Differential Equations of Genus Four Hyperelliptic $\wp$ Functions

In order to find higher dimensional integrable models, we study differential equations of hyperelliptic $\wp$ functions up to genus four. For genus two, differential equations of hyperelliptic $\wp$ functions can be written in the Hirota form. If the genus is more than one, we have KdV equation. If the genus is more than two, we have KdV and another KdV equations. If the genus becomes more than three, there appear differential equations which cannot be written in the Hirota form, which means that the Hirota form is not enough to characterize the integrable differential equations. We have shown that some differential equations are satisfied for general genus. We can obtain differential equations for general genus step by step.

In our previous papers, by using the Lie group structure as our guiding principle, we have revealed that the two dimensional integrable models such as KdV/mKdV/sinh-Gordon are the consequence of the SO(2,1)-SL(2,R) Lie group structure [21][22][23][24][25] 1 Here we would like to to study higher-dimensional integrable models. KdV/mKdV/sinh-Gordon equations and KP equations are typically understood as two-and three-dimensional integrable models, respectively. First, we would like to know whether there exists a universality of the integrable models, that is, whether any two-and three-dimensional integrable models always contain KdV/mKdV/sinh-Gordon equations and KP equations, respectively.
For higher-dimensional integrable models, there is a soliton type approach of Kyoto school [16][17][18][19] where they use the special fermion, which generates N -soliton solutions. Starting with the fermionic bilinear identity of glp8, Rq, they have obtained KP hierarchy and finite higherdimensional Hirota forms by the reduction of KP hierarchy. Another systematic approach to high-dimensional integrable models is to find differential equations for higher genus hyperelliptic functions by using the analogy of differential equation of Weierstrass ℘ function. By solving the Jacobi's inversion problem, the integrability of hyperelliptic functions are automatically guaranteed, since the integrability condition and the single-valuedness are equivalent for hyperelliptic functions. So far, only for genus one, two [28] and three [29][30][31] cases are studied because it becomes difficult to solve the Jacobi's inversion problem and obtain differential equations for higher genus cases.
In 1981, Dubrobin found the hyperelliptic solution of the KdV or the KP equation for genus two and three cases [32]. While, we want to find what kinds of and how many Weierstrass type differential equations such as d 2 ℘pxq{dx 2 " 6℘ 2 pxq´g 2 {2 for various genus are there, which give us the clue to understand the Lie group structure of hyperelliptic ℘ functions. Matsutani pointed out in 2001 [33], that Baker had already found hyperelliptic solution for KdV and KP equation for genus three case in 1897 [29].
In this paper, we study to obtain differential equations of genus four case. In the approach, we would like to examine the connections between i) higher-dimensional integrable differential equations, ii) higher-rank Lie group structure and iii) higher genus hyperelliptic functions.

Formulation of Differential Equations in General
Genus and the Review of Genus Two and Three Cases

Formulation of differential equations in general genus
We summarize the formulation of hyperelliptic ℘ function according to Baker's work [26][27][28][29]. We consider the genus g hyperelliptic curve The Jacobi's inversion problem consists of solving the following system From these equations, we have by using the relation We define F pxq " g ź i"1 px´x i q and denote F 1 px i q as F 1 px i q " dF pxq dxˇˇˇx"x i . For example, F 1 px 1 q " px 1´x2 qpx 1´x3 q¨¨¨px 1´xg q . For χ g´j px i ; x 1 , x 2 ,¨¨¨, x g q, we first define the following generalized function where h j px 1 ,¨¨¨, x p q is the j-th fundamental symmetric polynomial basis of tx 1 ,¨¨¨, x p u, i.e.
Putting p " g and x " x k in χ g´j px; x 1 , x 2 ,¨¨¨, x p q, we have χ g´j px i ; x 1 , x 2 ,¨¨¨, x g q in the following form For example i`h 2 px 1 , x 2 ,¨¨¨, x g qx g´2 i`¨¨¨`p´1 q g h g px 1 , x 2 ,¨¨¨, x g q " 0. (2.8) The ζ j functions are given from the hyperelliptic curve in the following way [26] dp´ζ j q " where q x j denotes that the x j variable is missing. The overall factor of ζ j is different from the textbook of Baker [26] and papers of Buchstaber et al. [30,31]. We changed it to make it easier to see the dual symmetry of differential equations in §4.1. In this expression, we can show dp´ζ 0 q " 0 in the following way where we use χ g´1 px i ; x 1 , x 2 ,¨¨¨, q x i ,¨¨¨, x g q " F 1 px i q. In addition, an identity dp´ζ g`1 q " 0 can be derived easily as follows dp´ζ g`1 q " (2.11) because there is no such sum of k and we have χ´2 " 0. These ζ j pu 1 , u 2 ,¨¨¨, u g q satisfy the integrability condition B p´ζ j pu 1 , u 2 ,¨¨¨, u g qq Bu k " B p´ζ k pu 1 , u 2 ,¨¨¨, u g qq Bu j . (2.12) In the Baker's textbook [26], the expression of the second term of the r.h.s of Eq.(2.9) is misleading. 2 ℘ jk pu 1 , u 2 ,¨¨¨, u g q functions are given from the above ζ j pu 1 , u 2 ,¨¨¨, u g q functions in the form ℘ jk pu 1 , u 2 ,¨¨¨, u g q " ℘ kj pu 1 , u 2 ,¨¨¨, u g q " B p´ζ j pu 1 , u 2 ,¨¨¨, u g qq Bu k .

Differential equations of genus two hyperelliptic ℘ functions
We here review the genus two hyperelliptic ℘ function. The hyperelliptic curve in this case is given by The Jacobi's inversion problem consists of solving the following system Then we have In this case, For these ζ 1 and ζ 2 , we have checked the integrability condition Bζ 1 {Bu 2 " Bζ 2 {Bu 1 . We use the useful functions p ℘ 22 , p ℘ 21 and p ℘ 11 of the form In order that the differential equation becomes the polynomial type of p ℘ 22 and p ℘ 21 but not infinite series of these, we must put λ 6 " 0 4 Then we have By using the analogy of the differential equation of Weierstrass ℘ function in the form d 2 ℘pxq{dx 2 " 6℘ 2 pxq´g 2 {2, with the help of REDUCE, we have the following differential equations [28] 1q In addition to λ 6 " 0, which is necessary to obtain differential equations of polynomial type, we can always put If we differentiate Eq.(2.35) with u 2 , and identify ℘ 22 pu 1 , u 2 q Ñ upx, tq, du 2 Ñ dx and We can eliminate λ 4 u x by the constant shift of u Ñ u´λ 4 {3, which gives the KdV equation In the standard form of λ 0 " 0, as the result of the dual symmetry, by identifying ℘ 11 pu 1 , u 2 q Ñ upx, tq, du 1 Ñ dx and du 2 Ñ dt, we have another KdV equation We must notice that upx, tq " ℘ xx px, tq " B 2 x p´log σpx, tqq, expressed with the genus two hyperelliptic σ function, is the solution but not the wave type solution, because x and t comes in the combination X " x´vt pv : const.q in the wave type solution.
In this way, we have the KdV equation and another KdV equation. As the Lie group structure of genus two hyperelliptic differential equations, we have sub structure of SO(2,1) and another SO(2,1) because each KdV equations have the SO(2,1) Lie group structure,

Differential equations of genus three hyperelliptic ℘ functions
We now move to the genus three case. The hyperelliptic curve in this case is given by The Jacobi's inversion problem consists of solving the following system Then we have (2.44) and tx 1 , x 2 , x 3 u, ty 1 , y 2 , y 3 u cyclic permutation. In this case, (2.47) For these ζ 3 , ζ 2 and ζ 1 , we have checked the integrability condition Bζ i {Bu j " Bζ j {Bu i , p1 ď i ă j ď 3q. Just as the same as the genus two case, in order that differential equations become of the polynomial type, we must put λ 8 " 0. In this case, we have Then we have the following differential equations [29][30][31] 1q While if we take λ 1 " 0 as the standard form, by identifying ℘ 11 Ñ u, du 1 Ñ dx, du 2 Ñ dy and du 3 Ñ dt, by differentiating Eq.(2.66) with u 1 twice, we have KP equatioǹ u xxx´3 uu x´λ2 u x´4 λ 0 u t˘x "´3λ 0 u yy . By differentiating Eq.(2.62) with u 2 twice, we have the following three variables differential equation`u xxx´3 uu x´λ4 u x´λ5 u t˘x " 3∆ xx´3 λ 6 u tt`λ3 u xy´3 λ 2 u yy , (2.68) by identifying ℘ 22 Ñ u, du 1 Ñ dt, du 2 Ñ dx and du 3 Ñ dy. If we consider the special hyperelliptic curve with λ 6 " 0 and λ 3 " 0, Eq.(2.67) becomes the KP equation except ∆ xx term in the form`u and we have checked that ∆ xx ‰ 0 even for this special hyperelliptic curve. Then we have three variables new type integrable differential equation, which is KP type but is different from KP equation itself.
In the standard form of λ 0 " 0, the differential equation of Eq. By differentiating Eq.(3.33) with u 3 twice, we have four variables differential equation, which is KP type equation except the term p∆ 2 q xx p‰ 0q in the form u xxx´3 uu x´λ7 u t´λ6 u x˘x " 3p∆ 2 q xx´3 λ 9 u zt`4 λ 8 u zx´3 λ 8 u tt`λ5 u xy´3 λ 4 u yy , (3.48) by identifying ℘ 33 Ñ u, du 1 Ñ dz, du 2 Ñ dt, du 3 Ñ dx and du 4 Ñ dy. Then we have four variables KP type new integrable differential equation. Eq.(3.43) is four variables another KP type differential equation.

Some dual symmetry for the set of differential equations
In the previous sections, we have explained the symmetry of differential equations, that is, in the standard form of λ 2g`2 " 0 and λ 0 " 0 in the hyperelliptic curve, the set of differential equations have some dual symmetry under ℘ jk Ø ℘ g`1´j,g`1´k , ℘ jklm Ø ℘ g`1´j,g`1´k,g`1´l,g`1´m , λ k Øλ k " λ 2g`2´k . (4.1) The standard form of the hyperelliptic curve is given by If we change variables in the formx i " ,λ k " λ 2g`2´k , we can rewrite the curve in the formC Then we have that is, dũ g "´du 1 , dũ g´1 "´du 2 ,¨¨¨, dũ 2 "´du g´1 , and dũ 1 "´du g . From the curve Eq.(4.3), we construct hyperelliptic sigma functionσ. While we construct σ from the curve Eq.(4.2). But the difference between Eq.(4.2) and Eq.(4.3) is only the choice of the local variable, so that σ function andσ function is essentially the same, then we have Bp´logσq Bũ j " Bp´log σq Bũ j " p´ζjq. Then du j Ø´dũ j is equivalent to ℘ jk Ø p´1q 2 ℘jk " ℘jk and ℘ jklm Ø p´1q 4 ℘jklm " ℘jklm. Therefore, we conclude that the set of differential equations have some dual symmetry under (4.1).

Some differential equations for general genus
Buchstaber et al. have shown the quite interesting result that one family of differential equations always exist for general genus [30,31], and it is really the family of KdV equation in 1997 [30]. We sketch the proof of their result. We start from dp´ζ g´1 q " where we have used x i " p ℘ gg and we denote p ℘ ggg " ℘ ggg {λ 2g`1 . Then we have d p2 p ℘ ggg`p´ζg´1 qq " where we have used du j " . In the right-hand side of Eq.(4.6), by using Eq.(2.19), we reduce the power of x i in the range x j´1 i , pj " 1, 2,¨¨¨, gq and comparing the coefficients of left-and right-hand side of Eq.(4.6), we have following differential equations for general genus where we write differential equations with ℘ jk and ℘ jklm instead of p ℘ jk , and p ℘ jklm to compare with our result. Then in the standard form of λ 0 " 0, another KdV equation `λ 2 ℘ 1,g`1´j`1 2 λ 1 λ 3 δ g`1´j,1 , p1 ď j ď gq, (4.8) is satisfied for general genus.
We can obtain other differential equations for general genus recursively. For example, we start from dp´ζ g´2 q " (4.9) If we notice the relation i¯.
(4.11) By using Eq.(2.19), we reduce the power of x i in the range x j´1 i , pj " 1, 2,¨¨¨, gq and comparing coefficients of left-and right-hand side of Eq.(4.11), we have the following differential equations for general genus This is another type of differential equations than Eqs. Another example of differential equation can be derived starting from the following equation, (4.13) If we notice relations

Hirota form differential equations
For genus two case, all differential equations Eqs.(2.35)-(2.39) are written in the Hirota form, that is, bilinear differential equation with Hirota derivative. For genus three and four cases, though the left hand side can be written in the Hirota form, but differential equations which contain ∆ are not written in the Hirota form. As it is quite natural, Baker already has used the Hirota derivative for the derivative of p´log σq, that is, ℘ jk and ℘ jklm [28]. We use following relations where D x , D y , D z and D t are Hirota derivatives. Just as the Weierstrass ℘ function solution in the KdV equation, we identify the τ function in such a way as p´log τ q is proportional to p´log σq [24]. Then we put ℘ jk " p´log σq jk " αp´log τ q jk with constant α. We show that I " τ 2´℘ xyzt´1 2 p℘ xy ℘ zt`℘xz ℘ yt`℘xt ℘ yz q¯can be written in the Hirota form in the following way plog τ q xyzt`α 2´p log τ q xy plog τ q zt`p log τ q xz plog τ q yt`p log τ q xt plog τ q yz¯ı "´α 2 (4. 19) where in the last step we choose α " 4. For more general form, we have  [24]. Equations which contain ∆ and ∆ i cannot be written as the Hirota bilinear differential form.

Summary and Discussions
In order to find higher dimensional integrable models, we have explicitly studied how to obtain differential equations of genus four hyperelliptic ℘ function.
In the standard form of λ 0 " 0, we have KdV and another KdV equations for genus being more than two. In the standard form of λ 1 " 0, if genus is three, we have KP equation. The universality of integrable model is guaranteed up to three dimensional integrable models. As the two-and three-dimensional integrable models, KdV equation and KP equation come out, respectively.
If genus is two, all differential equations are written in the Hirota form. However, we obtain differential equations which cannot be written in the Hirota form, if genus is more than three. This means that the Hirota form or the fermionic bilinear form is not sufficient to characterize higher dimensional integrable models.
From the series of investigations of genus two, three, and four, differential equations for general genus will not be so complicated, but only the quadratic term ∆ j of ℘ jk becomes complicated.
We have also shown, in the standard form of λ 0 " 0, some duality for the set of differential equations, which gives that KdV and another KdV equations always exist for genus being more than two. In the standard form of λ 1 " 0, there also exist duality for the KP equation for genus three and four. We expect that the same expression Eq.(2.66) and/or Eq.(3.47) will be satisfied for the general genus.
Since we have KdV, another KdV equation, and pg´2q pieces of KP type differtial equations in the standard form of λ 0 " 0, where KP type equations are similar to the KdV equation, we expect that genus g hyperelliptic ℘ functions have rank g Lie group structure. Actually, hyperelliptic ℘ functions can be defined through hyperelliptic ϑ functions instead of hyperelliptic curves. For the genus two case, hyperelliptic theta functions are written in the form ϑ " a c b d  pu 1 , u 2 ; τ 11 , τ 22 , τ 12 q, where a, b, c, d " 0 or 1{2. If we take the special limit of τ 12 Ñ 0, they reduce to the product of genus one theta functions in the form ϑ " a c b d  pu 1 , u 2 ; τ 11 , τ 22 , 0q " ϑ " a b  pu 1 ; τ 11 qϑ " c d  pu 2 ; τ 22 q. In this special limit, we have two independent one dimensional KdV equations (static KdV equations), which correspond to Eq.(2.35) and Eq.(2.39). As we know that there is an SO(2,1) structure in each one dimensional KdV equation [22], we have SO(2,1) b SO(2,1) structure in the genus two hyperelliptic ℘ functions. For the general genus cases, the situation is the same and we expect that there is rank g Lie grroup structure for the genus g hyperelliptic ℘ functions.
In some special cases, we have given some differential equations for general genus by using our method step by step. The differential equations of hyperelliptic ℘ functions are integrable, then we expect that Lax pairs for these differential equations exist. Parshin have succeeded in generalizing the KP hierarchy with the Lax pair [35], but it is the genus 0 case, that is, the solution is expressed by polynomial, exponential and trigonometric functions. The Lax pair of genus two will contain three potentials, ℘ 22 pu 1 , u 2 q, ℘ 21 pu 1 , u 2 q, and ℘ 11 pu 1 , u 2 q, will give the set of 5 equations but not the single KP equation, so that it seems difficult to construct the Lax pair in this case.