Algebro-geometric Constructions of a Hierarchy of Integrable Semi-discrete Equations

A hierarchy of integrable semi-discrete equations is deduced in terms of the discrete zero curvature equation as well as its bi-Hamiltonian structure is gotten through the trace identity. The above hierarchy is separated into soluble ordinary differential equations according to the relationship between the elliptic variables and the potentials, from which the continuous flow is straightened out via the Abel–Jacobi coordinates resorting to the algebraic curves theory. Eventually, the meromorphic function and the Baker–Akhiezer function are introduced successively on the hyperelliptic curve and the algebro-geometric solutions which are expressed as Riemann theta function can be obtained through the two functions mentioned above.


Introduction
It can be said with certainly that solving soliton equations is a crucial subject of theoretical research of solitons and integrable system, and that ever-growing number of methods for solving the soliton equations are proposed with the efforts of researchers. Algebro-geometric solution is an important category of solutions to nonlinear equations, which can not only reveal the intrinsic structure of solutions, but also characterize the quasi-periodic behavior of the nonlinear phenomena.
During the past five decades, numbers of researchers made significant contributions to solving soliton equations with the algebro-geometric approach. The modern origin of this method is that the researchers Burchnall and Chanundy found that there is a connection between linear differential operators and algebraic curves with the aid of the study of the linear differential operators [1,2]. In the mid-twentieth century, Novikov and Krichever proposed a systematic algebraic geometric approach to solve nonlinear equations with the aim of obtaining quasi-periodic solutions of the Riemann theta function [3][4][5]. Moreover thrilling results appeared later. In 1988, Cao managed to use the nonlinear features of Lax pairs to study algebro-geometric solutions of finite-dimensional integrable Hamiltonian systems and it was applied to many soliton equations successfully [6,7]. Then Geng generalized a new method called finite-order expansion of the Lax matrix, which has obtained great achievements and gave an alternative way of solving the soliton equations [8,9]. In the 1990s, F. Gestesy and H. Holden developed a polynomial recursion method that successfully extended algebro-geometric solutions from a single equation to a hierarchy [10,11]. In view of the current situation, the study of solving soliton equations using algebraic geometric methods attracted increasing interest and it has been applied successfully to soliton equations connected with 2 × 2 and 3 × 3 discrete matrix spectral problems [12][13][14][15][16]. From the perspective of development, there is hope that we can apply algebro-geometric methods to solving the soliton equations linked to the 4 × 4 discrete matrix spectral problems [17].
It is universally acknowledged that solving the soliton equations connected with 2 × 2 matrix spectral problem is a prerequisite and basis for solving the soliton equations linked to higher-order matrix spectral problem. The whole solving process can be divided into four main steps as follows.
Step 1. Generating an integrable hierarchy with bi-Hamiltonian structure We first introduce a discrete spectral parameter and then the hierarchy can be derived with the aid of the discrete zero curvature equation. Subsequently, the bi-Hamiltonian structure of the hierarchy can be obtained through the trace identity.
Step 2. Separating the above hierarchy into soluble ordinary differential equations (ODEs) The Lenard recursion gradients and a Lax matrix are introduced, from which the relationship between the elliptic variables and the potentials can be established, so we can separate the corresponding hierarchy into soluble ODEs.
Step 3. Straightening out the of continuous flow For the sake of completing this step, we need to introduce the theory of hyperelliptic Riemann surface, then the flow can be straightened successfully by the Abel-Jacobi coordinates.
Step 4. Getting the algebro-geometric solutions In order to solve the soliton equation, the main tool is to straighten out the discrete flow to linearize the equations and then express the linearized equations as the Riemann theta function. There are two ways to straighten out it. One way is to define a fundamental solution matrix of soliton equations, using the expression of Dubrovin-Novikov's type to straighten out the discrete flow to linearize the soliton equations [18]. The other way is to establish the meromorphic function Φ , then analyze the asymptotic properties of them, the solutions can be obtained if we can get the Riemann theta function representations of the function Φ in a concise form [19], if not, we need to define the Baker-Akhiezer function Ψ on the basis of the function Φ and analyze its asymptotic behavior, the explicit Riemann theta function representations of the functions Φ and Ψ can be generated after that [20].
Though using algebro-geometric method to solve soliton equations is systematically provided in the case of 2 × 2 matrix spectral problem [21][22][23][24], inferring the solutions is challenging because of concerning the knowledge of the discrete variables and algebraic curve. For this paper, we will discuss the following integrable hierarchy of semi-discrete equations which has been studied explicit solutions by Darboux transformation [25], where u n and v n are two potentials and t is the time variable. As an application, the main aim of this paper is to use algebraic geometric methods to solve the hierarchy (1).
The outline of the present paper is as follows. In Sect. 2, we derive a hierarchy of semi-discrete equations (1) by use of the discrete zero curvature equation and its bi-Hamiltonian system is constructed through the trace identity. In Sect. 3, the Lax matrix and elliptic variables are introduced, through which we can decompose the hierarchy (1) into soluble ODEs. In Sect. 4, we straighten out the continuous flow on the basis of the hyperelliptic Riemann surface and the Abel-Jacobi coordinates. In Sect. 5, we define the stationary meromorphic function and analyze its asymptotic behavior, then in the time-dependent case, the algebro-geometric constructions of the hierarchy (1) are proposed owing to the Riemann theta function and the asymptotic behavior of the functions. In Sect. 6, the conclusions and remarks of this paper are summarized.

The Integrable Hierarchy and Its Bi-Hamiltonian Structure
To get the corresponding hierarchy (1), we design the following isospectral problem where n = ( 1 n , 2 n ) T , E is a shift operator (Eu n = u n+1 ) and is a constant spectral parameter ( t = 0) . Then, we introduce the stationary discrete zero curvature equation Let us suppose that a solution V [m] n of (3) is given by V [m] n = a n b n c n − a n , where a n , b n , c n are polynomials of the spectral with Taking and substituting the expanding expressions (5) into (4), the recurrence relation can be derived as follows and Under the initial-value conditions of a (0) n,± , b (0) n,± and c (0) n,± , the recursion relations (5) uniquely decide a (m) n,± , b (m) n,± , c (m) n,± , m ≥ 1 and the first few quantities are given by −u n (a n+1 + a n ) + u n c n+1 − c n = 0, Since the formula (6) and (7) imply that V [m] n unsatisfied the compatibility condition, so we must take an addictive constant to adjust the expression of a n . For m ≥ 0 (m ∈ Z) , we select and make a modification According to (12), the semi-discrete equations are implied as follows a (1) n,+ =u n (1 + v n−1 ), b (1) n+1,+ = 1 + v n , c (1) n,+ = u n , a (2) n, so the discrete zero curvature representation of (13) is (12). It is easy to verify that the first nonlinear lattice equation in (13) is (1), namely if and only if m = 1 and t 1 = t.
Next, we will build up the bi-Hamiltonian structure and prove its Liouville integrability. Depends on the discrete trace identity we have here and Using (5), we derive the recursion constructions the recursion operator is with and Therefore, it is not difficult to prove the Liouville integrability of the hierarchy (1).

Elliptic Variables and Soluble Ordinary Differential Equations
Firstly, the hierarchy (1) should be converted to soluble ordinary differential equations. Suppose that (3) has two essential solutions (n) = ( (1) (n), (2) (n)) and (n) = ( (1) (n), (2) (n)) , we then give a Lax matrix W n with three functions f(n), h(n) and g(n) of , and W n ought to satisfy It is easy to see that (21) can be extended as and Suppose the following functions are polynomials of finite order in : Then, the Lenard gradient sequences are listed as follows Taking (24) into (22) and defining G j, ± (n) = (h j,± (n), g j,± (n), f j,± (n)) T , we have where the matrix differential operators J n ,K n are given by It is evident that where 0,± are constants. Taking J −1 nKn and K −1 nJn into (27), we have so According to (27) and (28), we can obtain the values of G , ± (n) ( = 0, 1, 2) and S 0,± (n) as 0,± = 1,

3
By deduction, (24) can also be represented as by (j ∈ ℕ 0 ), The corresponding homogeneous polynomials can also be defined as by (s ∈ ℕ), and with (29) h 0,+ (n) = 0, g 0, , We write g(n) and h(n) in the following expression where the elements j (n) and j (n) named elliptic variables. Owing to (36), contrasting the coefficients of in g(n) and h(n), we can get As det W n is a (2N + 2) th-order polynomial of and its coefficients are constants of the n-flow and t m -flow, we obtain According to (38), we have Based on (24) and (39), we obtain which means

3
To sum up, we have separated the hierarchy of semi-discrete equations into soluble ordinary differential equations.

Straightening Out of the Continuous Flow
In the chapter, we introduce the Riemann surface K N and then give a list of independent canoncial basis cycles: r 1 , … , r N ;z 1 , … , z N , which have the following intersection numbers For K N , we select the differentials of holomorphic and define where L and M are period matrices. Now, on the basis of matrices L and M , we define two new matrices N and to be of the following form and the can be proved to be symmetric ( ij = ji ) and its imaginary part Im > 0.
Substituting ̃ s into the new basis j : which meets .
In order to finalize the straightening out of the continuous flow, we need to first introduce the following equations and lemmas. We start with a few elementary results, let = ( 1 , … , 2N+2 ),̄0( ) = 0 ( ) = 1 , then one can get and Next, we can deduce that N -dependent summation constants 0,+ , ⋯ , N � ,+ and 0,− , ⋯ , N � −1,− in f(n), g(n), and h(n) with 0 ( ) ±1 = 1. Hence, the relationships between the homogeneous and nonhomogeneous can be derived as follows, where s = 0, ..., N � − ± , and the ± follows the following convention, In addition, we introduce the symmetric functions according to the relevant formula of the Lagrange interpolation, with Lemma 1 Assuming that are N distinct complex numbers. Then, where Proof From (24) and (36), we obtain Using (56) and noticing that h 0,+ =h 0,+ = 0 and 0,± = 1 , we arrive According to (34), we obtain So, we can get the following recursion formula by (57), Considering the case 1 ≤ m ≤ N ′ , using (35), (58) and H 0,± = 0 , we can continue the transformation of the above equations (61) Using (49) and (58), we obtain Therefore, the equation (59) can be proven and we can get (60) in a similar way. In summary, we have straightened out of the continuous flow. ◻

Algebro-geometric Solutions
For this paper, we will choose the second way of the Step 4 mentioned above to get the solutions. During this section, we will analyze the asymptotic behavior of the function Φ(x, n) , and then we will introduce the function Ψ(x, n) to analyze its asymptotic behavior since we can not get the simple Riemann theta function representations of the function Φ(x, n) . Eventually, the algebro-geometric solutions of the time-dependent case can be deduced.

The Stationary Meromorphic Function
In what follows, we will give the stationary meromorphic function of the integrable hierarchy (1). According to (38), the hyperelliptic curve K N of arithmetic genus N can be denoted by From (38) and (67), we know that and Then, the stationary function Φ(x, n) on K N can be defined as the following two forms and The curve K N can be compacted by adding two points at x ∞+ and x ∞− . Based on the definition of (67), we can define the points j (n) and j (n) of K N Furthermore, the points x 0,+ and x 0,− can be defined through with and we introduce the holomorphic sheet exchange map * , where y j ( ), j = 0, 1. Next, we define the corresponding Baker-Akhiezer function Ψ(x, n, n 0 ) by and y 2 N � 2 = f 2 (n) + g(n)h(n), n)). The stationary meromorphic function Φ(x, n) on K N is expressed by According to (3) and (73), the function Φ(x, n) meets the following Riccati-type which also satisfies the system through (69) In addition, we introduce the expressions of F ,G,H and according to (69), we can get Assuming that (76) have the following asymptotic behavior. As x → x ∞± , we have and as x → x 0 ± , we have of which is the local coordinate and = −1 ∕ beside x ∞± ∕x 0± . (73) (1 + u(n � )Φ(x, n � )), n ≥ n 0 + 1, 1, n = n 0 , (1 + u(n � )Φ(x, n � )) −1 ), n ≤ n 0 − 1, n). (76) Otherwise, (f s , ± ,g s , ± ,h s , ± ) and (f s , ± ,ḡ s , ± ,h s , ± ) have the same recursion relations, so we can obtain Lemma 3 Suppose that u and v meet the discrete hierarchy (1) and define x = ( , y) ∈ K N ⧵ {x ∞± , x 0± }. Based on these, the meromorphic function has the following asymptotic behavior where u n , v n satisfy the hierarchy of nonlinear semi-discrete equations (1).
In the following, we will express the functions Φ, Ψ 2 , u(n) and v(n) in the form of (48). For the third kind holomorphic on K N ⧵ {x + , x − } , we give one regular differential w (3) x + ,x − and it has poles at x ± and residues ±1 . Particularly, we can get So, we inference the following asymptotic expansions. For x → x 0± , we have for x → x ∞± , we have Similar to the previous step, for the second kind holomorphic, we select w (2) x ∞± ,q and w (2) x 0± ,q be the regular differentials with a distinctive pole at x ∞± and x 0± . Between them, the major part as follow, Moreover, we define where ,+ are the summation constants in b n .