Construction of two parametric deformation of KdV-hierarchy and solution in terms of meromorphic functions on the sigma divisor of a hyperelliptic curve of genus 3

Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we constructed the field of meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3 and solutions of the systems for $g=3$ by these functions. In this paper, as an application of our previous results, we construct two parametric deformation of the KdV-hierarchy. This new system is integrated in the meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3.


Introduction
Let V g be a hyperelliptic curve of genus g defined by V g = {(X, Y ) ∈ C 2 | Y 2 = X 2g+1 +y 4 X 2g−1 −y 6 X 2g−2 +· · ·+y 4g X −y 4g+2 , y i ∈ C}. (1) A meromorphic function on the Jacobian of V g is called hyperelliptic function. The theory of hyperelliptic functions has deep relations with that of KdV-hierarchy. The KdV-hierarchy is an infinite system of differential equations defined by U t k = χ k U, k = 1, 2, . . . , for a function U = U(t 1 , t 2 , . . . , ). The functions χ k U are determined by the recursion with the initial condition χ 1 = ∂/∂t 1 , where R is the Lenard operator R = 1 4 where (∂/∂t 1 ) −1 implies an integral with respect to t 1 . The KdV-equation is obtained for k = 2 In the theory of hyperelliptic functions associated with the model (1), the hyperelliptic sigma functions play an important role. The hyperelliptic sigma functions σ(w 1 , w 3 , . . . , w 2g−1 ) are entire functions of g complex variables, which are originally introduced by Klein as a generalization of the Weierstrass elliptic sigma functions. Baker made a significant contribution of the theory of sigma functions: for hyperelliptic curves of genera 2 and 3, he obtained explicit expressions for higher logarithmic derivatives of sigma functions of many variables in the form of polynomials in the second and the third logarithmic derivatives of these functions [2,3,4]. Relatively recently it was shown that these differential polynomials give the fundamental equations of mathematical physics, including KdV-hierarchy and KP-equations (see [5,7,11]). The surface determined by the equation σ(w 1 , . . . , w 2g−1 ) = 0 in the Jacobian of V g is called the sigma divisor and denoted by (σ). Let F ((σ)) be the field of meromorphic functions on the sigma divisor of the hyperelliptic curves of genus 3. The functions f ∈ F ((σ)) are considered as meromorphic functions on C 3 whose restrictions to the sigma divisor (σ) are 6-periodic. In [9], the polynomial dynamical systems in C 4 with two polynomial integrals are constructed on the basis of commuting vector fields on the symmetric square of the hyperelliptic curves V g . In [1], for g = 3, the solutions of the systems are constructed in terms of the functions of F ((σ)).
For g = 2, the dynamical systems of [9] are related to the KdV-equation [5,10]. In this paper we consider the case of g = 3 and construct two parametric deformation of the KdV-hierarchy by using the dynamical systems of [9] (Theorem 5.1). We construct a solution of the new system in terms of functions of F ((σ)) (Theorem 7.1). If y 12 = y 14 = 0, then the new system goes to the system of the KdV-hierarchy in [5] Theorem 5.2 (Proposition 6.4). The result of this paper is one of the applications of the results in [1]. In [12], an extention of the sine-Gordon equation is given and a solution is constructed in terms of the al-function on the subvariety in the hyperelliptic Jacobian. The results of this paper can be regarded as an analog of the results of [12] for the KdV-hierarchy. In Section 8, we consider the rational case (y 4 , . . . , y 14 ) = (0, . . . , 0) and derive a rational solution of the KdV-hierarchy. This solution is equal to the solution obtained by the rational limit of the hyperelliptic functions of genus 2. This result would give an insight into the degeneration of the sigma functions.

Rational functions on the symmetric square
In [1], for g = 3, the structure of the field of rational functions on the symmetric square of the curve V 3 is described explicitly. These results can be extended for any genus similarly. In this section we describe the structure of the field of rational functions on the symmetric square of the curve V g .
Let F (V 2 g ) be the field of rational functions on V 2 g and let J g be the ideal in . We denote the quotient field of an integral domain R by R . We have Let Sym 2 (C 2 ) be the symmetric square of C 2 and let F (Sym 2 (C 2 )) be the set of be the symmetric square of the curve V g and let F (Sym 2 (V g )) be the set of elements h ∈ F (V 2 g ) such that there exists a representative h ∈ F (Sym 2 (C 2 )) of h. In [9, 1], the following elements of F (Sym 2 (C 2 )) are used: Note that the elements a, b, c, and d are algebraically independent and generate the field and N g = −N g /2 + aM g . For example, for g = 2, we obtain and for g = 3, we obtain 1 Let A g be the ideal generated by the polynomials M g and N g in the ring C[a, b, c, d] and let u i , i = 2, 4, 2g − 1, 2g + 1, denote the elements of F (V 2 g ) such that u 2 = a, u 4 = b, u 2g−1 = c, and u 2g+1 = d in the field C(X 1 , Y 1 , X 2 , Y 2 ), i.e., u 2 , u 4 , u 2g−1 , and u 2g+1 are the equivalence classes of a, b, c, and d in F (V 2 g ), respectively. Note that u 2 , u 4 , u 2g−1 , and u 2g+1 are contained in F (Sym 2 (V g )). Consider the homomorphism Then we have Ker(Γ g ) = A g and the isomorphism ([1], Lemma 3.3 and Theorem 3.4) The following two commuting derivations acting on the field F (V 2 g ) were used in [9,1]: In [9] Lemmas 16 and 17, L i u j is expressed as a polynomial of u 2 , u 4 , u 2g−1 , and u 2g+1 whose coefficients are in Q[y 4 , y 6 , . . . , y 4g+2 ] for any i = 2g − 3, 2g − 1 and j = 2, 4, 2g − 1, 2g + 1 These can be regarded as polynomial dynamical systems in C 4 with coordinates u 2 , u 4 , u 2g−1 , and u 2g+1 . We assume g = 3.
Theorem 3.1. ( [9] Lemmas 16 and 17) In the space C 4 with coordinates u 2 , u 4 , u 5 , and u 7 , we have the following families of dynamical systems with constant parameters y 4 , y 6 , y 8 , and y 10 : The systems (I) and (II) have common first integrals H 12 := M 3 (u 2 , u 4 , u 5 , u 7 ) + y 12 and H 14 := N 3 (u 2 , u 4 , u 5 , u 7 ) + y 14 ( [9], [10],[1] Theorem 7.1). Moreover, the system (I) is a Hamiltonian system with the Hamiltonian H 12 and the Poisson structure determined by The system (II) is a Hamiltonian system with the Hamiltonian H 14 and the Poisson structure determined by {u 2 , ). These Hamiltonians are in involution with respect to the Poisson structures and the systems are Liouville integrable ( [10]).

Meromorphic functions on the sigma divisor
In [1], for g = 3, the field of meromorphic functions on the sigma divisor of the curve V 3 is described. In this section we recall these results.
We assume g = 3. Fix any constant vector (y 4 , y 6 , y 8 , y 10 , y 12 , y 14 ) ∈ B 3 . Let F be the field of all meromorphic functions on C 3 and let F [(σ)] be the set of meromorphic functions f ∈ F satisfying the following two conditions: • for any point w ∈ W , there exist an open neighborhood U 1 ⊂ C 3 of this point and two holomorphic functions g and h on U 1 such that the function h does not identically vanish on U 1 ∩ W and f = g/h on U 1 ; • f (w + Ω) = f (w) for any w ∈ W and Ω ∈ Λ 3 .
Note that F [(σ)] is a subring in F , but it is not generally a field. Let us consider the Abel-Jacobi map The Abel-Jacobi map I 3 induces a ring holomorphism Let J * be the set of meromorphic functions f ∈ F [(σ)] identically vanishing on W . Thus, we have Ker I * 3 = J * . We set F ((σ)) = F [(σ)]/J * . Then F ((σ)) is a field and, by construction, there is an isomorphism of fields (see [1] Section 4) The following meromorphic functions on C 3 are introduced in [1]: We have F i ∈ F [(σ)] and I * 3 (F i ) = u i for i = 2, 4, 5, 7 (see [1] Proposition 4.1). In [1], the following derivations acting on the field F ((σ)) are introduced: 3 and L hold.

Two parametric deformation of KdV-hierarchy
We assume g = 3 and consider the following derivations:

.
From [1] Lemma 6.2, the commutation relation [T 1 , T 3 ] = 0 holds in the Lie algebra of derivations of F . Since the operators L 3 and L 5 are the derivations of the field F ((σ)) and f 2 , f −1 4 ∈ F [(σ)], the operators T 1 and T 3 are also the derivations of the field F ((σ)). We consider the following derivations acting on the field F (V 2 g ) 5 ] = 0, the direct calculation shows the proposition. Proposition 5.2. We have , Proof. The direct calculation shows the proposition.
Let u = 4u 2 and v = 2(u 4 − u 2 2 ). For any w ∈ F (Sym 2 (V 3 )), we use the notation w ′ = T 1 w andẇ = T 3 w. Then we obtain the main result of this paper.
Theorem 5.1. We obtain the following new system that can be called two parametric deformed KdV-hierarchy: Proof. From Proposition 5.3, the direct calculation shows where in the above calculation we deleted the terms u 5 u 7 and u 2 7 by using the relations M 3 (u 2 , u 4 , u 5 , u 7 ) = N 3 (u 2 , u 4 , u 5 , u 7 ) = 0 in F (Sym 2 (V 3 )) (see Section 3). By differentiating the both sides of (11) with respect to T 1 , we obtain Therefore we obtain the equation (7). By differentiating the both sides of (11) with respect to T 3 , we obtaiṅ Therefore we obtain the equation (8). From Proposition 5.3, we obtain the equations (9) and (10).
In Section 3, we noted F (Sym 2 (C 2 )) = C (a, b, c, d). The map Sym(ψ) transforms the generators a, b, c, and d as follows.
Proof. By the direct calculation we can check T 1 •ψ 1 (X i ) = ψ 1 •L (2) 1 (X i ) and T 1 •ψ 1 (Y i ) = ψ 1 • L (2) 1 (Y i ) for i = 1, 2. Therefore we obtain T 1 • ψ 1 = ψ 1 • L (2) 1 . Similary, we obtain We assume (y 4 , y 6 , y 8 , y 10 ) ∈ B 2 . Let us consider the Abel-Jacobi map of the curve V 2 Let F (Jac(V 2 )) be the field of meromorphic functions on the Jacobian Jac(V 2 ). The Abel-Jacobi map I 2 induces the isomorphism of the fields: As derivations of F (V 2 2 ), the derivations L 1 and L 3 can be expressed as ([1] Section 6) 2 we regard X i and Y i as meromorphic functions on V 2 2 , and dX i and dY i are the total differentials of X i and Y i for i = 1, 2. Let us describe the action of these operators in more detail. For g(P 1 , P 2 ) ∈ F (V 2 2 ), d P i (g) is the total differential of g as a meromorphic function of P i . Then d P i (g)/dX i is the meromorphic function on V 2 2 determined uniquely by d P i (g) = (d P i (g)/dX i ) · dX i . We consider the following derivations of F (Jac(V 2 )) Lemma 6.1. We have L 1 and L 3 . Proof. Set h ∈ F (Jac(V 2 )) and w = I 2 ((P 1 , P 2 )). We have  3 are proved similarly.
2 In [1], these expressions are given for g = 3 and we can prove them for any g similarly.
Theorem 7.1. The functions U and V satisfy the two parametric deformed KdVhierarchy Proof. From Lemma 5.1, 7.1, and Theorem 5.1, we obtain the theorem.