Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation
- DOI
- 10.1080/14029251.2020.1819608How 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],
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,
Technically, it is more convenient to use the derivative version
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
The compatibility condition
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
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 (ℝ2N, dp ∧ dq) as the phase space. The symplectic coordinate is defined as (p, q) = (p1,..., pN, q1,...,qN). Let A = diag(α1,...,αN) with distinct, non-zero
∀ (ξ,η) = (ξ1,...,ξN,η1,...,η N) ∈ ℝ2N.
The generating function ℱλ = detL(λ; p, q) is a rational function of the argument ζ = λ2 ,
The expansion
They are exactly N copies of Eq. (1.7) with distinct λ = αj and the constraint
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,...,FN given by (2.3) are first integrals of the phase flow with Hamiltonian function H1, i.e., {Fj, H1} = 0 (j = 1,...,N), where {·, ·} denotes the Poisson bracket on the phase space. The involution and functional independence between F1,...,FN guarantee that the Hamiltonian system (H1) is completely integrable.
Consider the Hamiltonian system (ℱλ),
As a corollary, we have
Actually, by Eq. (2.7), (d/dtλ)L2(μ) = [W(λ, μ), L2(μ)]. Since L2(μ) = −Iℱμ, where I is the identity matrix, we have dℱμ/dtλ = 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 ℱλ has a factorization
Putting μ = νk, with
Eq. (2.15) is rewritten in a simple form and gives rise to
Proposition 2.1.
The ℱλ- and the Fl-flow are linearized by ϕ′s as
In particular, {ϕ′s, F1} = 0, 1 ⩽ s ⩽ g.
Proposition 2.2.
The Hamiltonian system (H1) is integrable, possessing N integrals F1,...,FN, involutive with each other and functionally independent in the dense, open subset 𝒪 = {(p, q) ∈ ℝ2N : F1 ≠ 0}.
Proof.
Fl is an integral since {H1, Fl} = (1/2){F1, Fl} = 0 by Eq. (2.9). It needs only to prove that dF1,...,dFN are linearly independent in
By Eq. (2.18), the coefficient matrix is non-degenerate,
Thus c2 = ··· = cN = 0 and c1dF1 = 0. We have c1 = 0 since dF1 ≠ 0 at 𝒪. Otherwise,
Hence αjqj + 〈p, q〉pj = 0, ∀j; and F1 = 0. This is a contradiction.
3. The Integrable Symplectic Map 𝒮γ
As a non-linearization of Eq. (1.8), define a map 𝒮γ : ℝ2N → ℝ2N,
Lemma 3.1.
Let P(γ)(b; p, q) = b2L21(γ) + 2bL11(γ) − L12(γ). Then
Proof.
By Eq. (3.1), we get
Based on these preparations, we calculate the left-hand side of Eq. (3.2),
By using Eq. (3.1), we obtain
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,
Actually γb can be written as a meromorphic function on ℛ,
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 𝒮γ : ℝ2N → ℝ2N,
Proof.
Since P(γ)(b) = 0, by Eq. (3.2) and (3.3) we have
Taking the determinant of Eq. (3.8), we obtain
By Eq. (3.7), the discrete flow
By Eq. (3.4), they have the relation
By induction we have
Lemma 3.2.
The following functions are polynomials of the argument ζ = λ2:
Besides, as λ → ∞,
By Eq. (3.13), the solution space ℰλ of Eq. (3.14) is invariant under the action of the linear operator Lm(λ), which has two eigenvalues
They define a meromorphic function
Putting m = 0, we solve
Lemma 3.3 (Formula of Dubrovin-Novikov type).
Proof.
Using Eq. (3.16), we calculate the left-hand side of Eq. (3.25),
With the help of Eq. (2.12) and (3.25), Eq. (3.26) is verified by some calculations.
Lemma 3.4.
As λ → ∞,
Proof.
Since
Thus we obtain Eq. (3.28) by solving
From Eq. (3.20) we have
By (Lemma 3.2) and the discussion on
Proposition 3.2.
H(2)(2k, 𝔭) and H(2)(2k + 1, 𝔭) have the divisors respectively,
Proof.
From Eq. (3.26) and (3.29) we obtain
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
This endows Eq. (3.33) with a clear geometric explanation.
Proposition 3.3.
In the Jacobi variety J(ℛ) = ℂg
/𝒯, the discrete flow
The meromorphic function H(2)(2k, 𝔭) is expressed by its divisor up to a constant factor
We introduce a new variable vm by
Cancelling the constant factor in Eq. (3.38), we arrive at
Similarly, considering the analytic expression for H(2)(2k + 1, 𝔭) leads to
Proposition 3.4.
The finite genus potential vm, defined by Eq. (3.11) and (3.40), has an explicit evolution formula along the discrete flow
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 𝒮γ1 and 𝒮γ2, sharing the same set of integrals {Fl}. 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,
Proof.
By the commutativity of
From Eq. (3.12) we obtain
By Eq. (3.6), χj = (pj(m, n), qj(m, n))T solves simultaneously
Thus umn satisfies Eq. (1.1) by Eq. (1.11). In order to prove that vmn 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
Proposition 4.2.
The lSKdV equation (1.1) has finite genus solutions
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.
References
Cite this article
TY - JOUR AU - Xiaoxue Xu AU - Cewen Cao AU - Guangyao Zhang PY - 2020 DA - 2020/09/04 TI - Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation JO - Journal of Nonlinear Mathematical Physics SP - 633 EP - 646 VL - 27 IS - 4 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1819608 DO - 10.1080/14029251.2020.1819608 ID - Xu2020 ER -