# Journal of Nonlinear Mathematical Physics

Volume 27, Issue 4, September 2020, Pages 697 - 704

# Trigonal Toda Lattice Equation

Authors
Shigeki Matsutani
Electrical Engineering and Computer Science, Graduate School of Natural Science & Technology, Kanazawa University, Kakuma Kanazawa, Ishikawa 920-1192, Japan,s-matsutani@se.kanazawa-u.ac.jp
Received 16 December 2019, Accepted 8 March 2020, Available Online 4 September 2020.
DOI
10.1080/14029251.2020.1819622How to use a DOI?
Keywords
Toda lattice equation; directed 6-regular graph; Eisenstein integers
Abstract

12d3dt3q(t)=eq+1(t)+eq+ζ3(t)++eq+ζ32(t)3eq(t),
for a lattice point ∈ 𝕑[ζ3] as a directed 6-regular graph where ζ3=e2π1/3, and its elliptic solution for the curve y(ys) = x 3, (s ≠ 0).

Open Access

## 1. Introduction

The elliptic functions have high symmetries and generate many interesting relations. In the celebrated paper [11] , Toda derived the Toda lattice equation based on the addition formula of the elliptic functions. Using the addition formulae of hyperelliptic curves [3] , 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(ys) = x3 , which is called the equiharmonic elliptic curve [10] . The curve E has the automorphism, the cyclic group action of order three as a Galois action [4] , i.e., ζ^3(x,y)=(ζ3x,y), where ζ3=e2π1/3.

In this article, we use the new addition formula on E in [4] 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., [1] , 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

y(ys)=x3,(2.1)
which is called the equiharmonic elliptic curve [4, 10]. E has the automorphism associated with the cyclic group of order three as the Galois action on E; ζ^3:EE, ζ^3((x,y))=(ζ3x,y) where ζ3=e2π1/3; the action ζ^3 on E is invariant. We call it the trigonal cyclic symmetry. The affine equation is expressed by
(ys2)2=(x+s243)(x+ζ3s243)(x+ζ32s243).
By letting ej=ζ31js243, it corresponds to the Weierstrass standard form
()2=43g3=4(e1)(e2)(e3),
where g3 = −4s2 using the Weierstrass 𝒫-function,
(u)=x(u),(u)=2y(u)s,y(u)=12((u)+s),
for the elliptic integral
u=(x,y)du,du=dx2ys.

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,

(u)=d2du2logσ(u),(y(u)=12(d3du3logσ(u)+s)).
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,
σ(ζ3u)=ζ3σ(u),(ζ3u)=ζ3(u),(ζ3u)=(u).
Eilbeck, Matsutani and Ônishi showed an addition formula of the elliptic sigma function of the curve E [4] ,
σ(uv)σ(uζ3v)σ(uζ32v)σ(u)3σ(v)3=(y(u)y(v)).(2.2)

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,

ωs=(0,s)du,ω0=(0,0)du,du=dx2ys.

The complete elliptic integrals of the first and the second kinds are given by

ωi=eidu,ηi=eixdu,(i=1,2,3).

Following Weierstrass’ convention

ω=ω1,ω=ω3,η=η1,η=η3,
they satisfy the relations [5, 10]
ω=ζ3ω,η=ζ32ηω1+ω2+ω3=0,η1+η2+η3=0,ηωηω=π12.

Further for the branch points (0, 0) and (s, 0), the following relations hold:

### Lemma 2.1.

x(ω0)=x(ωs)=0,y(ω0)=0,y(ωs)=s,ωs=1ζ3232ω,ω0=ωs,ω=321ζ3ζ32Γ(13)2s1/3Γ(23),η=π133s1/3Γ(23)Γ(13)2.

### Proof.

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

𝒵v,u:=𝕑[ζ3]v+u:={1v+2ζ3v+u|1,2𝕑}.(2.3)

Noting ζ6 = ζ3 + 1 for ζ6:=e2π1/6, 𝕑[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 L𝒵2ω′,0, the Jacobian 𝒥E of the curve E is given by

𝒥E=/𝒵2ω=0,κ:𝒥E.

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 13𝒵2ω,0.

## 3. The trigonal Toda lattice equation

The addition formula (2.2) gives the following lemma:

### Lemma 3.1.

The quantity

q(u,v):=log(y(u)y(v)),(y(u)y(v)=eq(u,v))
satisfies the relation
12d3du3q(u,v)=eq(uv,v)+eq(uζ3v,v)+eq(uζ32v,v)3eq(u,v).

### Proof.

We consider the logarithm of both sides of (2.2) and differentiate both side three times with respect to u. Then we obtain

d3du3log(y(u)y(v))=2(y(uv)+y(uζ3v)+y(uζ32v))6(y(u)).
The 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 [6] ; 𝒢v0,u0 is illustrated in Figure 2. Every node n in 𝒩v0,u0 is labeled by an Eisenstein integer ∈ 𝕑[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:

### Lemma 3.2.

1. (1)

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

2. (2)

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,

Ω:={Q:×𝒩v0,u0𝕇|meromorphic},logΩ:={q:×𝒩v0,u0𝕇|eqΩ}.
For an Eisenstein integer ∈ 𝕑[ζ3] or n𝒩v0,u0, t ∈ ℂ and fixed u0, v0 ∈ ℂ, let us consider an element in log Ω,
q(t;u0,v0):=q(t+u0v0,v0)=log(y(t+u0v0)y(v0)),(3.1)
which is denoted by q(t) for brevity.

Lemma 3.1 gives the nonlinear relation on log Ω:

### Proposition 3.1.

For n𝒩v0,u0, t ∈ ℂ and fixed u0, v0 ∈ ℂ, q(t) = q(t;u0, v0) satisfies the relation,

12d3dt3q(t)=eq+1(t)+eq+ζ3(t)+eq+ζ32(t)3eq(t).(3.2)

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 [11] 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 n𝒩v0,u0, let us consider subgraph 𝒢𝒢v0,u0 given by its nodes 𝒩:={n,n+1,n+ζ6,n+ζ62,,n+ζ65}(𝒩v0,u0); 𝒩 consists of the center point n with a hexagon n+ζ6i (i = 0, 1,..., 5). The submatrix of the incoming adjacency matrix 𝒜in for 𝒢 is given by

𝒜in|𝒢=(0101010001000110000000010100100000000001011000000).
The incoming degree matrix is given by the diagonal matrix 𝒟in whose diagonal element is three. Thus we define the incoming Laplacian [6] ,
Δin:=𝒟in𝒜in.
Let us consider the functions q ∈ log Ω and eq ∈ Ω whose components at n are given by q(t) and eq(t). We regard them as column vectors for each ∈ 𝕑[ζ3]. Then the Laplacian acts on the vector spaces.

Using the incoming Laplacian, Proposition 3.1 is reduced to the following formula.

### Proposition 3.2.

Using the above notations, (3.2) is written by

d3dt3q(t)=Δineq(t).
It 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 [11] , 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:

### Remark 3.1.

1. (1)

The domain of the solution eq of (3.1) is the Jacobian 𝒥E; for L𝒵2ω′,0, eq(t + L) = eq(t). From Lemma 2.1, the periods 2ω′ and 2ω″ are scaled by s1/3. Further in the projection π : E → 𝕇, (π((x, y)) = y), which determines the three special points (0, s, ∞) in 𝕇, the range of the solution eq 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 eq and thus we may fix s as a finite real number.

2. (2)

For v0(≠ 0) such that v0ω0, q(u, v0) as a function with respect to u diverges only at the points in 𝒵2ω′,0 and i=02(ζ3iv0+𝒵2ω,0). Their union is denoted by 𝒮v0. It means that for an ∈ 𝕑[ζ3], if the orbit of q(t;u0, v0) in t avoids 𝒮v0, the value of q(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 eα1×𝒵v0,u0 for a certain unit direction eα1 (i.e., |eα1|=1),

For the case (2) in Lemma 3.2, there are infinite many points at which |q(t;u0, v0)| is greater than for every given positive number 1/ε. Thus we should employ the case (1) in Lemma 3.2.

3. (3)

Let us assume that K := |𝒩v0,u0/𝒵2ω′,0| is finite. For a certain direction eα1 in the complex plane and u0 ≠ 0, we find the subspace eα1 in ℂ such that every q(treα1;u0,v0) does not diverge for each ∈ 𝕑[ζ3] and t ∈ ℝ, and satisfies the trigonal Toda lattice equation,

d3dtr3q(treα1)=e3α1Δineq(treα1).(3.3)

The conditions on eα1 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, eα1=ω/|ω| and s = 1.0.

Let us consider the continuum limit of the the trigonal Toda lattice equation as follows:

### Remark 3.2.

1. (1)

It is noted that q(u, v0) diverges for the limit v0 → 0 and thus for this elliptic function solution q(t), we cannot obtain the continuum limit of the graph Laplacian Δin and of the trigonal Toda lattice equation.

2. (2)

The elliptic curve E becomes the three rational curves for the limit s → 0 as in [5, Appendix C], and y behaves like y=1u3+12s2u3+o(s3) [4] . In the limit, the trigonal Toda lattice equation does not satisfy.

## 4. Discussion

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 [1] , 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 [2] . 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 [5] and q(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 ζ^3 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 [3] and more general curves with a trigonal cyclic group [8] .

## Acknowledgments

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.

Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
27 - 4
Pages
697 - 704
Publication Date
2020/09/04
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2020.1819622How to use a DOI?
Open Access

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  -