(1.10)

with

D^(γ1)D^(γ2)−D˜(γ2)D(γ1)=1ϒ((u−u˜)(u−u^)λ(u˜^−u˜−u^+u)0−(u˜−u˜^)(u^−u˜^))Ξ,(1.11)

where ϒ=(u−u˜)(u−u^)(u˜−u˜^)(u^−u˜^) and Ξ is defined by Eq. (1.1).

The paper is organised as follows. In Sec. 2, a finite-dimensional Hamiltonian system which is a nonlinear version of the spectral problem (1.7) is presented. In Sec. 3, resorting to the Hamiltonian system, we construct an integrable symplectic map. In addition, with the help of the Burchnall-Chaundy theory, the discrete potential is expressed in terms of theta functions. In Sec. 4, based on the discrete version of the Liouville-Arnold theorem, the finite genus solutions of lSKdV equation (1.1) are obtained through the commutativity of integrable maps [7].

2. The Integrable Hamiltonian System (H₁)

Take the symplectic manifold (ℝ²^N, dp ∧ dq) as the phase space. The symplectic coordinate is defined as (p, q) = (p₁,..., p_N, q₁,...,q_N). Let A = diag(α₁,...,α_N) with distinct, non-zero α12,…,αN2. Define a Lax matrix

L(λ;p,q)=σ++12∑j=1N(ɛjλ−αj+σ3ɛjσ3λ+αj)=(λQλ(p,q)1−Qλ(Ap,p)Qλ(Aq,q)−λQλ(p,q)),(2.1)

where σ₊, σ₃ are the usual Pauli matrices, and

ɛj=(pjqj−pj2qj2−pjqj), Qλ(ξ,η)=∑j=1Nξjηjλ2−αj2,

∀ (ξ,η) = (ξ₁,...,ξ_N,η₁,...,η _N) ∈ ℝ^2N.

The generating function ℱ_λ = detL(λ; p, q) is a rational function of the argument ζ = λ² ,

ℱλ(p,q)=(Qλ(Ap,p)−1)Qλ(Aq,q)−λ2Qλ2(p,q).(2.2)

The expansion ℱλ=∑l=1∞Flζ−1 gives rise to a set of quantities on phase space as

F1=−〈Aq,q〉−〈p,q〉2,Fl=−〈A2l−1q,q〉+∑j+k=lj,k⩾1〈A2j−1p,p〉〈A2k−1q,q〉−∑j+k=l+1j,k⩾1〈A2j−2p,q〉〈A2k−2p,q〉,(2.3)

(l = 1, 2,...), where 〈ξ,η〉=∑j=1Nξjηj. Consider the Hamiltonian system (H₁), defined by the Hamiltonian function

H1=F12=−12〈Aq,q〉−12〈p,q〉2,∂x(pjqj)=(−∂H1/∂qj∂H1/∂pj)=(〈p,q〉αj0−〈p,q〉)(pjqj), 1⩽j⩽N.(2.4)

They are exactly N copies of Eq. (1.7) with distinct λ = α_j and the constraint

u=fU(p,q)=〈p,q〉.(2.5)

In this context (H₁) is called a non-linearization of the linear spectral problem (1.7).

According to the Liouville-Arnold theory [2], we shall discuss the coefficients F₁,...,F_N given by (2.3) are first integrals of the phase flow with Hamiltonian function H₁, i.e., {F_j, H₁} = 0 (j = 1,...,N), where {·, ·} denotes the Poisson bracket on the phase space. The involution and functional independence between F₁,...,F_N guarantee that the Hamiltonian system (H₁) is completely integrable.

Consider the Hamiltonian system (ℱ_λ),

ddtλ(pjqj)=(−∂ℱλ/∂qj∂ℱλ/∂pj)=W(λ,αj)(pjqj),W(λ,μ)=2λ2−μ2(λL11(λ)μL12(λ)μL21(λ)−λL11(λ))=L(λ)λ−μ+σ3L(λ)σ3λ+μ,(2.6)

where L(λ) is the abbreviation of L(λ; p, q) and L^ij(λ), i, j = 1, 2 are entries of the matrix L(λ). Hence we obtain dε_j/dt_λ = [W(λ, α_j), ε_j], where [·, ·] stands for the matrix commutator. Based on this formula, it is easy to derive the following basic equation,

ddtλL(μ)=[W(λ,μ),L(μ)], ∀λ,μ∈ℂ.(2.7)

As a corollary, we have

{ℱμ,ℱλ}=0, ∀λ,μ∈ℂ;(2.8)

{Fj,Fk}=0, j,k=1,2,….(2.9)

Actually, by Eq. (2.7), (d/dt_λ)L²(μ) = [W(λ, μ), L²(μ)]. Since L²(μ) = −Iℱ_μ, where I is the identity matrix, we have dℱ_μ/dt_λ = 0. According to the definition of Poisson bracket [2], this is exactly Eq. (2.8), whose power series expansion gives rise to Eq. (2.9).

The generating function ℱ_λ has a factorization

ℱλ=F1Z(ζ)α(ζ)=R(ζ)ζα2(ζ),(2.10)

with α(ζ)=∏j=1N(ζ−αj2), Z(ζ)=∏k=1N−1(ζ−ζk), R(ζ) = ζα(ζ)Z(ζ), where F₁ is given by Eq. (2.3). The spectral curve is defined as

ℛ:ξ2−R(ζ)=0,(2.11)

which is hyperelliptic with genus g = N − 1 and has two points at infinity, ∞₊, ∞₋. At the branch point 𝔬 = (ζ = 0, ξ = 0), ℛ has a local coordinate λ = ζ^1/2. The generic point on ℛ is given as

𝔭(ζ)=(ζ,ξ=R(ζ)), (τ𝔭)(ζ)=(ζ,ξ=−R(ζ)),

where τ : ℛ → ℛ is the hyperelliptic involution. The variables {νj2} defined as the roots of the equation

L21(λ)=∑j=1Nαjqj2λ2−αj2=〈Aq,q〉𝔫(ζ)α(ζ)=0, 𝔫(ζ)=∏j=1g(ζ−νj2),(2.12)

give an elliptic coordinate system [17]. By Eq. (2.7) we have

ddtλL21(μ)=2(W21(λ,μ)L11(μ)−W11(λ,u)L21(μ)).(2.13)

Putting μ = ν_k, with L11(νk)=−F1⋅R(νk2)/(νkα(νk2)) from Eq. (2.10), we get the evolution of the elliptic variables along the ℱ_λ-flow,

12R(νk2)⋅d(νk2)dtλ=−2−F1α(ζ)⋅𝔫(ζ)(ζ−νk2)𝔫′(νk2), 1⩽k⩽g,(2.14)

∑k=1g(νk2)g−s2R(νk2)⋅d(νk2)dtλ=−2−F1α(ζ)⋅ζg−s, 1⩽s⩽g,(2.15)

where the interpolation formula of polynomials is used. With the help of the quasi-Abel-Jacobi variables

φs′=∑k=1g∫𝔭0𝔭(νk2)ωs′, ωs′=ζg−sdζ2R(ζ), 1⩽s⩽g,(2.16)

Eq. (2.15) is rewritten in a simple form and gives rise to

Proposition 2.1.

The ℱ_λ- and the F_l-flow are linearized by ϕ′_s as

dφs′dtλ={φs′,ℱλ}=−2−F1α(ζ)⋅ζg−s, 1⩽s⩽g,(2.17)

dφs′dtl={φs′,ℱl}=−2−F1⋅Al−s−1, l=1,2,…,(2.18)

where A₀ = 1; A−_j = 0 (j = 1, 2,...); while A_j (j = 1, 2,...), are defined by

ζNα(ζ)=1∏k=1N(1−αk2ζ−1)=∑j=0∞Ajζ−j.

In particular, {ϕ′_s, F1} = 0, 1 ⩽ s ⩽ g.

Proposition 2.2.

The Hamiltonian system (H₁) is integrable, possessing N integrals F₁,...,F_N, involutive with each other and functionally independent in the dense, open subset 𝒪 = {(p, q) ∈ ℝ^2N : F₁ ≠ 0}.

Proof.

F_l is an integral since {H₁, F_l} = (1/2){F₁, F_l} = 0 by Eq. (2.9). It needs only to prove that dF₁,...,dF_N are linearly independent in T(p,q)*ℝ2N at (p, q) ∈ 𝒪. Suppose ∑j=1NcjdFj=0. Then

c2{φs′,F2}+⋯+cN{φs′,FN}=0, 1⩽s⩽N−1.

By Eq. (2.18), the coefficient matrix is non-degenerate,

({φ1′,F2}⋯{φ1′,FN}⋮⋱⋮{φg′,F2}⋯{φg′,FN})=−2−F1⋅(1A1A2⋯Ag−11A1⋯Ag−2⋱…⋮1A11).

Thus c₂ = ··· = c_N = 0 and c₁dF₁ = 0. We have c₁ = 0 since dF₁ ≠ 0 at 𝒪. Otherwise,

−12dF1=∑j=1N(〈p,q〉qjdpj+(αjqj+〈p,q〉pj)dqj)=0.

Hence α_jq_j + 〈p, q〉p_j = 0, ∀_j; and F₁ = 0. This is a contradiction.

3. The Integrable Symplectic Map 𝒮_γ

As a non-linearization of Eq. (1.8), define a map 𝒮_γ : ℝ²^N → ℝ²^N, (p,q)↦(p˜,q˜) by

(p˜jq˜j)=(αj2−γ2)−1/2D(γ)(αj,b)(pjqj), 1⩽j,N,(3.1)

where a constraint b = f_γ(p, q) is to be chosen so that 𝒮_γ is integrable and symplectic.

Lemma 3.1.

Let P(^γ)(b; p, q) = b²L²¹(γ) + 2bL¹¹(γ) − L¹²(γ). Then

L(λ;p˜,q˜)D(γ)(λ,b)−D(γ)(λ,b)L(λ;p,q)=−γb−1P(γ)(b;p,q)σ3,(3.2)

∑j=1N(dp˜j∧dq˜j−dpj∧dqj)=12γb−2dP(γ)(b;p,q)∧db.(3.3)

Proof.

By Eq. (3.1), we get ɛ˜jD(γ)(αj)−D(γ)(αj)ɛj=0. Besides, we have σ32=I and

σ3D(γ)(λ)σ3=−D(γ)(−λ),D(γ)(±λ)−D(γ)(αj)=±(λ∓αj)I.

Based on these preparations, we calculate the left-hand side of Eq. (3.2),

[σ+D(γ)(λ)]+12∑j=1N(ɛ˜jD(γ)(λ)−D(γ)(λ)ɛjλ−αj+σ3ɛ˜jσ3D(γ)(λ)−D(γ)(λ)σ3ɛjσ3λ+αj)=γb−1σ3+12∑j=1N((ɛ˜j−ɛj)+σ3(ɛ˜j−ɛj)σ3)=(γb−1+〈p˜,q˜〉−〈p,q〉)σ3.

By using Eq. (3.1), we obtain

γb−1+〈p˜,q˜〉−〈p,q〉=−γb−1P(γ)(b;p,q).(3.4)

This proves Eq. (3.2). Eq. (3.3) is obtained through direct calculations.

Consider the quadratic equation P(^γ)(b) = 0, whose roots give the constraint on b,

b=fγ(p,q)1Qγ(Aq,q)(−γQγ(p,q)±−ℱγ(p,q)).(3.5)

Actually γb can be written as a meromorphic function on ℛ,

𝔟(𝔭)=1Qγ(Aq,q)(−γ2Qγ(p,q)+−F1ξα(γ)).

Though doubled-valued as a function of β ∈ ℂ, it is single-valued as a function of 𝔭(β²) ∈ ℛ. Hence we obtain

Proposition 3.1.

The map 𝒮_γ : ℝ²^N → ℝ²^N, (p,q)↦(p˜,q˜), defined as

(p˜jq˜j)=(αj2−γ2)−1/2(αjpj+γbqjγb−1pj+αjqj)|b=fγ(p,q), 1⩽j,N,(3.6)

is symplectic and integrable, possessing the Liouville set of integrals

Fl(p˜,q˜)=Fl(p,q), 1⩽j⩽N.(3.7)

Proof.

Since P(^γ)(b) = 0, by Eq. (3.2) and (3.3) we have

L(λ;p˜,q˜)D(γ)(λ,fγ(p,q))−D(γ)(λ,fγ(p,q))L(λ;p,q)=0,(3.8)

∑j=1Ndp˜j∧dq˜j=∑j=1Ndpj∧dqj.(3.9)

Taking the determinant of Eq. (3.8), we obtain ℱλ(p˜,q˜)=ℱλ(p,q), hence Eq. (3.7).

By Eq. (3.7), the discrete flow (p(m),q(m))=𝒮γm(p0,q0) has constants of motion {F_l}. Define finite genus potentials as

b(m)=bm=fγ(p(m),q(m)),(3.10)

u(m)=um=fU(p(m),q(m))=〈p(m),q(m)〉.(3.11)

By Eq. (3.4), they have the relation

bm=γ/(um−um+1),(3.12)

which meets the requirement of Eq. (1.10). Along the m-flow, Eq. (3.8) is rewritten as

Lm+1(λ)Dm(γ)(λ)=Dm(γ)(λ)Lm(λ),(3.13)

where L_m(λ) = Lλ; p(m), q(m), Dm(γ)(λ)=D(γ)(λ,bm). Now we calculate u_m with the help of the following spectral problem and its fundamental solution matrix M_γ(m, λ),

hγ(m+1,λ)=Dm(γ)(λ)hγ(m,λ);(3.14)

Mγ(m+1,λ)=Dm(γ)(λ)Mγ(m,λ), Mγ(0,λ)=I.(3.15)

By induction we have

Mγ(m,λ)=Dm−1(γ)(λ)Dm−2(γ)(λ)⋯D0(γ)(λ),detMγ(m,λ)=(λ2−γ2)m,Lm(λ)Mγ(m,λ)=Mγ(m,λ)L0(λ).(3.16)

Lemma 3.2.

The following functions are polynomials of the argument ζ = λ²:

Mγ11(2k,λ),λ−1Mγ12(2k,λ),λ−1Mγ21(2k,λ)Mγ22(2k,λ),λ−1Mγ11(2k+1,λ),Mγ12(2k+1,λ),Mγ21(2k+1,λ),λ−1Mγ22(2k+1,λ).(3.17)

Besides, as λ → ∞,

Mγ(m,λ)=(λm[1+O(λ−2)]O(λm−1)O(λm−1)λm[1+O(λ−2)]).(3.18)

By Eq. (3.13), the solution space ℰ_λ of Eq. (3.14) is invariant under the action of the linear operator L_m(λ), which has two eigenvalues ρλ±=±ρλ,

ρλ=−ℱλ=−F1⋅R(ζ)λα(ζ).(3.19)

They define a meromorphic function 𝔠(𝔭)=−F1ξ(𝔭)/α(ζ(𝔭)) on ℛ with 𝔠(𝔭(λ2))=λρλ+, 𝔠((τ𝔭)(λ2))=λρλ−. The corresponding eigenvectors satisfy

h±(m,λ)=(h±(1)(m,λ)h±(2)(m,λ))=Mγ(m,λ)(cλ(±)1),(3.20)

(Lm(λ)−ρλ±)h±(m,λ)=0.(3.21)

Putting m = 0, we solve

cλ±=L011(λ)±ρλL021(λ)=−L012(λ)L011(λ)∓ρλ,(3.22)

cλ+cλ−=−L012(λ)L021(λ),(3.23)

defining a meromorphic function 𝔠(𝔭) with 𝔠(𝔭(λ2))=λcλ+, 𝔠((τ𝔭)(λ2))=λcλ−. As λ → ∞,

cλ±=〈p,q〉±−F1〈Aq,q〉|〈p0,q0〉λ[1+O(λ−2)].(3.24)

Lemma 3.3 (Formula of Dubrovin-Novikov type).

(h+(1)h−(1)h+(1)h−(2)h+(2)h−(1)h+(2)h−(2))|(m,λ)=(λ2−γ2)mL021(λ)(−Lm12(λ)Lm11(λ)+ρλLm11(λ)−ρλLm21(λ)),(3.25)

h+(2)(m,λ)h−(2)(m,λ)=〈Aq,q〉m〈Aq,q〉0(ζ−γ2)m∏j=1gζ−νj2(m)ζ−νj2(0).(3.26)

Proof.

Using Eq. (3.16), we calculate the left-hand side of Eq. (3.25),

LHS=Mγ(m,γ)(cλ+cλ−cλ+cλ−1)MγT(m,λ)=1L021(λ)Mγ(m,λ)[L0(λ)+ρλI](01−10)MγT(m,λ)=1L021(λ)[Lm(λ)+ρλI]Mγ(m,λ)(01−10)MγT(m,λ)=1L021(λ)[Lm(λ)+ρλI](01−10)detMγ(m,λ)=RHS.

With the help of Eq. (2.12) and (3.25), Eq. (3.26) is verified by some calculations.

Lemma 3.4.

As λ → ∞,

h±(1)(m,λ)=〈p,q〉0±−F1〈Aq,q〉0λm+1[1+O(λ−2)],(3.27)

h±(2)(m,λ)=〈p,q〉m∓−F1〈p,q〉0∓−F1λm[1+O(λ−2)].(3.28)

Proof.

Since h±(1)(m,λ)=Mγ11(m,λ)cλ±+Mγ12(m,λ), we have Eq. (3.27) in virtue of Eq. (3.18) and (3.24). By Eq. (3.25) we get

h+(1)h−(2)|(m,λ)=(λ2−γ2)mLm11(λ)+ρλL021(λ)=〈p,q〉m+−F1〈Aq,q〉0λ2m+1[1+O(λ−2)],h+(2)h−(1)|(m,λ)=(λ2−γ2)mLm11(λ)−ρλL021(λ)=〈p,q〉m−−F1〈Aq,q〉0λ2m+1[1+O(λ−2)].

Thus we obtain Eq. (3.28) by solving h±(2) and using Eq. (3.27).

From Eq. (3.20) we have

h±(2)(2k,λ)=(λcλ±)λ−1Mγ21(2k,λ)+Mγ22(2k,λ),λh±(2)(2k+1,λ)=(λcλ±)Mγ21(2k+1,λ)+λMγ22(2k+1,λ).

By (Lemma 3.2) and the discussion on λcλ±, two meromorphic functions (the Baker functions) H⁽²⁾(2k, 𝔭) and H⁽²⁾(2k + 1, 𝔭) are defined on ℛ, respectively, with

H(2)(2k,𝔭(λ2))=h+(2)(2k,λ),H(2)(2k,(τ𝔭)(λ2))=h−(2)(2k,λ),H(2)(2k+1,𝔭(λ2))=λh+(2)(2k+1,λ),H(2)(2k+1,(τ𝔭)(λ2))=λh−(2)(2k+1,λ).(3.29)

Proposition 3.2.

H⁽²⁾(2k, 𝔭) and H⁽²⁾(2k + 1, 𝔭) have the divisors respectively,

∑j=1g[𝔭(νj2(2k))−𝔭(νj2(0))]+2k𝔭(γ2)−k(∞++∞−),∑j=1g[𝔭(νj2(2k+1))−𝔭(νj2(0))]+(2k+1)𝔭(γ2)+𝔬−(k+1)(∞++∞−).(3.30)

Proof.

From Eq. (3.26) and (3.29) we obtain

H(2)(2k,𝔭)H(2)(2k,τ𝔭)=〈Aq,q〉2k〈Aq,q〉0(ζ−γ2)2k∏j=1gζ−νj2(2k)ζ−νj2(0),H(2)(2k+1,𝔭)H(2)(2k+1,τ𝔭)=〈Aq,q〉2k+1〈Aq,q〉0ζ(ζ−γ2)2k+1∏j=1gζ−νj2(2k+1)ζ−νj2(0),(3.31)

where 𝔭 = 𝔭(ζ). As 𝔭 → ∞_±, by Eq. (3.28) and (3.29) we have

H(2)(2k,𝔭)=〈p,q〉2k∓−F1〈p,q〉0∓−F1ζk[1+O(ζ−1)],H(2)(2k+1,𝔭)=〈p,q〉2k+1∓−F1〈p,q〉0∓−F1ζk+1[1+O(ζ−1)].(3.32)

By these formulas it is easy to calculate the divisors.

By using the technique developed by Toda [28], based on the meromorphic differentials dlnH⁽²⁾(2k, 𝔭) and dlnH⁽²⁾(2k + 1, 𝔭), immediately we get

∑j=1g∫𝔭(νj2(0))𝔭(νj2(2k))ω→+k(∫∞+𝔭(γ2)ω→+∫∞−𝔭(γ2)ω→)≡0, (mod𝒯),∑j=1g∫𝔭(νj2(0))𝔭(νj2(2k+1))ω→+(k+1)(∫∞+𝔭(γ2)ω→+∫∞−𝔭(γ2)ω→)+∫𝔭(γ2)𝔬ω→≡0, (mod𝒯),(3.33)

where ω→=(ω1,…,ωg)T are the normalized basis of holomorphic differentials on ℛ, while 𝒯 is the basic lattice spanned by the periodic vectors of ℛ [8, 11]. With the help of the Abel map 𝒜(𝔭)=∫𝔭0𝔭ω→, the Abel-Jacobi variable is defined as

φ(m)=𝒜(∑j=1g𝔭(νj2(m))).(3.34)

This endows Eq. (3.33) with a clear geometric explanation.

Proposition 3.3.

In the Jacobi variety J(ℛ) = ℂ^g /𝒯, the discrete flow 𝒮γm is linearized by the Abel-Jacobi variable

φ(m)≡φ(0)+mΩγ+δmΩ0γ, (mod𝒯),(3.35)

where δ₂_k = 0, δ₂_k₊₁ = 1, and

Ωγ=12(∫𝔭(γ2)∞+ω→+∫𝔭(γ2)∞−ω→), Ω0γ=Ωγ+∫0𝔭(γ2)ω→.(3.36)

The meromorphic function H⁽²⁾(2k, 𝔭) is expressed by its divisor up to a constant factor

H(2)(2k,𝔭)=const⋅θ[−𝒜(𝔭)+φ(2k)+K]θ[−𝒜(𝔭)+φ(0)+K]⋅exp{k∫𝔭0𝔭ω[𝔭(γ2),∞+]+ω[𝔭(γ2),∞−]},(3.37)

where K is the Riemann constant vector and w[𝔭, 𝔮] is an Abel differential of the third kind, possessing only two simple poles at 𝔭, 𝔮 with residues +1, −1, respectively. Resorting to Eq. (3.32), by the asymptotic behaviors of Eq. (3.37) near ∞_± we obtain

u2k−−F1u0−−F1=const⋅θ[−𝒜(∞+)+φ(2k)+K]θ[−𝒜(∞+)+φ(0)+K](rγ+rγ,−+)k,u2k+−F1u0+−F1=const⋅θ[−𝒜(∞−)+φ(2k)+K]θ[−𝒜(∞−)+φ(0)+K](rγ−rγ,+−)k,(3.38)

where

rγ±=lim𝔭→∞±1ζ(𝔭)exp∫𝔭0𝔭ω[𝔭(γ2),∞±], rγ,∓±=exp∫𝔭0∞±ω[𝔭(γ2),∞∓].(3.39)

We introduce a new variable v_m by

vm=um−−F1um+−F1, um=−F11+vm1−vm.(3.40)

Cancelling the constant factor in Eq. (3.38), we arrive at

v2k=v0⋅θ[2kΩγ+Ω+K(0)]⋅θ[K(0)]θ[2kΩγ+K(0)]⋅θ[Ω+K(0)]⋅e2kRγ,(3.41)

where Ω=∫∞+∞−ω→ and

−𝒜(∞−)=∫∞−𝔭0ω→=η−,−𝒜(∞+)=Ω+η−,K(m)=φ(m)+K+η−,Rγ=12ln(rγ+rγ,−+rγ−rγ,+−).(3.42)

Similarly, considering the analytic expression for H⁽²⁾(2k + 1, 𝔭) leads to

v2k+1=v0⋅θ[(2k+1)Ωγ+Ω0γ+Ω+K(0)]⋅θ[K(0)]θ[(2k+1)Ωγ+Ω0γ+K(0)]⋅θ[Ω+K(0)]⋅e(2k+1)Rγ+R0γ,(3.43)

where

R0γ=Rγ+lnr0γ, r0γ=exp∫∞−∞+ω[𝔬,𝔭(γ2)].(3.44)

Proposition 3.4.

The finite genus potential v_m, defined by Eq. (3.11) and (3.40), has an explicit evolution formula along the discrete flow 𝒮γm,

vm=v0⋅θ[mΩγ+δmΩ0γ+K(0)+Ω]⋅θ[K(0)]θ[mΩγ+δmΩ0γ+K(0)]⋅θ[K(0)+Ω]⋅emRγ+δmR0γ,(3.45)

where the vectors K(m), Ω_γ, Ω₀_γ and Ω are given by Eq. (3.36) and (3.42), while the constants R_γ, R₀_γ are defined by Eq. (3.42) and (3.44); moreover, δ₂_k = 0, δ₂_k₊₁ = 1, for all k.

4. Solutions of lSKdV equation (1.1)

Let γ₁, γ₂ be the two constants given in Eq. (1.1). By (Proposition 3.1), setting γ = γ₁, γ₂ in the above we have two symplectic maps 𝒮_γ₁ and 𝒮_γ₂, sharing the same set of integrals {F_l}. Resorting to the discrete version of Liouville-Arnold theorem [25,27,29], they commute. Thus we have well-defined functions with two discrete arguments m and n,

(p(m,n),q(m,n))=𝒮γ1m𝒮γ2n(p0,q0),bmn=fγ(p(m,n),q(m,n)),umn=fU(p(m,n),q(m,n))=〈p(m,n),q(m,n)〉,vmn=(umn−−F1)/(umn+−F1).(4.1)

Proposition 4.1.

Both the functions umn and vmn, defined by Eq. (4.1), solve Eq. (1.1).

Proof.

By the commutativity of 𝒮γ1m and 𝒮γ2n, we have

(p(m,n),q(m,n))=𝒮γ1m(p(0,n),q(0,n))=𝒮γ2n(p(m,0),q(m,0)).(4.2)

From Eq. (3.12) we obtain

bmn=γ1/(u−u˜)=γ1/(u−u^).(4.3)

By Eq. (3.6), χ_j = (p_j(m, n), q_j(m, n))^T solves simultaneously

χ˜j=(αj2−γ12)−1/2D(γ1)(αj,bmn)χj, bmn=γ1/(u−u˜),χ^j=(αj2−γ22)−1/2D(γ2)(αj,bmn)χj, bmn=γ2/(u−u^).(4.4)

Thus u_mn satisfies Eq. (1.1) by Eq. (1.11). In order to prove that v_mn is also a solution, it is sufficient to notice that (i) F₁ is a constant of motion which is independent of m and n; (ii) Eq. (1.1) is invariant under the Möbius transformation u ↦ v given by Eq. (4.1).

Apply Eq. (3.45) to the flow 𝒮γ1m and 𝒮γ2n successively. By v₀₀ → v_m₀ → v_mn we obtain

Proposition 4.2.

The lSKdV equation (1.1) has finite genus solutions

vmn=v00⋅θ[mΩγ1+nΩγ2+δmΩ0γ1+δnΩ0γ2+K00+Ω]⋅θ[K00]θ[mΩγ1+nΩγ2+δmΩ0γ1+δnΩ0γ2+K00]⋅θ[K00+Ω] ⋅exp(mRγ1+nRγ2+δmR0γ1+δnR0γ2),(4.5)

and umn=−F1(1+vmn)/(1−vmn). Further, any Möbius transformation wmn = (a₁₁v_mn + a₁₂)/(a₂₁v_mn + a₂₂) solves Eq. (1.1), where ajk are constants.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant Nos. 11426206; 11501521), State Scholarship Found of China (CSC No. 201907045035), and Graduate Student Education Research Foundation of Zhengzhou University (Grant No. YJSXWKC201913). We would like to thank Prof. Frank W. Nijhoff and Prof. Da-jun Zhang for helpful discussions.

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Journal: Journal of Nonlinear Mathematical Physics
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Journal of Nonlinear Mathematical Physics

Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation

1. Introduction

Lemma 1.1.

2. The Integrable Hamiltonian System (H1)

Proposition 2.1.

Proposition 2.2.

Proof.

3. The Integrable Symplectic Map 𝒮γ

Lemma 3.1.

Proof.

Proposition 3.1.

Proof.

Lemma 3.2.

Lemma 3.3 (Formula of Dubrovin-Novikov type).

Proof.

Lemma 3.4.

Proof.

Proposition 3.2.

Proof.

Proposition 3.3.

Proposition 3.4.

4. Solutions of lSKdV equation (1.1)

Proposition 4.1.

Proof.

Proposition 4.2.

Acknowledgments

References

Cite this article

2. The Integrable Hamiltonian System (H₁)

3. The Integrable Symplectic Map 𝒮_γ