Algebro-Geometric Solutions of the Sine-Gordon Hierarchy

On the basis of two sets of Lenard recursion sequences and zero-curvature equation associated with a matrix spectral problem, we derive the entire sine-Gordon hierarchy, which is composed of all the positive and negative flows. Using the theory of hyperelliptic curves, the Abel-Jacobi coordinates are introduced, from which the corresponding positive and negative flows are linearized. The algebro-geometric solutions of the entire sine-Gordon hierarchy are constructed by using the asymptotic properties of the meromorphic function.


Introduction
Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the integrability of soliton equations has been found by the inverse scattering transformation method [1][2][3], which has greatly promoted the universal understanding of soliton equations, thus making the soliton theory get rapid development. Subsequently, several systematic approaches have been developed to obtain algebro-geometric solutions of soliton equations associated with 2 × 2 matrix spectral problems. For example, an analog of the inverse scattering theory for Hill's equation was developed, from which ones gave an explicit formula of the periodic potentials with the finite number of gaps in the spectrum and derived Xue Geng and Liang Guan contributed equally to this work.  [4][5][6][7]. Another effective method is the algebraic curve method that combine the spectral theory of differential equations, the algebraic curve of compact Riemann surfaces and their Jacobians, the Riemann theta functions and inverse problems (see, e.g., [8][9][10][11] and references therein), by which algebro-geometric solutions of many soliton equations associated with 2 × 2 matrix spectral problems are obtained such as the KdV equation, nonlinear Schrödinger equation, mKdV equation, sine-Gordon equation, relativistic Toda lattice, and so on [4][5][6][7][8][9][10][11]. Matveev and Yavor gave complex finite-band multiphase solutions of the Kaup-Boussinesq equation and used the degeneracy procedure to find multisoliton solutions [12]. The nonlinearization approach of Lax pairs in Ref. [13] has been developed and applied to construct the analytical Riemann theta function solutions of soliton equations [14][15][16][17][18][19][20][21], which is generalized to the spatial part and the temporal parts of the Lax pairs and the adjoint Lax pairs are constrained as finite-dimensional Liouville integrable Hamiltonian systems and integrable symplectic maps [22,23]. Gesztesy and Ratnaseelan have proposed an alternative systematic approach based on elementary algebraic methods to get analytical Riemann theta function solutions of the AKNS hierarchy [24]. In addition, this method was further developed to deal with soliton equations such as the modified Boussinesq, the Kaup-Kupershmidt, the coupled modified KdV hierarchies, and so on [25][26][27][28][29][30].
The main aim of the present paper is to construct analytical Riemann theta function solutions of the entire local sine-Gordon hierarchy, which is composed of all the positive and negative flows, on the basis of the algebraic curve method. It is well known that the higher-order sine-Gordon equations in the sine-Gordon hierarchy are nonlocal in the light-cone coordinates, which makes the problem extremely complicated. To overcome this difficulty, Boiti et al. use the fact that a nonlinear evolution equation in one field can be written as a system of coupled nonlinear evolution equations in two fields. They found that the suitable spectral problem to realize their idea is the Boiti-Tu spectral problem [31] with some compatiable reductions [32]. They derive a hierarchy of local nonlinear evolution equations generated by a recursion operator and its explicit inverse. This hierarchy satisfies a canonical geometrical scheme and contains the sine-Gordon equation and Liouville equation for special cases in laboratory coordinates. A generalization of the Bäcklund transformation and a nonlinear superposition formula for the sine-Gordon equation are also obtained. The positive and negative order hierarchy generated from the Boiti-Tu spectral problem was first proposed by Qiao in 1994 [33]. Regarding the negative flows of the sine-Gordon hierarchy is also given by Qiao in [34]. This paper is organized as follows. In Sect. 2, we introduce the Lenard gradient sequences and derive the entire local sine-Gordon hierarchy, which is composed of all the positive and negative flows, with the aid of the zero-curvature equation associated with a 2 × 2 matrix spectral problem. In Sect. 3, we define a Lax matrix from which the elliptic variables and the hyperelliptic Riemann surface of arithmetic genus 2N + 1 are derived, the meromorphic function is introduced and its properties on some points are discussed. In Sect. 4, under the Able-Jacobi coordinates, the spatial and temporal flows of the entire local sine-Gordon hierarchy are linearized. Finally, analytical Riemann theta function solutions of the entire local sine-Gordon hierarchy are constructed by using the asymptotic properties of the meromorphic function .

The Entire Local Sine-Gordon Hierarchy
In this section, we shall derive the entire local sine-Gordon hierarchy associated with the Boiti-Tu spectral problem [31] where u, v, s are potentials with the condition s 2 − v 2 = s 2 0 , s 0 is a nonzero constant and is a constant spectral parameter. To this end, we introduce the Lenard gradient sequences with starting points where S j,± = (c j,± , b j,± , a j,± ) T , and two operators K and J are defined by It is easy to see that where ± are arbitrary constants. Then S j,+ (j ≥ 0) and S j,− (j ≤ 0) are uniquely determined by (2) and (3), respectively, if choosing all constants of integration to be zero. For example, the first two members are   (7) is reduced to which is the sine-Gordon equation in laboratory coordinates.
Then the second nontrivial member in the hierarchy reads

Elliptic Variables
Let = ( 1 , 2 ) T and = ( 1 , 2 ) T be two basic solutions of (1) and (4). We introduce a Lax matrix which satisfies the Lax equations Therefore, detW is a constant independent of x and t m . Equation (8) can be written as   and   Let where g, f and h are finite-order polynomials in and −1 : Substituting (12) into (9) yields It is easy to see that the equations JE 0,+ = 0 and have the general solution where 0,+ and 0,− are constants. Acting with the operators (J −1 K) k and (K −1 J) k respectively, on (14) and (15), we obtain from (13) and (2) that where 1,± , 2,± , … , k,± are constants of integration. The first member in (16) is By inspection of (11) and (12) , we write G, F and H as follows: Comparing the coefficients of with the same power, we arrive at From the relation above, one can easily find that H( ) = −F(− ) . So we can write F and H as finite products which take the form: where { j } 2N+1 j=1 are called elliptic variables. Since detW depends only on , whose coefficients are constants independent of x and t m , we have from which one is naturally led to introduce the hyperelliptic curve K 2N+1 of arithmetic genus 2N + 1 defined by The curve K 2N+1 can be compactified by joining two points at infinity, P ∞± , where P ∞+ ≠ P ∞− . For notational simplicity the compactification of the curve K 2N+1 is also denoted by K 2N+1 . Here we assume that j of R( ) in (18) are mutually distinct, which means j ≠ k , for j ≠ k , 1 ≤ j, k ≤ 2N + 2 . Then the hyperelliptic curve K 2N+1 becomes nonsingular. According to the definition of K 2N+1 , we can lift the roots j to K 2N+1 by introducing We also introduce the points P 0± by where we emphasize that P 0± and P ∞± are not necessarily on the same sheet of K 2N+1 .
From (18) and (19) we can define the meromorphic function (P, Proof According to (9) and (21), we can get that satisfies the Riccati-type equation By inserting the ansatz into (24), a comparison of powers of then proves the first line in (22). Similarly, after the ansatz is inserted into (24), comparing powers of proves the second line in (22). In exactly the same manner, inserting the ansatz into (24) immediately yields (23). ◻ Hence the divisor of (P, x, t m ) is The following elementary results are benefit for our main work.

Straightening Out the Flows
In order to straightening out the spatial and temporal flows of the full local sine-Gordon hierarchy, we consider the Riemann surface K 2N+1 and equip K 2N+1 with canonical basis cycles: 1 , … , 2N+1 ; 1 , … , 2N+1 , which are independent and having intersection numbers as follows For the present, we will choose our basis as the following set [22,23] f l,

3
which are 2N + 1 linearly independent homomorphic differentials on K 2N+1 . Then the period matrices A and B can be constructed from It is possible to show that matrices A and B are invertible [35,36]. Now we define the matrices C and by C = A −1 , = A −1 B . The matrix can be shown to be symmetric ( kj = jk ), and it has positive definite imaginary part (Im > 0 ). If we normalize ̃l into the new basis j , then we obtain Let T be the lattice generated by 4N + 2 vectors j , From (20), we obtain Journal of Nonlinear Mathematical Physics (2023) 30:114-134 Noticing (9), (10) and (17), we get hence we arrive at the evolution of j along the x and t m flow: Assuming N ∈ ℕ to be fixed and introducing one defines the symmetric functions According to the Lagrange interpolation theorem, we have the following important facts (these lemmas were proven in detail in [26], Lemma E.2 and Lemma E.3).
−s − s 0 v = N(x, t m )exp( 0+ 0 ) (z(P 0+ , −̂(x, t m ))) (z(P 0+ ,̂(x, t m ))) , and by direct calculation, we can derive (36). Hence, we prove this theorem on M . The extension of all these results from M to M then follows by continuity of the Abel map and the nonspecial nature of D̂( x,t m ) on M. ◻

Conclusion
In this paper, algebro-geometric solutions are constructed for the entire sine-Gordon hierarchy, which is composed of all the positive and negative flows. The results are different from the sine-Gordon hierarchy in light-cone coordinates which contains complicated nonlocal higher order sine-Gordon equations. Because this work involves hyperelliptic curves, it is very difficult to study the algebro-geometric solutions of the entire sine-Gordon hierarchy. Using the theory of hyperelliptic curves, the Abel-Jacobi coordinates are introduced, from which the corresponding positive and negative flows are linearized. The algebro-geometric solutions of the entire sine-Gordon hierarchy are constructed by using the asymptotic properties of the meromorphic function.

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