Trigonal Toda Lattice Equation
- 10.1080/14029251.2020.1819622How to use a DOI?
- Toda lattice equation; directed 6-regular graph; Eisenstein integers
In this article, we give the trigonal Toda lattice equation,for a lattice point ℓ ∈ [ζ3] as a directed 6-regular graph where , and its elliptic solution for the curve y(y − s) = x 3, (s ≠ 0).
- © 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/).
The elliptic functions have high symmetries and generate many interesting relations. In the celebrated paper  , Toda derived the Toda lattice equation based on the addition formula of the elliptic functions. Using the addition formulae of hyperelliptic curves  , the hyperelliptic quasi-periodic solutions of the Toda lattice equation are also obtained as in [7, 9]. The derivation in [7, 9] can be regarded as a natural generalization of Toda’s original one. The addition formulae for the Toda lattice equation are essential.
Recently Eilbeck, Matsutani and Ônishi introduced a new addition formula for the Weierstrass 𝒫 functions on an elliptic curve E, y(y − s) = x3 , which is called the equiharmonic elliptic curve  . The curve E has the automorphism, the cyclic group action of order three as a Galois action  , i.e., , where .
In this article, we use the new addition formula on E in  and derive a non-linear differential and difference equation following the derivation in [7, 9, 11] as shown in Proposition 3.1. Thus we call it the trigonal Toda lattice equation. The trigonal Toda lattice equation consists of the third order differential and the trigonal difference operators, which reflects the cyclic symmetry of the curve. The difference operator agrees with the graph Laplacian of a directed 6-regular graph associated with Eisenstein integers [ζ3]. The trigonal Toda lattice equation is defined over the infinite directed 6-regular graph c.f. Proposition 3.2. It means that we provide the trigonal Toda lattice equation and its elliptic function solution as a special solution. Since the lattice given by the infinite directed 6-regular graph appear in models in statistical mechanics, e.g.,  , the new Toda lattice equation might show a nonlinear excitation of such models.
The contents are as follows. In Section 2, we show the properties of the elliptic curve E. We derive a new differential-difference equation, or the trigonal Toda lattice equation, and its elliptic function solution as an identity in the meromorphic functions on E in Section 3. We give some discussions in Section 4.
2. Properties of the equiharmonic elliptic curve
Let us consider an elliptic curve E given by the affine equation
It is known that since the image of the incomplete elliptic integral agrees with the complex plane ℂ, the 𝒫-function (and thus x and y) is expressed by the Weierstrass sigma function,It means that x(u) and y(u) are considered as meromorphic functions on ℂ. The trigonal cyclic symmetry induces the action on u and sigma function, i.e., for u ∈ ℂ, Eilbeck, Matsutani and Ônishi showed an addition formula of the elliptic sigma function of the curve E  ,
In this article, we consider the curve E and this formula (2.2).
Let the elliptic integral from the infinity point ∞ to (x, y) = (0, s) denoted by ωs, and similarly that to (0, 0) by ω0,
The complete elliptic integrals of the first and the second kinds are given by
Following Weierstrass’ conventionthey satisfy the relations [5, 10]
Further for the branch points (0, 0) and (s, 0), the following relations hold:
See [5, Appendix C].
The image of the incomplete elliptic integrals is acted by SL(2, ) and the cyclic group ζ3. For u, v ∈ ℂ (v ≠ 0), we define a lattice
Noting ζ6 = ζ3 + 1 for , [ζ3] = [ζ6]. Since 𝒵2ω′,0 agrees with the lattice of the periodicity, i.e., x(u + L) = x(u), y(u + L) = y(u) for L ∈ 𝒵2ω′,0, the Jacobian 𝒥E of the curve E is given by
These points of the integrals for the branch points of the curve E are illustrated in Figure 1. We also regard x and y as meromorphic functions on 𝒥E due to their periodicity.
Further Lemma 2.1 shows that ωs and ω0 belong to .
3. The trigonal Toda lattice equation
The addition formula (2.2) gives the following lemma:
The quantitysatisfies the relation
We consider the logarithm of both sides of (2.2) and differentiate both side three times with respect to u. Then we obtainThe right hand side in Lemma 3.1 should be expressed by a difference operator. In order to express it, we prepare the geometry associated with Lemma 3.1.
We fix the complex numbers u0 and v0. We regard 𝒵v0,u0 as the set of nodes 𝒩v0,u0 of an infinite directed (oriented) 6-regular graph 𝒢v0,u0 whose incoming degree and outgoing degree at each node are three, 𝒩v0,u0 = 𝒵v0,u0  ; 𝒢v0,u0 is illustrated in Figure 2. Every node nℓ in 𝒩v0,u0 is labeled by an Eisenstein integer ℓ ∈ [ζ3].
It is noted that every v0 ∈ ℂ is decomposed to v0 = v′0ω′ + v″0ω″ using v′0, v″0 ∈ ℝ. Further the quotient set of the lattice points modulo 𝒵2ω′,0 is denoted by 𝒩v0,u0/𝒵2ω′,0 = κ(𝒩v0,u0). The following are obvious:
For v0 = v′0ω′ + v″0ω″ of v′0, v″0 ∈ ∩ [0, 2], the cardinality |𝒩v0,u0/𝒵2ω′,0| is finite for every u0 ∈ ℂ, and
for v0 = v′0ω′ +v″0ω″ of v′0, v″0 ∈ (ℝ\)∩[0, 2], 𝒩v0,u0/𝒵2ω′,0 is dense in 𝒥E for every u0 ∈ ℂ.
Let us introduce the function spaces, Ω and log Ω,For an Eisenstein integer ℓ ∈ [ζ3] or nℓ ∈ 𝒩v0,u0, t ∈ ℂ and fixed u0, v0 ∈ ℂ, let us consider an element in log Ω,
Lemma 3.1 gives the nonlinear relation on log Ω:
For nℓ ∈ 𝒩v0,u0, t ∈ ℂ and fixed u0, v0 ∈ ℂ, qℓ(t) = qℓ(t;u0, v0) satisfies the relation,
It is emphasized that (3.2) can be regarded as a differential-difference non-linear equation and its special solution is given by (3.1) for the elliptic curve (2.1). Its derivation is basically the same as Toda’s original derivation of Toda lattice equation  and that in [7, 9]. Further it is related to the infinite graph 𝒢v0,u0. Thus we will call this relation the trigonal Toda lattice equation.
We recall ζ6 = ζ3 + 1. For a given nℓ ∈ 𝒩v0,u0, let us consider subgraph 𝒢ℓ ⊂ 𝒢v0,u0 given by its nodes ; 𝒩ℓ consists of the center point nℓ with a hexagon (i = 0, 1,..., 5). The submatrix of the incoming adjacency matrix 𝒜in for 𝒢ℓ is given byThe incoming degree matrix is given by the diagonal matrix 𝒟in whose diagonal element is three. Thus we define the incoming Laplacian  , Let us consider the functions q ∈ log Ω and eq ∈ Ω whose components at nℓ are given by qℓ(t) and eqℓ(t). We regard them as column vectors for each ℓ ∈ [ζ3]. Then the Laplacian acts on the vector spaces.
Using the incoming Laplacian, Proposition 3.1 is reduced to the following formula.
Using the above notations, (3.2) is written byIt turns out that the trigonal Toda lattice equation consists of the third order differential operators and trigonal graph Laplacian, which is a natural generalization of the original Toda lattice equation  , though it has not ever obtained as far as we know.
As we obtain the equation, we will consider its solutions (3.1), especially their initial condition u0 and the configurations 𝒢v0,u0:
The domain of the solution eqℓ of (3.1) is the Jacobian 𝒥E; for L ∈ 𝒵2ω′,0, eqℓ(t + L) = eqℓ(t). From Lemma 2.1, the periods 2ω′ and 2ω″ are scaled by s−1/3. Further in the projection π : E → , (π((x, y)) = y), which determines the three special points (0, s, ∞) in , the range of the solution eqℓ as a meromorphic function on E is also parameterized by s via y and the governing equation (2.1). It is easy to find the s-dependence of eqℓ and thus we may fix s as a finite real number.
For v0(≠ 0) such that v0 ≠ ω0, q(u, v0) as a function with respect to u diverges only at the points in 𝒵2ω′,0 and . Their union is denoted by 𝒮v0. It means that for an ℓ ∈ [ζ3], if the orbit of qℓ(t;u0, v0) in t avoids 𝒮v0, the value of qℓ(t;u0, v0) is finite.
Let us consider its orbit whose value is finite value. We restrict its domain ℂ × 𝒵v0,u0 to its real subspace for a certain unit direction (i.e., ),
Let us assume that K := |𝒩v0,u0/𝒵2ω′,0| is finite. For a certain direction in the complex plane and u0 ≠ 0, we find the subspace in ℂ such that every does not diverge for each ℓ ∈ [ζ3] and t ∈ ℝ, and satisfies the trigonal Toda lattice equation,
The conditions on and u0 correspond to the conditions on the embedding ι of ℝK into 𝒥E such that the image of ι is compact and disjoint from 𝒮v0/𝒵2ω′,0. Under these conditions, we have the complex valued finite solutions of the trigonal Toda lattice equation (3.3).
An elliptic function solution of this equation is illustrated in Figure 3 for v0 = (1 + ζ6)ω′/13, u0 = v0/2, and s = 1.0.
Let us consider the continuum limit of the the trigonal Toda lattice equation as follows:
It is noted that q(u, v0) diverges for the limit v0 → 0 and thus for this elliptic function solution qℓ(t), we cannot obtain the continuum limit of the graph Laplacian Δin and of the trigonal Toda lattice equation.
We derived the trigonal Toda lattice equation in Propositions 3.1 and 3.2 based on the addition formula (2.2) for the curve E associated with the automorphism of the curve. It is associated with the lattice, or the directed 6-regular graph, given by the Eisenstein integers [ζ3]. It means that we have an nonlinear equation on the lattice and its elliptic function solution. Since there are physical models based on the triangle lattice given by the infinite 6-regular graph  , this trigonal Toda lattice equation might describe a nonlinear excitation in the models.
The third order differential equation reminds us of the Chazy equation, which is a third order ordinary differential equation and posses Painlevé property  . However the trigonal Toda lattice equation cannot have a non-trivial continuum limit because E becomes the three rational curves for the limit s → 0  and qℓ(t;u0, v0) diverges for the limit v0 → 0 as in Remark 3.2. In other words, we could not directly argue the integrablity of the trigonal Toda lattice equation using the Chazy equation, even though both elliptic function solutions are closely related. It means, in this stage, that it is not obvious whether the trigonal Toda lattice equation is an integrable equation as a time-development equation, and thus it is an open problem to determine the behavior of its solution for every initial state as an initial value problem.
However the addition theorem in [3, (A.3)] for the genus three curve can be regarded as a generalization of the addition formula (2.2) for the cyclic action on curves. Thus it is expected that the trigonal Toda lattice equation might have algebro-geometric solutions of algebraic curves of higher genus. Further this approach could be generalized to more general curves, e.g., the genus three curve  and more general curves with a trigonal cyclic group  .
I would like to thank Yuji Kodama for helpful comments and pointing out the Chazy equation, and Yoshihiro Ônishi for valuable discussions. Further I am grateful to the two anonymous referees for their helpful comments and suggestions.
Cite this article
TY - JOUR AU - Shigeki Matsutani PY - 2020 DA - 2020/09/04 TI - Trigonal Toda Lattice Equation JO - Journal of Nonlinear Mathematical Physics SP - 697 EP - 704 VL - 27 IS - 4 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1819622 DO - 10.1080/14029251.2020.1819622 ID - Matsutani2020 ER -