An Elementary Introduction to Sato Theory

Yasuhiro OHTA, Junkichi SATSUMA, Daisuke TAKAHASHI and Tetsuji TOKIHIRO

Department of Applied Physics, Faculty of Engineering,

University of Tokyo, Bunkyo-ku, Tokyo 113

Abstract

An elementary introduction to Sato theory is given. Starting with an ordinary differential equation, introducing an infinite number of time variables, and imposing a certain time dependence on the solutions, we obtain the Sato equation which governs the time development of the variable coefficients. It is shown that the generalized Lax equation, the Zakharov-Shabat equation and the IST scheme are generated from the Sato equation. It is revealed that the τ-function becomes the key function to express the solutions of the Sato equation. By using the results of the representation theory of groups, it is shown that the τ-function is governed by the partial differential equations in the bilinear forms which are closely related to the Plücker relations.

§1. Introduction

It has been more than two decades since the inverse scattering transform (IST) method was discovered by Gardner, Green, Kruskal and Miura^{1)} to solve the initial value problems for the Korteweg-de Vries (KdV) equation,

4u_{t} = 12uu_{x} + u_{xxx} .  (1.1)

Though the method was at first thought to be applicable to very restricted class of nonlinear wave equations, it has been revealed that it actually applies to a wide class of equations, i.e., the soliton equations. The Lax scheme plays an important role in extending the applicability.^{2)}

If we introduce the differential operators,

L = ∂^{2} + 2u ,  (1.2)
B = ∂^{3} + 3u∂ + \frac{3}{2}u_{x} ,  (1.3)

where ∂ denotes ∂/∂x, then the inverse scattering scheme for the KdV equation may be written by

Lψ = λψ ,  (1.4)
\frac{∂ψ}{∂t} = Bψ .  (1.5)

If the eigenvalue λ is independent of x and t, the compatibility condition of Eqs. (1.4) and (1.5) yields

\frac{∂L}{∂t} = [B, L] = BL − LB ,  (1.6)

which reduces to the KdV equation (1.1). We shall call Eq. (1.6) the Lax equation.

Zakharov and Shabat succeeded in solving the nonlinear Schrödinger equation by extending L to a non-selfadjoint operator.^{3)} Ablowitz, Kaup, Newell and Segur gave a unified way of giving IST scheme for various equations including the sine-Gordon equation by means of the expansion in eigenvalues.^{4)} We can in principle solve the initial value problems of soliton equations by the IST method.

A formal extension of the IST scheme has also been given by Zakharov and Shabat.^{5)} From the compatibility condition between ∂ψ/∂t_{m} = B_{m}ψ and ∂ψ/∂t_{n} = B_{n}ψ, we obtain

\frac{∂B_{n}}{∂t_{m}} − \frac{∂B_{m}}{∂t_{n}} + [B_{n}, B_{m}] = 0 .  (1.7)

By choosing suitable differential operators B_{n} and B_{m}, we can reduce Eq. (1.7) to several soliton equations. The Kadomtsev-Petviashvili (KP) equation,

(4u_{t} − 12uu_{x} − u_{xxx})_{x} − 3u_{yy} = 0 ,  (1.8)

is one example of such an equation. We shall call Eq. (1.7) the Zakharov-Shabat equation. In the IST method, N-soliton solutions are obtained by solving the algebraic equations derived from the Gel’fand-Levitan-Marchenko equation. The solutions are expressed in determinant forms.

Hirota’s method is another powerful way of obtaining soliton solutions.^{6)} In this method, we introduce dependent variable transformations to reduce the soliton equations into bilinear forms. Then by using a kind of perturbation technique, N-soliton solutions can be constructed in a systematic way. For example the KP equation is reduced to

(4D_{x}D_{t} − D_{x}^{4} − 3D_{y}^{2}) τ · τ = 0 ,  (1.9)

by the dependent variable transformation,

u = (log τ)_{xx} ,  (1.10)

where the operators D_{x} etc. are called Hirota’s operators and are defined by

D_{x}^{n}a(x) · b(x) = (\frac{∂}{∂x} − \frac{∂}{∂x′})^{n}a(x)b(x′)|_{x=x′} = \frac{∂^{n}}{∂s^{n}} a(x + s)b(x − s)|_{s=0} .  (1.11)

If we apply a formal expansion on τ, we obtain the N-soliton solution in the form of a polynomial in exponential functions. In this method, the transformed dependent variable τ becomes a key function. We shall call it the τ-function hereafter. Hirota’s method has been used to obtain not only soliton solutions but several types of special solutions for many nonlinear evolution equations.

There is another way of expressing soliton solutions. It is by means of the Wronski determinant.^{7)} Freeman and Nimmo have given the Wronskian representation of soliton solutions for various equations.^{8)}

The diversity of expressing solutions reflects the richness of algebraic structures which the soliton equations possess in common. It is Sato that unveiled the structures by means of the method of algebraic analysis. He noticed that the τ-function of the KP equation is closely related to the Plücker coordinates appearing in the theory of Grassmann manifolds and then discovered that the totality of solutions of the KP equation as well as of its generalization constitutes an infinite dimensional Grassmann manifold.^{9)} In the concrete, it has been shown that the τ-functions of soliton equations are governed in common by the Plücker relations and can be expressed in terms of determinants with the Wronskian structure.

Shortly after this discovery, Date, Jimbo, Kashiwara and Miwa extended Sato’s idea and developed the theory of transformation groups for soliton equations.^{10)} All these results make it possible to understand the soliton theory from a unified point of view. For example, the relationship among the IST method, Hirota’s method and the Bäcklund transformation is clearly explained by the infinite dimensional Lie algebra and its representation on a function space.

A lot of important results have already been presented on these subjects.^{11)} However, they are rather mathematical and we think it is not so easy to have full understanding of the grand theory. Moreover, the progress after Sato’s discovery has been so fast that the detail of original work seems not to be popular even at present. Though there are several lecture notes based on the talks given by Sato^{12)}, they are mostly written in Japanese. There are only two short papers which Sato himself wrote in English.^{9),13)}

Considering this situation, we attempt in this paper to explain Sato theory in an elementary fashion. Almost all the results included in the following are due to Sato. Since our aim is to introduce the theory in plain language, we make no attempt to be mathematically rigorous. We hope that this paper may serve as an entrance to this magnificent theory.

In §2, we introduce the microdifferential operator and discuss the relationship between the solutions and the variable coefficients of an ordinary differential equation. In §3, we impose an assumption that the solutions also depend on an infinite number of time variables. By giving the dependency explicitly, we obtain an equation governing the time development of the variable coefficients. We call it the Sato equation, in which the τ-function plays an important role to express the solutions. It is shown in §4 that a generalization of the Lax equation and the Zakharov-Shabat equation can be derived from the Sato equation. In §5 we introduce a linear system associated with the generalized Lax equation. We see that the system reduces to the IST scheme if a certain condition is imposed. The structure of the τ-function is investigated in §6. It has a close relationship to the representation theory of groups. We find that, as a result, the τ-function satisfies partial differential equations which relate to the Plücker relations. The KP equation in its bilinear form comes up as the simplest nontrivial one of such differential equations. In §7 we explicitly express the solutions of the Sato equation in terms of the τ-function. The result also gives the explicit expression of the solutions of the generalized Lax equation and the eigenfunction of its associated linear scheme in terms of the τ-function. The concluding remarks are given in §8. Finally in the appendix, we present some results of the representation theory of groups which are useful for discussing the structures of the τ-function.

§2. Basis of Sato theory

Let us introduce a microdifferential operator,

W = 1 + w_{1}∂^{−1} + w_{2}∂^{−2} + w_{3}∂^{−3} + ⋯,  (2.1)

where w_{j} (j = 1, 2, …) are functions of x and ∂^{−n} is formally defined by

∂^{−n} = (\frac{d}{dx})^{−n} .  (2.2)

If we employ the Leibniz rule,

∂^{n}f(x) = ∑_{r=0}^{∞} \frac{n(n − 1) … (n − r + 1)}{r!} \frac{d^{r}f}{dx^{r}} ∂^{n−r} ,  (2.3)

then ∂^{n} can be a well-defined operator even for negative integer n. For example, we have

\begin{aligned}∂^{−1}f & = & f∂^{−1} & − & f′∂^{−2} & + & f′′∂^{−3} & − & ⋯, \\ ∂^{−2}f & = & f∂^{−2} & − & 2f′∂^{−3} & + & 3f′′∂^{−4} & − & ⋯, \end{aligned}

where the prime denotes the differentiation in x.

For Eq. (2.1), the inverse operator W^{−1} exists and is written by

W^{−1} = 1 + v_{1}∂^{−1} + v_{2}∂^{−2} + v_{3}∂^{−3} + ⋯,  (2.4)

where

v_{1} = −w_{1} ,  (2.5a)
v_{2} = −w_{2} + w_{1}^{2} ,  (2.5b)
v_{3} = −w_{3} + 2w_{1}w_{2} − w_{1}w′_{1} − w_{1}^{3} ,  (2.5c)
· · · .

Though the general theory is developed for W = ∑_{n=0}^{∞} w_{n}(x)∂^{−n}, w_{0} = 1, we confine ourselves to

W_{m} = 1 + w_{1}∂^{−1} + w_{2}∂^{−2} + ⋯ + w_{m}∂^{−m} ,  (2.6)

for simplicity.^{14)} It is noted that the essence of the general theory is still kept in this simplification.

Let us consider the ordinary differential equation,

W_{m}∂^{m}f(x) = (∂^{m} + w_{1}∂^{m−1} + ⋯ + w_{m})f(x) = 0 ,  (2.7)

which has m linearly independent solutions, f^{(1)}(x), f^{(2)}(x), …, f^{(m)}(x). We assume that f^{(j)}(x), j = 1, 2, …, m, are analytic, i.e.,

f^{(j)}(x) = ξ_{0}^{(j)} + \frac{1}{1!}ξ_{1}^{(j)}x + \frac{1}{2!}ξ_{2}^{(j)}x^{2} + ⋯,  j = 1, 2, …, m .  (2.8)

Then the rank of the ∞ × m matrix,

Ξ = ( \begin{matrix}ξ_{0}^{(1)} & ξ_{0}^{(2)} & … & ξ_{0}^{(m)} \\ ξ_{1}^{(1)} & ξ_{1}^{(2)} & … & ξ_{1}^{(m)} \\ ⋮ & ⋮ & ⋯ & ⋮ \end{matrix} ) ,  (2.9)

is m and Ξ satisfies

W_{m}∂^{m}(1, \frac{x}{1!}, \frac{x^{2}}{2!} , …)Ξ = 0 .  (2.10)

For an m×m regular constant matrix R, ΞR also satisfies Eq. (2.10), i.e., Ξ is only unique up to a multiplicative factor in GL(m, C). Hence Ξ can be regarded as an element of

{∞ × m matrix of rank m}/GL(m, C) ,

which is called the Grassmann manifold GM(m, ∞).

We introduce a shift operator,

Λ = ( \begin{matrix}0 1 0 1 0 & 1 & 0 \\ 0 & · & \begin{aligned}· \\ · & · \\ & · \end{aligned} \end{matrix} ) .  (2.11)

Then, we have

exp(xΛ) = I + xΛ + \frac{x^{2}}{2!} Λ^{2} + ⋯

= ( \begin{matrix}1 & x & x^{2}/2! & x^{3}/3! & … \\ & 1 & x & x^{2}/2! & … \\ & & 1 & x & … \\ & 0 & & 1 & … \\ . & . & . \end{matrix} ) .  (2.12)

Moreover, we find

H(x) ≡ exp(xΛ) Ξ = ( \begin{matrix}f^{(1)} & f^{(2)} & … & f^{(m)} \\ ∂f^{(1)} & ∂f^{(2)} & … & ∂f^{(m)} \\ ∂^{2}f^{(1)} & ∂^{2}f^{(2)} & … & ∂^{2}f^{(m)} \\ . . . . . . . . . . & . . . . . . . . . . & ⋯ & . . . . . . . . . . \end{matrix} ) .  (2.13)

We now consider the problem to determine W_{m} from the solutions, f^{(1)}, f^{(2)}, …, f^{(m)}.

Equation (2.7) yields

(∂^{m−1}f^{(1)})w_{1} + (∂^{m−2}f^{(1)})w_{2} + ⋯ + f^{(1)}w_{m} = −∂^{m}f^{(1)} ,
⋯
(∂^{m−1}f^{(m)})w_{1} + (∂^{m−2}f^{(m)})w_{2} + ⋯ + f^{(m)}w_{m} = −∂^{m}f^{(m)} .

By solving these simultaneous equations, we find

w_{j} = \begin{matrix}| & \begin{matrix}∂^{m−1}f^{(1)} & … & −∂^{m}f^{(1)} & … & f^{(1)} \\ ⋮ & ⋯ & ⋮ & ⋯ & ⋮ \\ ∂^{m−1}f^{(m)} & … & −∂^{m}f^{(m)} & … & f^{(m)} \end{matrix} & | \\ | & \begin{matrix}∂^{m−1}f^{(1)} & … & ∂^{m−j}f^{(1)} & … & f^{(1)} \\ ⋮ & ⋯ & ⋮ & ⋯ & ⋮ \\ ∂^{m−1}f^{(m)} & … & ∂^{m−j}f^{(m)} & … & f^{(m)} \end{matrix} & | \end{matrix} ,  (2.14)

or (from Eq. (2.6)),

W_{m} = \begin{matrix}| & \begin{matrix}f^{(1)} & … & f^{(m)} & ∂^{−m} \\ ⋮ & ⋯ & ⋮ & ⋮ \\ ∂^{m−1}f^{(1)} & … & ∂^{m−1}f^{(m)} & ∂^{−1} \\ ∂^{m}f^{(1)} & … & ∂^{m}f^{(m)} & 1 \end{matrix} & | \\ | & \begin{matrix}f^{(1)} & … & f^{(m)} \\ ⋮ & ⋯ & ⋮ \\ ∂^{m−1}f^{(1)} & … & ∂^{m−1}f^{(m)} \end{matrix} & | \end{matrix} .  (2.15)

In Eq. (2.15), the operator ∂^{j} has to be put in the rightmost position when we evaluate the determinant of the numerator. The denominator of Eq. (2.14) or (2.15) is the determinant of the matrix consisting of the first m rows of H(x), and is a Wronskian.

§3. Sato equation

We here assume that w_{j} (j = 1, 2, …) in Eq. (2.1) are also the functions of an infinite number of time variables t = (t_{1}, t_{2}, t_{3}, …). Then the solutions of Eq. (2.7), f^{(j)}, include t_{1}, t_{2}, t_{3}, … as parameters,

f^{(j)} = f^{(j)}(x; t) = f^{(j)}(x; t_{1}, t_{2}, …) ,

and hence H defined by Eq. (2.13) may also be written as H(x; t).

Let us impose the condition that H(x; t) evolves in time as

H(x; t) = exp xΛ exp η(t, Λ) Ξ ,  (3.1)

where

η(t, Λ) = ∑_{n=1}^{∞} t_{n}Λ^{n} .  (3.2)

If we formally expand the operator,

exp{(x + t_{1})Λ + t_{2}Λ^{2} + t_{3}Λ^{3} + ⋯} = ∑_{n=0}^{∞} p_{n}Λ^{n} ,  (3.3)

we find

p_{n}(x + t_{1}, t_{2}, t_{3}, …) = ∑_{{ν_{0}+ν_{1}+2ν_{2}+3ν_{3}+···=n \atop ν_{0},ν_{1},ν_{2},ν_{3},...≥0}} \frac{x^{ν_{0}}t_{1}^{ν_{1}}t_{2}^{ν_{2}} ⋯}{ν_{0}!ν_{1}!ν_{2}! ⋯} .  (3.4)

The first few p_{n}’s are

p_{0} = 1 ,  (3.5a)
p_{1} = x + t_{1} ,  (3.5b)
p_{2} = \frac{1}{2}(x + t_{1})^{2} + t_{2} ,  (3.5c)
p_{3} = \frac{1}{6}(x + t_{1})^{3} + (x + t_{1})t_{2} + t_{3} .  (3.5d)

These polynomials have the property

\frac{∂p_{n}}{∂t_{m}} = p_{n−m} ,  (p_{n} = 0 for n < 0) ,  (3.6)

and, especially,

\frac{∂p_{n}}{∂x} = p_{n−1} .  (3.7)

We remark that the p_{n}’s play an important role in the representation theory of groups (see Appendix). By means of these polynomials, H(x; t) is expressed by

H(x; t) = ( \begin{matrix}1 & p_{1} & p_{2} & p_{3} & … \\ & 1 & p_{1} & p_{2} & … \\ & & 1 & p_{1} & … \\ & 0 & & 1 & … \\ . & . & . \end{matrix} ) ( \begin{matrix}ξ_{0}^{(1)} & ξ_{0}^{(2)} & … & ξ_{0}^{(m)} \\ ξ_{1}^{(1)} & ξ_{1}^{(2)} & … & ξ_{1}^{(m)} \\ ξ_{2}^{(1)} & ξ_{2}^{(2)} & … & ξ_{2}^{(m)} \\ ⋮ & ⋮ & ⋯ & ⋮ \end{matrix} )

≡ ( \begin{matrix}h_{0}^{(1)}(x; t) & h_{0}^{(2)}(x; t) & … & h_{0}^{(m)}(x; t) \\ h_{1}^{(1)}(x; t) & h_{1}^{(2)}(x; t) & … & h_{1}^{(m)}(x; t) \\ ⋮ & ⋮ & ⋯ & ⋮ \end{matrix} ) .  (3.8)

The elements of the above matrix, h_{n}^{(j)}(x; t), have the following properties:

h_{0}^{(j)}(x; 0) = f^{(j)}(x) ,  (3.9)

and

h_{n}^{(j)}(x; t) = \frac{∂h_{0}^{(j)}(x; t)}{∂t_{n}} = \frac{∂^{n}h_{0}^{(j)}(x; t)}{∂x^{n}} .  (3.10)

Namely, h_{0}^{(j)}(x; t) is the solution of a set of linear partial differential equations,

(\frac{∂}{∂t_{n}} − \frac{∂^{n}}{∂x^{n}})h(x; t) = 0 ,  n = 1, 2, …,  (3.11)

with the initial value

h(x; 0) = f^{(j)}(x) .  (3.12)

We consider that W_{m} ( and w_{j} ) in Eq. (2.6) also depend on t so that

W_{m}(x; t)∂^{m}h_{0}^{(j)}(x; t)
= (∂^{m} + w_{1}(x; t)∂^{m−1} + ⋯ + w_{m}(x; t))h_{0}^{(j)}(x; t)
= 0 ,  j = 1, 2, …, m .  (3.13)

In the similar way to §2, we see that w_{j}(x; t) and W_{m}(x; t) are expressed by

w_{j}(x; t) = \begin{matrix}| & \begin{matrix}h_{m−1}^{(1)} & … & −h_{m}^{(1)} & … & h_{0}^{(1)} \\ ⋮ & ⋯ & ⋮ & ⋯ & ⋮ \\ h_{m−1}^{(m)} & … & −h_{m}^{(m)} & … & h_{0}^{(m)} \end{matrix} & | \\ | & \begin{matrix}h_{m−1}^{(1)} & … & h_{m−j}^{(1)} & … & h_{0}^{(1)} \\ ⋮ & ⋯ & ⋮ & ⋯ & ⋮ \\ h_{m−1}^{(m)} & … & h_{m−j}^{(m)} & … & h_{0}^{(m)} \end{matrix} & | \end{matrix} ,  (3.14)

and

W_{m}(x; t) = \begin{matrix}| & \begin{matrix}h_{0}^{(1)} & … & h_{0}^{(m)} & ∂^{−m} \\ ⋮ & ⋯ & ⋮ & ⋮ \\ h_{m−1}^{(1)} & … & h_{m−1}^{(m)} & ∂^{−1} \\ h_{m}^{(1)} & … & h_{m}^{(m)} & 1 \end{matrix} & | \\ | & \begin{matrix}h_{0}^{(1)} & … & h_{0}^{(m)} \\ ⋮ & ⋯ & ⋮ \\ h_{m−1}^{(1)} & … & h_{m−1}^{(m)} \end{matrix} & | \end{matrix} ,  (3.15)

respectively. In §7, we will show that the numerators of w_{j}(x; t) can be expressed by certain derivatives of the denominator. Hence, the denominator of Eq. (3.14) or (3.15) becomes an important function which yields all the w_{j}’s. We shall also see in §6 that it actually gives the τ-function mentioned in §1.

We now pose a problem to determine the time evolution equation for W_{m}(x; t). Differentiating (3.13) by t_{n} and employing the property (3.10), we have

(\frac{∂W_{m}}{∂t_{n}} ∂^{m} + W_{m}∂^{m}∂^{n})h_{0}^{(j)}(x; t) = 0 ,  (3.16)

which is the ordinary differential equation of (m+n)-th order and shares the same linearly independent solutions as Eq. (3.13). Therefore the differential operators in Eq. (3.16) has to be factorized as

\frac{∂W_{m}}{∂t_{n}} ∂^{m} + W_{m}∂^{m}∂^{n} = B_{n}W_{m}∂^{m} ,  (3.17)

where B_{n} is a differential operator. By applying ∂^{−m}W_{m}^{−1} from the right, we have

B_{n} = \frac{∂W_{m}}{∂t_{n}} W_{m}^{−1} + W_{m}∂^{n}W_{m}^{−1} .  (3.18)

From the properties of the operators W_{m} and W_{m}^{−1}, we know that the first term of the right hand side of Eq. (3.18) consists only of the terms with ∂^{−n}(n = 1, 2, …). Hence we have

B_{n} = (W_{m}∂^{n}W_{m}^{−1})^{+} ,  (3.19)

where ( )^{+} denotes the differential part of the operator. Consequently, it is shown that the time evolution of W_{m}(x; t) is governed by

\frac{∂W}{∂t_{n}} = B_{n}W − W ∂^{n} ,  (3.20)
B_{n} = (W ∂^{n}W^{−1})^{+} ,  (3.21)

which we call the Sato equation hereafter. The first few of B_{n} are explicitly given by

B_{1} = ∂ ,  (3.22a)
B_{2} = ∂^{2} − 2w_{1,x} ,  (3.22b)
B_{3} = ∂^{3} − 3w_{1,x}∂ − 3w_{2,x} + 3w_{1}w_{1,x} − 3w_{1,xx} ,  (3.22c)
B_{4} = ∂^{4} − 4w_{1,x}∂^{2} − (4w_{2,x} + 6w_{1,xx} − 4w_{1}w_{1,x})∂
− 4w_{3,x} − 6w_{2,xx} − 4w_{1,xxx} + 4w_{1}w_{2,x} + 4w_{1,x}w_{2}
+ 6w_{1}w_{1,xx} + 8w_{1,x}^{2} − 4w_{1}^{2}w_{1,x} ,  (3.22d)

where w_{j,\underbrace{x … x}_{ℓ}} denotes ∂^{ℓ}w_{j}/∂x^{ℓ}.

Since the similar argument is possible for the operator W defined by Eq. (2.1), we have dropped the suffix m in Eqs. (3.20) and (3.21). It is an important fact in the following discussion that the solutions of Eqs. (3.20) and (3.21) are given in the form of Eq. (3.15). From the results in this section, we see that t_{1} plays the same role as x. Hence, we will use t_{1} or x without distinction hereafter.

§4. Generalized Lax equation

We have shown in the preceding section that the time evolution of W (x; t) is governed by the Sato equation (3.20) and (3.21), and the coefficients in W (x; t) are expressed in terms of the set of linearly independent solutions of the ordinary differential equations (3.13). In this section we show that a generalization of the Lax equation (1.6) and the Zakharov-Shabat equation (1.7) can be derived from the Sato equation.

We define an operator L by

L = W ∂W^{−1} ,  (4.1)

which can be written as

L = ∂ + u_{2}∂^{−1} + u_{3}∂^{−2} + u_{4}∂^{−3} + · · · .  (4.2)

Substituting Eqs. (2.1) and (2.4) into Eq. (4.1), we find that the coefficients u_{j} in Eq. (4.2) are related to w_{j} as

u_{2} = −w_{1,x} ,  (4.3a)
u_{3} = −w_{2,x} + w_{1}w_{1,x} ,  (4.3b)
u_{4} = −w_{3,x} + w_{1}w_{2,x} + w_{1,x}w_{2} − w_{1}^{2}w_{1,x} − w_{1,x}^{2} ,  (4.3c)
· · · .

Differentiating L by t_{n}, and using Eq. (3.20), we have

\frac{∂L}{∂t_{n}} = \frac{∂W}{∂t_{n}} ∂W^{−1} + W ∂ \frac{∂W^{−1}}{∂t_{n}}

= (B_{n}W − W ∂^{n})∂W^{−1} − W ∂W^{−1}(B_{n}W − W ∂^{n})W^{−1}

= B_{n}W ∂W^{−1} − W ∂W^{−1}B_{n} ,  (4.4)

where we have used the fact that

\frac{∂(W W^{−1})}{∂t_{n}} = W \frac{∂W^{−1}}{∂t_{n}} + \frac{∂W}{∂t_{n}} W^{−1} = 0 .  (4.5)

Consequently, we obtain

\frac{∂L}{∂t_{n}} = [B_{n}, L] = B_{n}L − LB_{n} ,  (4.6)

which may be called a generalized Lax equation.

It is clear from the definition of L that

L^{n} = W ∂^{n}W^{−1} ,  (4.7)

and therefore B_{n} may be written by

B_{n} = (L^{n})^{+} .  (4.8)

If we use u_{j} instead of w_{j}, Eqs. (3.22a)∼(3.22d) are expressed as

B_{1} = ∂ ,  (4.9a)
B_{2} = ∂^{2} + 2u_{2} ,  (4.9b)
B_{3} = ∂^{3} + 3u_{2}∂ + 3u_{3} + 3u_{2,x} ,  (4.9c)
B_{4} = ∂^{4} + 4u_{2}∂^{2} + (4u_{3} + 6u_{2,x})∂ + 4u_{4} + 6u_{3,x} + 4u_{2,xx} + 6u_{2}^{2} ,  (4.9d)

respectively.

We now show that an infinite number of equations are derived from the generalized Lax equation. For n = 2, Eq. (4.6) are written by

\frac{∂u_{2}}{∂t_{2}} ∂^{−1} + \frac{∂u_{3}}{∂t_{2}} ∂^{−2} + \frac{∂u_{4}}{∂t_{2}} ∂^{−3} + ⋯

= (∂^{2} + 2u_{2})(∂ + u_{2}∂^{−1} + u_{3}∂^{−2} + ⋯) − (∂ + u_{2}∂^{−1} + u_{3}∂^{−2} + ⋯)(∂^{2} + 2u_{2})
= (u_{2,xx} + 2u_{3,x})∂^{−1} + (u_{3,xx} + 2u_{4,x} + 2u_{2}u_{2,x})∂^{−2}
+ (u_{4,xx} + 2u_{5,x} − 2u_{2}u_{2,xx} + 4u_{2,x}u_{3})∂^{−3} + · · · .

By equating the coefficients of ∂^{−j}, j = 1, 2, …, we get

\frac{∂u_{2}}{∂t_{2}} = u_{2,xx} + 2u_{3,x} ,  (4.10a)
\frac{∂u_{3}}{∂t_{2}} = u_{3,xx} + 2u_{4,x} + 2u_{2}u_{2,x} ,  (4.10b)
\frac{∂u_{4}}{∂t_{2}} = u_{4,xx} + 2u_{5,x} − 2u_{2}u_{2,xx} + 4u_{2,x}u_{3} ,  (4.10c)
· · · .

Similarly from Eq. (4.6) with n = 3, we get

\frac{∂u_{2}}{∂t_{3}} = u_{2,xxx} + 3u_{3,xx} + 3u_{4,x} + 6u_{2}u_{2,x} ,  (4.11a)
\frac{∂u_{3}}{∂t_{3}} = u_{3,xxx} + 3u_{4,xx} + 3u_{5,x} + 6u_{2}u_{3,x} + 6u_{2,x}u_{3} ,  (4.11b)
\frac{∂u_{4}}{∂t_{3}} = u_{4,xxx} + 3u_{5,xx} + 3u_{6,x} − 3u_{2}u_{3,xx} − 3u_{2,xx}u_{3}
+ 3u_{2}u_{4,x} + 9u_{2,x}u_{4} + 6u_{3}u_{3,x} ,  (4.11c)
· · · .

Following this procedure for n = 4, 5, …, we get an infinite set of equations for the dependent variables u_{2}, u_{3}, u_{4}, . . .. If we eliminate u_{3}, u_{4} from Eqs. (4.10a), (4.10b) and (4.11a), we obtain an equation for u_{2},

\frac{∂}{∂x}(4\frac{∂u_{2}}{∂t_{3}} − 12u_{2} \frac{∂u_{2}}{∂x} − \frac{∂^{3}u_{2}}{∂x^{3}} ) − 3\frac{∂^{2}u_{2}}{∂t_{2}^{2}} = 0 ,  (4.12)

which is nothing but the KP equation (1.8). Because this equation is the simplest nontrivial one which is obtained by this formalism, we call the infinite set of equations the KP hierarchy.

It is readily shown that

\frac{∂L^{m}}{∂t_{n}} = [B_{n}, L^{m}] ,  (4.13)

for m, n = 1, 2, 3, . . .. Hence we have

\frac{∂L^{m}}{∂t_{n}} − \frac{∂L^{n}}{∂t_{m}} = [B_{n}, L^{m}] − [B_{m}, L^{n}] .  (4.14)

If we write

L^{n} = B_{n} − B_{n}^{c} ,  (4.15)

then B_{n}^{c} consists only of the terms with ∂^{−j}, j > 0. Substitution of Eq. (4.15) into Eq. (4.14) yields

\frac{∂L^{m}}{∂t_{n}} − \frac{∂L^{n}}{∂t_{m}} = [L^{n} + B_{n}^{c}, L^{m}] − [B_{m}, B_{n} − B_{n}^{c}]
= [B_{n}^{c}, B_{m} − B_{m}^{c}] − [B_{m}, B_{n} − B_{n}^{c}]
= [B_{n}, B_{m}] − [B_{n}^{c}, B_{m}^{c}] .  (4.16)

Taking the differential part of Eq. (4.16), we obtain

\frac{∂B_{m}}{∂t_{n}} − \frac{∂B_{n}}{∂t_{m}} = [B_{n}, B_{m}] ,  (4.17)

which is the Zakharov-Shabat equation (1.7). If we choose m = 2 and n = 3, then we recover the KP equation (4.12) from Eq. (4.17).

§5. Linear system associated with the generalized Lax equation

A linear system associated with the generalized Lax equation may be written as follows:

Lψ = λψ ,  (5.1)
\frac{∂ψ}{∂t_{n}} = B_{n}ψ ,  (5.2)
\frac{∂λ}{∂t_{n}} = 0 .  (5.3)

In fact, Eq. (4.6) is derived from the compatibility condition among Eqs. (5.1)∼(5.3). Equation (5.1) is considered as an eigenvalue problem for the microdifferential operator L and Eq. (5.2) describes the evolution of the eigenfunction for each of the time variables t_{1}, t_{2}, . . ..

In this section, we first investigate the structure of the eigenfunction. From the definition of L, we find that Eq. (5.1) is rewritten as

∂ψ_{0} = λψ_{0} ,  (5.4)

where

ψ_{0} = W^{−1}ψ .  (5.5)

Integration of Eq. (5.4) yields

ψ_{0} = g(t_{2}, t_{3}, …; λ)e^{λx} ,  (5.6)

or

ψ = (1 + \frac{w_{1}}{λ} + \frac{w_{2}}{λ^{2}} + ⋯)g(t_{2}, t_{3}, …; λ)e^{λx} ,  (5.7)

where g is an arbitrary function of the arguments. We may assume that g is analytic in λ. On the other hand, by noticing Eq. (4.15), we have from Eq. (5.2) that

\frac{∂ψ}{∂t_{n}} = (L^{n} + B_{n}^{c})ψ .  (5.8)

As mentioned in §4, B_{n}^{c} consists only of the terms with ∂^{−j}, j > 0. From the Leibniz rule (2.3), we see that, for j = 1, 2, …,

L^{−j} = ∂^{−j} − ju_{2}∂^{−j−2} + ⋯,  (5.9)

which means that ∂^{−j} can be expressed in terms of L as

∂^{−j} = ∑_{ℓ=j}^{∞} ṽ_{jℓ}L^{−ℓ} ,  (5.10)

where ṽ_{jℓ}’s are suitable functions of t_{1}, t_{2}, . . .. Hence, Eq. (5.8) may be written by

\frac{∂ψ}{∂t_{n}} = (L^{n} + v_{n1}L^{−1} + v_{n2}L^{−2} + ⋯)ψ ,  (5.11)

where v_{nj}’s are also suitable functions of t_{1}, t_{2}, . . ..

From Eq. (5.1), we have

L^{j}ψ = λ^{j}ψ .  (5.12)

Substituting Eq. (5.12) into Eq. (5.11), we obtain

\frac{∂ψ}{∂t_{n}} = (λ^{n} + \frac{v_{n1}}{λ} + \frac{v_{n2}}{λ^{2}} + ⋯)ψ ,  (5.13a)

or

\frac{∂}{∂t_{n}} log ψ = λ^{n} + \frac{v_{n1}}{λ} + \frac{v_{n2}}{λ^{2}} + · · · .  (5.13b)

From Eq. (5.13b), we obtain

log ψ = ∑_{j=1}^{∞} λ^{j}t_{j} + t_{0} + ∑_{j=1}^{∞} v_{j}λ^{−j} ,  (t_{0}; const.) ,  (5.14)

which may be considered as the Laurent expansion of log ψ at λ = ∞ and the first term of the right hand side corresponds to its principal part. In Eq. (5.14), v_{j}’s are again suitable functions of t_{1}, t_{2}, . . .. From Eq. (5.14), we have

ψ = exp(∑_{j=1}^{∞} v_{j}λ^{−j}) exp(t_{0} + λt_{1} + λ^{2}t_{2} + ⋯) .  (5.15)

Expanding exp(∑_{j=1}^{∞} v_{j}λ^{−j}) in 1/λ and imposing the condition that the resultant ψ coincides

with Eq. (5.7) at t_{2}, t_{3}, … = 0, we finally obtain

ψ = (1 + \frac{w_{1}}{λ} + \frac{w_{2}}{λ^{2}} + ⋯) exp(t_{0} + λt_{1} + λ^{2}t_{2} + ⋯) .  (5.16)

Consequently, the eigenfunction of the linear system (5.1) and (5.2) can be directly related to w_{j}’s which are included in the solution of the Sato equation, (3.20) and (3.21).

We now show that the linear system (5.1)∼(5.3) reduces to the IST scheme mentioned in §1, if a certain condition is imposed. Let us assume that L^{ℓ} consists only of its differential part B_{n} for a certain ℓ,

L^{ℓ} = B_{ℓ} .  (5.17)

This condition also implies that

L^{kℓ} = B_{kℓ} ,  k = 1, 2, ….  (5.18)

Then, Eq. (5.1) reduces to

B_{kℓ}ψ = λ^{kℓ}ψ ,  k = 1, 2, ….  (5.19)

Moreover, Eq. (5.2) gives

\frac{∂ψ}{∂t_{kℓ}} = λ^{kℓ}ψ ,  k = 1, 2, ….  (5.20)

Comparing Eq. (5.20) with Eq. (5.16), we find

\frac{∂w_{j}}{∂t_{kℓ}} = 0 ,  j, k = 1, 2, …,  (5.21)

which means that all of w_{j}’s are independent of t_{ℓ}, t_{2ℓ}, t_{3ℓ}, . . .. This situation is called ℓ-reduction.

In the case of 2-reduction, Eq. (5.19) with k = 1 is written by

(∂^{2} + 2u_{2})ψ = λ^{2}ψ ,  (5.22)

which is the Schrödinger-type eigenvalue problem. From Eq. (5.17), L^{2} = B_{2}, all of the coefficients of ∂^{−j}, j > 0 in L^{2} must be zero. Then we have, for example,

u_{3} = −\frac{1}{2}u_{2,x} ,  (5.23a)
u_{4} = \frac{1}{4}u_{2,xx} − \frac{1}{2}u_{2}^{2} ,  (5.23b)
u_{5} = −\frac{1}{8}u_{2,xxx} + \frac{3}{2}u_{2}u_{2,x} .  (5.23c)

Here, Eq. (5.2) with n = 3 gives

\frac{∂ψ}{∂t_{3}} = (∂^{3} + 3u_{2}∂ + \frac{3}{2}u_{2,x})ψ .  (5.24)

Equations (5.22) and (5.24) taken together are essentially the same as the IST scheme for the KdV equation (1.1).

If we consider

\frac{∂ψ}{∂t_{5}} = B_{5}ψ ,  (5.25)

instead of Eq. (5.24), we recover the IST scheme for the fifth-order KdV equation of the Lax-type.

In the case of 3-reduction, we have the third-order eigenvalue problem,

B_{3}ψ = (∂^{3} + 3u_{2}∂ + 3u_{3} + 3u_{2,x})ψ = λ^{3}ψ .  (5.26)

If we here consider

\frac{∂ψ}{∂t_{2}} = B_{2}ψ = (∂^{2} + 2u_{2})ψ ,  (5.27)

as the associated equation for the time development of ψ, then the compatibility condition of Eqs. (5.26) and (5.27) yields the Boussinesq-type equation,

\frac{∂^{2}u_{2}}{∂t_{2}^{2}} = −\frac{1}{3}u_{2,xxxx} − 2(u_{2}^{2})_{xx} .  (5.28)

Finally, in the case of 4-reduction, the eigenvalue problem, B_{4}ψ = λ^{4}ψ, is of the fourthorder. If ∂ψ/∂t_{3} = B_{3}ψ is considered as the associated equation, then the compatibility condition yields the coupled KdV equation discussed in Ref. 15).

§6. τ-function

In the preceding two sections we have shown that various analytical schemes to treat soliton equations may be derived from the Sato equation (3.20) and (3.21). Furthermore, we have seen that not only the solutions of soliton equations but the eigenfunctions in the IST scheme are directly related to the solutions of the Sato equation. As mentioned in §3, the latter solutions are expressed in terms of the τ-function,

τ(x; t) = | \begin{matrix}h_{0}^{(1)} & … & h_{0}^{(m)} \\ h_{1}^{(1)} & … & h_{1}^{(m)} \\ ⋮ & ⋯ & ⋮ \\ h_{m−1}^{(1)} & … & h_{m−1}^{(m)} \end{matrix} | .  (6.1)

In this context, we here study the structures of the τ-function in more detail.

Since the τ-function is the determinant of the matrix made of the first m rows of H(x; t), it may be written as

τ(t) = det [( \begin{matrix}1 & p_{1} & p_{2} & p_{3} & … \\ & 1 & p_{1} & p_{2} & … \\ & 0 & ... & ... & ⋯ \\ & & & 1 & … \end{matrix} ) ( \begin{matrix}ξ_{0}^{(1)} & ξ_{0}^{(2)} & … & ξ_{0}^{(m)} \\ ξ_{1}^{(1)} & ξ_{1}^{(2)} & … & ξ_{1}^{(m)} \\ ξ_{2}^{(1)} & ξ_{2}^{(2)} & … & ξ_{2}^{(m)} \\ . . . . . . . . . . & . . . . . . . . . . & ⋯ & . . . . . . . . . . \end{matrix} )]

= det (Ξ_{0}^{t} exp η(t, Λ) Ξ) ,  (6.2)

where Ξ_{0}^{t} is an m × ∞ matrix defined by

Ξ_{0}^{t} = ( \begin{matrix}1 & 0 & 0 & … \\ & 1 & 0 & 0 & … \\ & 0 & ... & ... & ... & ⋯ \\ & & & 1 & 0 & 0 & … \end{matrix} ) .  (6.3)

In this expression we have omitted x-dependence because t_{1} plays the same role as x. As a consequence, the p_{n}’s in Eq. (6.2) are defined by

p_{n}(t) = ∑_{ν_{1}+2ν_{2}+3ν_{3}+···=n} \frac{t_{1}^{ν_{1}}t_{2}^{ν_{2}}t_{3}^{ν_{3}} ⋯}{ν_{1}!ν_{2}!ν_{3}! ⋯} .  (6.4)

Using the expansion theorem on the determinant of product of matrices, τ(t) can be expanded as a sum of the products of determinants;

τ(t) = ∑_{0≤ℓ_{1}<ℓ_{2}<···<ℓ_{m}} | \begin{matrix}p_{ℓ_{1}} & p_{ℓ_{2}} & … & p_{ℓ_{m}} \\ p_{ℓ_{1}−1} & p_{ℓ_{2}−1} & … & p_{ℓ_{m}−1} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ p_{ℓ_{1}−m+1} & p_{ℓ_{2}−m+1} & … & p_{ℓ_{m}−m+1} \end{matrix} |

× | \begin{matrix}ξ_{ℓ_{1}}^{(1)} & ξ_{ℓ_{1}}^{(2)} & … & ξ_{ℓ_{1}}^{(m)} \\ ξ_{ℓ_{2}}^{(1)} & ξ_{ℓ_{2}}^{(2)} & … & ξ_{ℓ_{2}}^{(m)} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ ξ_{ℓ_{m}}^{(1)} & ξ_{ℓ_{m}}^{(2)} & … & ξ_{ℓ_{m}}^{(m)} \end{matrix} | ,  (6.5)

where the summation is taken over all possible combinations of m nonnegative numbers, (p_{n} = 0 for n < 0 as in Eq.(3.6)).

For each set of numbers (ℓ_{1}, ℓ_{2}, …, ℓ_{m}), it is possible to define a corresponding diagram. Let us prepare a chain of cells, each of which is numbered in numerical order (Fig. 1). We put Fermi particles in (i) all of the cells numbered less than and equal to −m, and (ii) each of the cells numbered ℓ_{1} −m+1, ℓ_{2} −m+1, …, ℓ_{m} −m+1. For example, if a set of numbers (2, 3, 5, 7) is given, then the cells are occupied by Fermi particles as Fig. 2. If no number is assigned (m = 0), the diagram is as Fig. 3, which may be considered to be the vacuum state. This type of diagram is called the Maya diagram after Sato. It is noted that the correspondence between the set of numbers and this diagram is one to one for fixed m.

Figs. 1∼3

There is also the one to one correspondence between a Maya diagram and a Young diagram. Suppose that we have a Maya diagram. If a cell is occupied by a particle in the diagram, assign a vertical line ↑, and if it is empty, assign a horizontal line →, respectively. Then we obtain a connected line. For the vacuum state we have a line with one corner. The diagram surrounded by these two lines gives the corresponding Young diagram. Figure 4 shows the procedure to construct the Young diagram for the Maya diagram of Fig. 2. The vacuum state itself corresponds to the Young diagram ϕ.

Fig. 4

Thus we find the one to one correspondence between a set of numbers and a Young diagram for fixed m via a Maya diagram. This correspondence is quite reasonable. In fact, the Young diagram is introduced to classify the irreducible representation of the symmetric group. Also the determinants composed of p_{j}’s in Eq. (6.5) are the Schur functions themselves which appear in symmetric group theory.

Since the correspondence is one to one, we may denote

S_{Y} (t) = | \begin{matrix}p_{ℓ_{1}} & p_{ℓ_{2}} & … & p_{ℓ_{m}} \\ p_{ℓ_{1}−1} & p_{ℓ_{2}−1} & … & p_{ℓ_{m}−1} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ p_{ℓ_{1}−m+1} & p_{ℓ_{2}−m+1} & … & p_{ℓ_{m}−m+1} \end{matrix} | ,  (6.6)

and

ξ_{Y} = | \begin{matrix}ξ_{ℓ_{1}}^{(1)} & ξ_{ℓ_{1}}^{(2)} & … & ξ_{ℓ_{1}}^{(m)} \\ ξ_{ℓ_{2}}^{(1)} & ξ_{ℓ_{2}}^{(2)} & … & ξ_{ℓ_{2}}^{(m)} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ ξ_{ℓ_{m}}^{(1)} & ξ_{ℓ_{m}}^{(2)} & … & ξ_{ℓ_{m}}^{(m)} \end{matrix} | ,  (6.7)

where the suffix Y means the Young diagram corresponding to the set of numbers (ℓ_{1}, ℓ_{2}, …, ℓ_{m}). It is noted that, although different sets of numbers may correspond to a certain Y if m is not fixed, the right-hand side of Eq. (6.6) gives the same function for those sets. With these notations, Eq. (6.5) may be written by

τ(t) = ∑ ; ϕ≤Y ≤m{ S_{Y} (t) ξ_{Y} ,  (6.8)

where the summation is taken over all Young diagrams which have less than m + 1 rows.

The case of m = 2 gives a simple example of this expansion:

τ(t) = det [( \begin{matrix}p_{0} & p_{1} & p_{2} & … \\ 0 & p_{0} & p_{1} & … \end{matrix} ) ( \begin{matrix}ξ_{0}^{(1)} & ξ_{0}^{(2)} \\ ξ_{1}^{(1)} & ξ_{1}^{(2)} \\ ⋮ & ⋮ \end{matrix} )]

= | \begin{matrix}p_{0} & p_{1} \\ 0 & p_{0} \end{matrix} | | \begin{matrix}ξ_{0}^{(1)} & ξ_{0}^{(2)} \\ ξ_{1}^{(1)} & ξ_{1}^{(2)} \end{matrix} | + | \begin{matrix}p_{0} & p_{2} \\ 0 & p_{1} \end{matrix} | | \begin{matrix}ξ_{0}^{(1)} & ξ_{0}^{(2)} \\ ξ_{2}^{(1)} & ξ_{2}^{(2)} \end{matrix} |

+ | \begin{matrix}p_{0} & p_{3} \\ 0 & p_{2} \end{matrix} | | \begin{matrix}ξ_{0}^{(1)} & ξ_{0}^{(2)} \\ ξ_{3}^{(1)} & ξ_{3}^{(2)} \end{matrix} | + ⋯

= S_{ϕ}ξ_{ϕ} + S ξ + S ξ + · · · .  (6.9)

As shown in Appendix, it is possible to expand an arbitrary analytic function f(t) in S_{Y} (t),

f(t) = ∑_{Y} S_{Y} (t) c_{Y} .  (6.10)

The coefficient c_{Y} is uniquely determined from the orthogonality condition as

c_{Y} = S_{Y} (˜∂_{t})f(t)|_{t=0} ,  (6.11)

where ˜∂_{t} is defined by

˜∂_{t} = (\frac{∂}{∂t_{1}} , \frac{1}{2} \frac{∂}{∂t_{2}} , \frac{1}{3} \frac{∂}{∂t_{3}} , …) .  (6.12)

Equation (6.8) is also considered to be the expansion of τ(t) in S_{Y} (t). Since the expansion (6.10) is unique, this is also unique.

For the coefficients ξ_{Y} ’s of τ-function, there exists an important property that they always have to satisfy constraints called the Plücker relations. Let us derive these relations in an elementary way. We first consider the simplest nontrivial case m = 2. For the τfunction (6.9), we choose four numbers, k, ℓ_{1} < ℓ_{2} < ℓ_{3}, and notice a trivial identity,

| \begin{matrix}ξ_{k}^{(1)} & ξ_{ℓ_{1}}^{(1)} & ξ_{ℓ_{2}}^{(1)} & ξ_{ℓ_{3}}^{(1)} \\ ξ_{k}^{(2)} & ξ_{ℓ_{1}}^{(2)} & ξ_{ℓ_{2}}^{(2)} & ξ_{ℓ_{3}}^{(2)} \\ 0 & ξ_{ℓ_{1}}^{(1)} & ξ_{ℓ_{2}}^{(1)} & ξ_{ℓ_{3}}^{(1)} \\ 0 & ξ_{ℓ_{1}}^{(2)} & ξ_{ℓ_{2}}^{(2)} & ξ_{ℓ_{3}}^{(2)} \end{matrix} | = | \begin{matrix}ξ_{k}^{(1)} & 0 & 0 & 0 \\ ξ_{k}^{(2)} & 0 & 0 & 0 \\ 0 & ξ_{ℓ_{1}}^{(1)} & ξ_{ℓ_{2}}^{(1)} & ξ_{ℓ_{3}}^{(1)} \\ 0 & ξ_{ℓ_{1}}^{(2)} & ξ_{ℓ_{2}}^{(2)} & ξ_{ℓ_{3}}^{(2)} \end{matrix} | = 0 .  (6.13)

By applying the Laplace expansion to the first determinant, we obtain

| \begin{matrix}ξ_{k}^{(1)} & ξ_{ℓ_{1}}^{(1)} \\ ξ_{k}^{(2)} & ξ_{ℓ_{1}}^{(2)} \end{matrix} | | \begin{matrix}ξ_{ℓ_{2}}^{(1)} & ξ_{ℓ_{3}}^{(1)} \\ ξ_{ℓ_{2}}^{(2)} & ξ_{ℓ_{3}}^{(2)} \end{matrix} | − | \begin{matrix}ξ_{k}^{(1)} & ξ_{ℓ_{2}}^{(1)} \\ ξ_{k}^{(2)} & ξ_{ℓ_{2}}^{(2)} \end{matrix} | | \begin{matrix}ξ_{ℓ_{1}}^{(1)} & ξ_{ℓ_{3}}^{(1)} \\ ξ_{ℓ_{1}}^{(2)} & ξ_{ℓ_{3}}^{(2)} \end{matrix} |
+ | \begin{matrix}ξ_{k}^{(1)} & ξ_{ℓ_{3}}^{(1)} \\ ξ_{k}^{(2)} & ξ_{ℓ_{3}}^{(2)} \end{matrix} | | \begin{matrix}ξ_{ℓ_{1}}^{(1)} & ξ_{ℓ_{2}}^{(1)} \\ ξ_{ℓ_{1}}^{(2)} & ξ_{ℓ_{2}}^{(2)} \end{matrix} | = 0 ,  (6.14)

which gives constraints among ξ_{Y} ’s. For example, if we choose (k, ℓ_{1}, ℓ_{2}, ℓ_{3}) = (0, 1, 2, 3), Eq. (6.14) is reduced to

ξ_{ϕ}ξ − ξ ξ + ξ ξ = 0 .  (6.15)

Similarly we have

ξ_{ϕ}ξ − ξ ξ + ξ ξ = 0 ,  for (k, ℓ_{1}, ℓ_{2}, ℓ_{3}) = (0, 1, 2, 4) ,  (6.16a)
ξ_{ϕ}ξ − ξ ξ + ξ ξ = 0 ,  for (k, ℓ_{1}, ℓ_{2}, ℓ_{3}) = (0, 1, 3, 4) ,  (6.16b)
ξ ξ − ξ ξ + ξ ξ = 0 ,  for (k, ℓ_{1}, ℓ_{2}, ℓ_{3}) = (0, 2, 3, 4) ,  (6.16c)
ξ ξ − ξ ξ + ξ ξ = 0 ,  for (k, ℓ_{1}, ℓ_{2}, ℓ_{3}) = (1, 2, 3, 4) ,  (6.16d)

respectively.

Generally for an arbitrary m, by choosing k_{1} < k_{2} < … < k_{m−1} and ℓ_{1} < ℓ_{2} < … < ℓ_{m+1}, and applying the Laplace expansion to

| 0  \begin{matrix}ξ_{ℓ_{1}}^{(1)} & ξ_{ℓ_{2}}^{(1)} & … & ξ_{ℓ_{m+1}}^{(1)} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ ξ_{ℓ_{1}}^{(m)} & ξ_{ℓ_{2}}^{(m)} & … & ξ_{ℓ_{m+1}}^{(m)} \end{matrix}^{\begin{matrix}ξ_{k_{1}}^{(1)} & ξ_{k_{2}}^{(1)} & … & ξ_{k_{m−1}}^{(1)} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ ξ_{k_{1}}^{(m)} & ξ_{k_{2}}^{(m)} & … & ξ_{k_{m−1}}^{(m)} \end{matrix} \begin{matrix}ξ_{ℓ_{1}}^{(1)} & ξ_{ℓ_{2}}^{(1)} & … & ξ_{ℓ_{m+1}}^{(1)} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ ξ_{ℓ_{1}}^{(m)} & ξ_{ℓ_{2}}^{(m)} & … & ξ_{ℓ_{m+1}}^{(m)} \end{matrix}} | = 0 ,  (6.17)

we obtain

∑_{i=1}^{m+1}(−1)^{δ}ξ_{Y_{1}}ξ_{Y_{2}} = 0 ,  (6.18)
δ = m + i − j ,  (6.19)

where Y_{1} is the Young diagram corresponding to (k_{1}, …, k_{j}, ℓ_{i}, k_{j+1}, …, k_{m−1}) and Y_{2} to (ℓ_{1}, …, ℓ_{i−1}, ℓ_{i+1}, …, ℓ_{m+1}), and where k_{j} < ℓ_{i} < k_{j+1} is satisfied. For all combinations of k_{ν} and ℓ_{ν}, Eq. (6.18) gives an infinite number of constraints on the ξ_{Y} ’s. These constraints are the Plücker relations.

On the other hand, if a function f(t) is given as f(t) = ∑_{Y} S_{Y} ξ_{Y}, and if the ξ_{Y} ’s

satisfy all the Plücker relations, then f(t) becomes the τ-function, that is, f(t) is written in the form of Eq. (6.2). For example, if ξ_{ϕ} = 1, the matrix Ξ in the τ-function may be expressed as

Ξ = \begin{aligned}& m \\ &  \\ ( & \begin{aligned}1 \\ & 1 & & 0 \\ . & . & . \\ 0 & & & 1 \\ … & ξ & −ξ & ξ \\ … & ξ & −ξ & ξ \\ ⋯ & ⋮ & ⋮ & ⋮ \end{aligned} & ) \end{aligned} .  (6.20)

We have shown that the ξ_{Y} ’s in Eq. (6.8) satisfy an infinite number of bilinear relations. We now show that the τ-function itself also satisfies partial differential equations with the same form as the Plücker relations.

From the definition of τ(t), Eq. (6.2), we may write

τ(t + s) = det(Ξ_{0}^{t}e^{η(t,Λ)} e^{η(s,Λ)} Ξ) .  (6.21)

If we denote

Ξ(s) = e^{η(s,Λ)} Ξ ,  (6.22)

Eq. (6.21) is rewritten as

τ(t + s) = ∑_{Y} S_{Y} (t) ξ_{Y} (s) ,  (6.23)

where the ξ_{Y} (s)’s are in the same form as Eq. (6.7) and satisfy all the Plücker relations with parameters s = (s_{1}, s_{2}, s_{3}, …). Applying S_{Y} (˜∂_{t}) to Eq. (6.23) and using the orthogonality condition of S_{Y} (see Appendix),

S_{Y} (˜∂_{t})S_{Y′}(t)|_{t=0} = δ_{Y Y′} ,  (6.24)

we obtain

ξ_{Y} (s) = S_{Y} (˜∂_{t})τ(t + s)|_{t=0}
= S_{Y} ( ˜∂_{s})τ(t + s)|_{t=0}
= S_{Y} ( ˜∂_{s})τ(s) .  (6.25)

Substitution of Eq. (6.25) into Eq. (6.18) yields

∑(−1)^{δ}{S_{Y_{1}}(˜∂_{t})τ(t)}{S_{Y_{2}}(˜∂_{t})τ(t)} = 0 ,  (6.26)

where δ is given by Eq. (6.19). Equation (6.26) is the bilinear equation for the τ-function. For example, Eq. (6.15) gives

S_{ϕ}(˜∂_{t})τ(t) S (˜∂_{t})τ(t) − S (˜∂_{t})τ(t) S (˜∂_{t})τ(t) + S (˜∂_{t})τ(t) S (˜∂_{t})τ(t) = 0 . (6.27)

Noticing that S_{ϕ} = 1, S = t_{1}, S = t_{1}^{2}/2 + t_{2}, S = t_{1}^{2}/2 − t_{2}, S = t_{1}^{3}/3 − t_{3} and S = t_{1}^{4}/12 − t_{1}t_{3} + t_{2}^{2}, we obtain

τ(t) \frac{1}{12}(\frac{∂^{4}}{∂t_{1}^{4}} − 4\frac{∂^{2}}{∂t_{1}∂t_{3}} + 3\frac{∂^{2}}{∂t_{2}^{2}} )τ(t) − \frac{∂}{∂t_{1}} τ(t) \frac{1}{3}(\frac{∂^{3}}{∂t_{1}^{3}} − \frac{∂}{∂t_{3}} )τ(t)
+ \frac{1}{2}(\frac{∂^{2}}{∂t_{1}^{2}} + \frac{∂}{∂t_{2}} )τ(t) \frac{1}{2}(\frac{∂^{2}}{∂t_{1}^{2}} − \frac{∂}{∂t_{2}} )τ(t) = 0 ,  (6.28)

which is essentially the same as the bilinear form of the KP equation.

It is also possible to directly derive the bilinear equations which are written by Hirota’s operators defined by Eq. (1.11). For the purpose, we evaluate

∑_{ℓ_{ν}} a_{1}^{ℓ_{1}}a_{2}^{ℓ_{2}} … a_{m}^{ℓ_{m}}S_{Y} (t) = ∑_{ℓ_{ν}} a_{1}^{ℓ_{1}}a_{2}^{ℓ_{2}} … a_{m}^{ℓ_{m}} | \begin{matrix}p_{ℓ_{1}} & p_{ℓ_{2}} & … & p_{ℓ_{m}} \\ p_{ℓ_{1}−1} & p_{ℓ_{2}−1} & … & p_{ℓ_{m}−1} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ p_{ℓ_{1}−m+1} & p_{ℓ_{2}−m+1} & … & p_{ℓ_{m}−m+1} \end{matrix} | ,

where a_{1}, a_{2}, …, a_{m} are arbitrary parameters and the summation is taken over all possible combinations of m nonnegative numbers. In the following, we assume that the set of numbers (ℓ_{1}, ℓ_{2}, …, ℓ_{m}) corresponding to the Young diagram Y need not satisfy the condition ℓ_{1} < ℓ_{2} < … < ℓ_{m}. Even though the assumption may change the signs of S_{Y} and ξ_{Y} defined by Eqs. (6.6) and (6.7), respectively, the essence of the argument below is still kept.

We have

∑_{ℓ_{ν}} a_{1}^{ℓ_{1}}a_{2}^{ℓ_{2}} … a_{m}^{ℓ_{m}}S_{Y} (t)

= ∑_{ℓ_{ν}} | \begin{matrix}a_{1}^{ℓ_{1}}p_{ℓ_{1}} & a_{2}^{ℓ_{2}}p_{ℓ_{2}} & … & a_{m}^{ℓ_{m}}p_{ℓ_{m}} \\ a_{1}a_{1}^{ℓ_{1}−1}p_{ℓ_{1}−1} & a_{2}a_{2}^{ℓ_{2}−1}p_{ℓ_{2}−1} & … & a_{m}a_{m}^{ℓ_{m}−1}p_{ℓ_{m}−1} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ a_{1}^{m−1}a_{1}^{ℓ_{1}−m+1}p_{ℓ_{1}−m+1} & a_{2}^{m−1}a_{2}^{ℓ_{2}−m+1}p_{ℓ_{2}−m+1} & … & a_{m}^{m−1}a_{m}^{ℓ_{m}−m+1}p_{ℓ_{m}−m+1} \end{matrix} |

= | \begin{matrix}∑_{ℓ_{1}=0}^{∞} a_{1}^{ℓ_{1}}p_{ℓ_{1}} & ∑_{ℓ_{2}=0}^{∞} a_{2}^{ℓ_{2}}p_{ℓ_{2}} & … & ∑_{ℓ_{m}=0}^{∞} a_{m}^{ℓ_{m}}p_{ℓ_{m}} \\ a_{1} ∑_{ℓ_{1}=0}^{∞} a_{1}^{ℓ_{1}−1}p_{ℓ_{1}−1} & a_{2} ∑_{ℓ_{2}=0}^{∞} a_{2}^{ℓ_{2}−1}p_{ℓ_{2}−1} & … & a_{m} ∑_{ℓ_{m}=0}^{∞} a_{m}^{ℓ_{m}−1}p_{ℓ_{m}−1} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ a_{1}^{m−1} ∑_{ℓ_{1}=0}^{∞} a_{1}^{ℓ_{1}−m+1}p_{ℓ_{1}−m+1} & a_{2}^{m−1} ∑_{ℓ_{2}=0}^{∞} a_{2}^{ℓ_{2}−m+1}p_{ℓ_{2}−m+1} & … & a_{m}^{m−1} ∑_{ℓ_{m}=0}^{∞} a_{m}^{ℓ_{m}−m+1}p_{ℓ_{m}−m+1} \end{matrix} | .

Noticing Eq. (3.3) and that p_{n} = 0 for n < 0, we find

∑_{ℓ_{ν}} a_{1}^{ℓ_{1}}a_{2}^{ℓ_{2}} … a_{m}^{ℓ_{m}}S_{Y} (t) = | \begin{matrix}e^{η(t,a_{1})} & e^{η(t,a_{2})} & … & e^{η(t,a_{m})} \\ a_{1}e^{η(t,a_{1})} & a_{2}e^{η(t,a_{2})} & … & a_{m}e^{η(t,a_{m})} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ a_{1}^{m−1}e^{η(t,a_{1})} & a_{2}^{m−1}e^{η(t,a_{2})} & … & a_{m}^{m−1}e^{η(t,a_{m})} \end{matrix} |
= Δ(a_{1}, a_{2}, …, a_{m}) exp ∑_{j=1}^{m} η(t, a_{j}) ,  (6.29)

where

Δ(a_{1}, a_{2}, …, a_{m}) ≡ ∏_{1≤i<j≤m}(a_{j} − a_{i})  (6.30)

is the Vandermonde’s determinant.

Let us introduce a function ζ(t; a_{1}, a_{2}, …, a_{m}) by

ζ(t; a_{1}, a_{2}, …, a_{m}) = Δ(a_{1}, a_{2}, …, a_{m}) (exp ∑_{j=1}^{m} η(˜∂_{t}, a_{j})) τ(t) .  (6.31)

From Eq. (6.29), we have

ζ(t; a_{1}, a_{2}, …, a_{m}) = ∑_{ℓ_{ν}} a_{1}^{ℓ_{1}}a_{2}^{ℓ_{2}} … a_{m}^{ℓ_{m}}S_{Y} (˜∂_{t})τ(t) .  (6.32)

By using Eq. (6.25), Eq. (6.32) is reduced to

ζ(t; a_{1}, a_{2}, …, a_{m}) = ∑_{ℓ_{ν}} a_{1}^{ℓ_{1}}a_{2}^{ℓ_{2}} … a_{m}^{ℓ_{m}}ξ_{Y} (t) ,  (6.33)

which shows that ζ plays the role of the generating function for ξ_{Y} (t).

From Eqs. (6.18) and (6.33), we have the following identity:

∑_{i=1}^{m+1}(−1)^{i−1}ζ(t; b_{1}, …, b_{m−1}, a_{i})ζ(t; a_{1}, …, a_{i−1}, a_{i+1}, …, a_{m+1})

= ∑_{i=1}^{m+1}(−1)^{i−1} ∑ b_{1}^{k_{1}} … b_{m−1}^{k_{m−1}}a_{i}^{ℓ_{i}}(−1)^{m−j−1}ξ_{Y_{1}}(t)
× ∑ a_{1}^{ℓ_{1}} … a_{i−1}^{ℓ_{i−1}}a_{i+1}^{ℓ_{i+1}} … a_{m+1}^{ℓ_{m+1}}ξ_{Y_{2}}(t)
= ∑ b_{1}^{k_{1}} … b_{m−1}^{k_{m−1}}a_{1}^{ℓ_{1}} … a_{m+1}^{ℓ_{m+1}} ∑_{i=1}^{m+1}(−1)^{δ}ξ_{Y_{1}}(t)ξ_{Y_{2}}(t)
=0 ,  (6.34)

where δ is defined by Eq. (6.19). It is noted that the Young diagrams Y_{1} and Y_{2} are the same as those in Eq. (6.18).

On the other hand, Eq. (6.31) gives

∑_{i=1}^{m+1}(−1)^{i−1}ζ(t; b_{1}, …, b_{m−1}, a_{i})ζ(t; a_{1}, …, a_{i−1}, a_{i+1}, …, a_{m+1})

= ∑_{i=1}^{m+1}(−1)^{i−1}Δ(b_{1}, …, b_{m−1}, a_{i}) exp( ∑_{n=1}^{m−1} η(˜∂_{t}, b_{n}) + η(˜∂_{t}, a_{i}))τ(t)
× Δ(a_{1}, …, a_{i−1}, a_{i+1}, …, a_{m+1}) exp( ∑_{n=1}^{m+1} η(˜∂_{t}, a_{n}) − η(˜∂_{t}, a_{i}))τ(t) . (6.35)

Hence we have

∑_{i=1}^{m+1}(−1)^{i−1}Δ(b_{1}, …, b_{m−1}, a_{i})Δ(a_{1}, …, a_{i−1}, a_{i+1}, …, a_{m+1})

× exp(\frac{1}{2} ∑_{n=1}^{m−1} η(˜∂_{t}, b_{n}) − \frac{1}{2} ∑_{n=1}^{m+1} η(˜∂_{t}, a_{n}) + η(˜∂_{t}, a_{i})) exp(\frac{1}{2} ∑_{n=1}^{m−1} η(˜∂_{t}, b_{n}) + \frac{1}{2} ∑_{n=1}^{m+1} η(˜∂_{t}, a_{n}))τ(t)

× exp(−\frac{1}{2} ∑_{n=1}^{m−1} η(˜∂_{t}, b_{n}) + \frac{1}{2} ∑_{n=1}^{m+1} η(˜∂_{t}, a_{n}) − η(˜∂_{t}, a_{i})) exp(\frac{1}{2} ∑_{n=1}^{m−1} η(˜∂_{t}, b_{n}) + \frac{1}{2} ∑_{n=1}^{m+1} η(˜∂_{t}, a_{n}))τ(t)

= 0 .  (6.36)

Noticing that

exp(\frac{1}{2} ∑_{n=1}^{m−1} η(˜∂_{t}, b_{n}) + \frac{1}{2} ∑_{n=1}^{m+1} η(˜∂_{t}, a_{n}))τ(t)

=τ(t_{1} + \frac{1}{2}( ∑_{n=1}^{m−1} b_{n} + ∑_{n=1}^{m+1} a_{n}), …, t_{j} + \frac{1}{2j} ( ∑_{n=1}^{m−1} b_{n}^{j} + ∑_{n=1}^{m+1} a_{n}^{j}), …) ,  (6.37)

and rewriting t_{j} for t_{j} + \frac{1}{2j} ( ∑_{n=1}^{m−1} b_{n}^{j} + ∑_{n=1}^{m+1} a_{n}^{j}), we have from Eq. (6.36) that

∑_{i=1}^{m+1}(−1)^{i−1}Δ(b_{1}, …, b_{m−1}, a_{i})Δ(a_{1}, …, a_{i−1}, a_{i+1}, …, a_{m+1})

× exp( ∑_{n=1}^{m−1} η(\frac{1}{2} ˜∂_{s}, b_{n}) + η(\frac{1}{2} ˜∂_{s}, a_{i})) exp( ∑_{n=1}^{m+1} η(−\frac{1}{2} ˜∂_{s}, a_{n}) + η(\frac{1}{2} ˜∂_{s}, a_{i}))τ(t + s)τ(t − s)|_{s=0}

=0 .  (6.38)

Then, by means of Eq. (6.29), we obtain

∑_{i=1}^{m+1}(−1)^{i−1} ∑ b_{1}^{k_{1}} … b_{m−1}^{k_{m−1}}a_{1}^{ℓ_{1}} … a_{m+1}^{ℓ_{m+1}}S_{Y_{1}}(\frac{1}{2} ˜∂_{s})S_{Y_{2}}(−\frac{1}{2} ˜∂_{s})

× τ(t + s)τ(t − s)|_{s=0} = 0 .  (6.39)

Since each of the coefficient of b_{1}^{k_{1}} … b_{m−1}^{k_{m−1}}a_{1}^{ℓ_{1}} … a_{m+1}^{ℓ_{m+1}} has to be zero, we get

∑_{i=1}^{m+1} (−1)^{i−1}S_{Y_{1}}(\frac{1}{2} ˜∂_{s})S_{Y_{2}}(−\frac{1}{2} ˜∂_{s})τ(t + s)τ(t − s)|_{s=0} = 0 .  (6.40)

If Hirota’s operators defined by Eq. (1.11) are employed, Eq. (6.40) reduces to

∑_{i=1}^{m+1}(−1)^{i−1}S_{Y_{1}}(\frac{1}{2}D̃_{t})S_{Y_{2}}(−\frac{1}{2}D̃_{t})τ(t) · τ(t) = 0 .  (6.41)

For example, the Plücker relation (6.15) gives

(4D_{t_{1}}D_{t_{3}} − D_{t_{1}}^{4} − 3D_{t_{2}}^{2})τ(t) · τ(t) = 0 ,  (6.42)

which is nothing but the bilinear form of the KP equation, Eq. (1.9). Similarly, Eqs. (6.16a)∼(6.16d) give

(D_{t_{1}}^{3}D_{t_{2}} − 3D_{t_{1}}D_{t_{4}} + 2D_{t_{2}}D_{t_{3}})τ(t) · τ(t) = 0 ,  (6.43a)
(D_{t_{1}}^{6} + 4D_{t_{1}}^{3}D_{t_{3}} − 9D_{t_{1}}^{2}D_{t_{2}}^{2} + 36D_{t_{2}}D_{t_{4}} − 32D_{t_{3}}^{2})τ(t) · τ(t) = 0 ,  (6.43b)
D_{t_{1}}(D_{t_{1}}^{2}D_{t_{4}} − 2D_{t_{1}}D_{t_{2}}D_{t_{3}} + D_{t_{2}}^{3})τ(t) · τ(t) = 0 ,  (6.43c)
(D_{t_{1}}^{8} − 20D_{t_{1}}^{5}D_{t_{3}} + 15D_{t_{1}}^{4}D_{t_{2}}^{2} − 72D_{t_{1}}^{2}D_{t_{2}}D_{t_{4}} + 64D_{t_{1}}^{2}D_{t_{3}}^{2}
+ 48D_{t_{1}}D_{t_{2}}^{2}D_{t_{3}} − 36D_{t_{2}}^{4})τ(t) · τ(t) = 0 , (6.43d)

respectively. These equations form a subset of the KP hierarchy.

§7. Explicit expression of solutions by the τ-function

As mentioned in §3, the solutions of the Sato equation can be written in terms of certain derivatives of the τ-function. Then, through the transformations (4.3), the solutions of the generalized Lax equation are expressed by the τ-function. Moreover, from Eq. (5.16), the eigenfunction of the linear system (5.1)∼(5.3) is also expressed by the τ-function. In this section, we give explicit expression of those functions.

Following Freeman and Nimmo^{8)}, we introduce the notation,

| ℓ_{1}, ℓ_{2}, …, ℓ_{m}| = | \begin{matrix}h_{ℓ_{1}}^{(1)} & h_{ℓ_{1}}^{(2)} & … & h_{ℓ_{1}}^{(m)} \\ h_{ℓ_{2}}^{(1)} & h_{ℓ_{2}}^{(2)} & … & h_{ℓ_{2}}^{(m)} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ h_{ℓ_{m}}^{(1)} & h_{ℓ_{m}}^{(2)} & … & h_{ℓ_{m}}^{(m)} \end{matrix} | ,  (7.1)

where h_{n}^{(j)} satisfies Eq. (3.10). Then, the τ-function (6.1) and the solutions (3.14) of the Sato equation are simply written by

τ(x; t) = | 0, 1, …, m − 1| ,  (7.2)

and

w_{j}(x; t) = (−1)^{j} \frac{| 0, 1, …, m − j − 1, m − j + 1, …, m|}{| 0, 1, …, m − 1|} ,  (7.3)

respectively.

By using the Vandermonde’s determinant defined by Eq. (6.30), the τ-function may also be expressed as

τ(x; t) = Δ(∂_{x_{1}}, ∂_{x_{2}}, …, ∂_{x_{m}}) h_{0}^{(1)}(x_{1}; t)h_{0}^{(2)}(x_{2}; t) … h_{0}^{(m)}(x_{m}; t)|_{x_{1},...,x_{m}=x} . (7.4)

If we differentiate Eq. (7.4) by t_{ℓ}, we obtain

\frac{∂τ}{∂t_{ℓ}} = ∑_{k=1}^{m} ∂_{x_{k}}^{ℓ}Δ(∂_{x_{1}}, …, ∂_{x_{m}})h_{0}^{(1)}(x_{1}; t) … h_{0}^{(m)}(x_{m}; t)|_{x_{1},...,x_{m}=x} ,  (7.5)

where we have used Eq. (3.10).

Let us operate S_{Y} (˜∂_{t}) on Eq. (7.4) for a certain Young diagram Y. From Eq. (A.20), we have

S_{Y} (˜∂_{t})τ = ∑_{ρ} χ_{ρ}^{Y} \frac{∂_{t_{1}}^{α_{1}} … ∂_{t_{m}}^{α_{m}}}{1^{α_{1}} … m^{α_{m}}α_{1}! ⋯ α_{m}!}τ .  (7.6)

By using Eqs. (7.5) and (A.5), Eq. (7.6) is reduced to

S_{Y} (˜∂_{t})τ = \frac{1}{m!} ∑_{ρ} χ_{ρ}^{Y} h_{ρ}(∑_{k=1}^{m} ∂_{x_{k}})^{α_{1}} … (∑_{k=1}^{m} ∂_{x_{k}}^{m})^{α_{m}}
× Δ(∂_{x_{1}}, …, ∂_{x_{m}})h_{0}^{(1)}(x_{1}; t) … h_{0}^{(m)}(x_{m}; t)|_{x_{1},...,x_{m}=x} .  (7.7)

If we take ϵ_{i} = ∂_{x_{i}} in Eq. (A.15), we have

(∑_{k=1}^{m} ∂_{x_{k}})^{α_{1}} … (∑_{k=1}^{m} ∂_{x_{k}}^{m})^{α_{m}} = ∑_{|Y |=m} χ_{ρ}^{Y} S̃_{Y} (∂_{x}) ,  (7.8)

where ∂_{x} means (∂_{x_{1}}, ∂_{x_{2}}, …, ∂_{x_{m}}). We then obtain

S_{Y} (˜∂_{t})τ =\frac{1}{m!} ∑_{|Y′|=m} ∑_{ρ} h_{ρ}χ_{ρ}^{Y} χ_{ρ}^{Y′}
× S̃_{Y′}(∂_{x})Δ(∂_{x_{1}}, …, ∂_{x_{m}}) h_{0}^{(1)}(x_{1}; t) … h_{0}^{(m)}(x_{m}; t)|_{x_{1},...,x_{m}=x} . (7.9)

The orthogonality condition for the irreducible characters, Eq. (A.4), yields

S_{Y} (˜∂_{t})τ = S̃_{Y} (∂_{x})Δ(∂_{x_{1}}, …, ∂_{x_{m}})h_{0}^{(1)}(x_{1}; t) … h_{0}^{(m)}(x_{m}; t)|_{x_{1},...,x_{m}=x} .  (7.10)

By taking ϵ_{i} = ∂_{x_{i}} in Eq. (A.13) and substituting S̃_{Y} (∂_{x}) into Eq. (7.10), we have

S_{Y} (˜∂_{t})τ = | \begin{matrix}∂_{x_{1}}^{ν_{m}} & ∂_{x_{1}}^{ν_{m−1}+1} & … & ∂_{x_{1}}^{ν_{1}+m−1} \\ ∂_{x_{2}}^{ν_{m}} & ∂_{x_{2}}^{ν_{m−1}+1} & … & ∂_{x_{2}}^{ν_{1}+m−1} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ ∂_{x_{m}}^{ν_{m}} & ∂_{x_{m}}^{ν_{m−1}+1} & … & ∂_{x_{m}}^{ν_{1}+m−1} \end{matrix} | h_{0}^{(1)}(x_{1}; t) … h_{0}^{(m)}(x_{m}; t)|_{x_{1},...,x_{m}=x}

= | ν_{m}, ν_{m−1} + 1, …, ν_{1} + m − 1| ,  (7.11)

where [ν_{1}, ν_{2}, …, ν_{m}] is the partition corresponding to the Young diagram Y. If we consider the partition [\underbrace{1, 1, …, 1}_{j}, \underbrace{0, 0, …, 0}_{m−j}] or the Young diagram \} j, then

Eq. (7.11) gives

S _{}j}(˜∂_{t}) τ = | 0, 1, …, m − j − 1, m − j + 1, …, m| .  (7.12)

Substituting Eq. (7.12) into Eq. (7.3), we find

w_{j} = (−1)^{j} \frac{1}{τ} S _{\}j}(˜∂_{t}) τ .  (7.13)

This is an explicit expression for the solutions of the Sato equation in terms of the τfunction.

The expression (7.13) may be simplified by using some properties of the Schur function. From Eq. (A.18), we have

S _{\}j}(˜∂_{t}) = \frac{1}{j!} ∑_{ρ} h_{ρ}χ_{ρ}^{\}j} (\frac{∂}{∂t_{1}} )^{α_{1}}(\frac{∂}{∂t_{2}} )^{α_{2}} ⋯ (\frac{∂}{∂t_{j}})^{α_{j}} ,  (7.14)

and

S_{j}(−˜∂_{t}) = \frac{1}{j!} ∑_{ρ} h_{ρ}χ\underbrace{j}_{ρ}
× (−1)^{α_{1}+α_{2}+···+α_{j}}(\frac{∂}{∂t_{1}} )^{α_{1}}(\frac{∂}{∂t_{2}} )^{α_{2}} ⋯ (\frac{∂}{∂t_{j}})^{α_{j}} .  (7.15)

By using the relation between the irreducible characters χ_{ρ} and χ_{ρ}, Eq. (A.2), and noticing that

S_{j}(t) = p_{j}(t) ,  (7.16)

we finally obtain

w_{j} = \frac{1}{τ} p_{j}(−˜∂_{t}) τ ,  (7.17)

which is the simplified expression for the solutions of the Sato equation. For example, we have

w_{1} = −\frac{1}{τ} \frac{∂τ}{∂x} ,  (7.18a)
w_{2} = \frac{1}{2τ} (\frac{∂^{2}τ}{∂x^{2}} − \frac{∂τ}{∂t_{2}} ) ,  (7.18b)
w_{3} = −\frac{1}{6τ} (\frac{∂^{3}τ}{∂x^{3}} − 3\frac{∂^{2}τ}{∂x∂t_{2}} + 2\frac{∂τ}{∂t_{3}} ) .  (7.18c)

The solutions of the generalized Lax equation can also be expressed in terms of the τ-function through the transformation (4.3). For example,

u_{2} = \frac{∂^{2}}{∂x^{2}} log τ ,  (7.19a)

u_{3} = \frac{1}{2}(−\frac{∂^{3}}{∂x^{3}} + \frac{∂^{2}}{∂x∂t_{2}} ) log τ ,  (7.19b)
u_{4} = \frac{1}{6}(\frac{∂^{4}}{∂x^{4}} − 3\frac{∂^{3}}{∂x^{2}∂t_{2}} + 2\frac{∂^{2}}{∂x∂t_{3}} ) log τ − (\frac{∂^{2}}{∂x^{2}} log τ)^{2} .  (7.19c)

Especially, Eq. (7.19a) is nothing but the dependent variable transformation (1.10) to reduce the KP equation to its bilinear form.

It is also possible to express the eigenfunction (5.16) in terms of the τ-function. Substitution of Eq. (7.17) into Eq. (5.16) yields

ψ = \frac{1}{τ} \{τ + \frac{p_{1}(−˜∂_{t})τ}{λ} + \frac{p_{2}(−˜∂_{t})τ}{λ^{2}} + ⋯\} e^{t_{0}+λt_{1}+λ^{2}t_{2}+···}

= \frac{1}{τ} \{( ∑_{ν_{1}=0}^{∞} \frac{1}{ν_{1}!}(\frac{−∂_{t_{1}}}{λ})^{ν_{1}} ∑_{ν_{2}=0}^{∞} \frac{1}{ν_{2}!}(\frac{−∂_{t_{2}}}{2λ^{2}} )^{ν_{2}} ⋯) τ\} e^{t_{0}+λt_{1}+λ^{2}t_{2}+···}

= \frac{1}{τ} \{exp(−\frac{1}{λ}∂_{t_{1}} − \frac{1}{2λ^{2}} ∂_{t_{2}} − ⋯)τ\} e^{t_{0}+λt_{1}+λ^{2}t_{2}+···} .  (7.20)

Thus we obtain

ψ = \frac{τ(t_{1} − \frac{1}{λ}, t_{2} − \frac{1}{2λ^{2}} , …)}{τ(t_{1}, t_{2}, …)}e^{t_{0}+λt_{1}+λ^{2}t_{2}+···} .  (7.21)

§8. Concluding remarks

We have made an introduction to Sato theory in the preceding sections. It has been shown that the Sato equation generates the generalized Lax equation, the Zakharov-Shabat equation and the IST scheme. It has also been shown that an infinite number of nonlinear evolution equations (the KP hierarchy), of which the KP equation is the simplest nontrivial one, share solutions. The τ-function is the key function to express the solutions. By employing the results of the representation theory of groups, we have shown that the partial differential equations governing the τ-function are closely related to the Plücker relations and may be written in bilinear form. The solutions of the Sato equation, and consequently those of the generalized Lax equation and the eigenfunction in the IST scheme, are explicitly expressed by the τ-function.

Since the aim of this paper is the elementary introduction to Sato theory, we did not include any further results. The extension to the multi-component systems includes the nonlinear evolution equations such as the nonlinear Schrödinger equation and the sine-Gordon equation. The theory also applies to discrete systems such as the Toda equation. Date, Jimbo, Kashiwara and Miwa have extended this theory by using the method of field theory. In this approach, the relationship between the soliton equations and the infinite dimensional Lie algebra becomes apparent. Moreover, the vertex operator is shown to have a close relation to the Bäcklund transformation. Interested readers may refer to Refs. 10)∼14) and 16).

Acknowledgements

The authors would like to express their sincere thanks to Professor R. Hirota for valuable discussions. Thanks are also due to Dr. J. Sachs for his editorial assistance.

Appendix : Results from the representation theory of groups

In this appendix, we present some results of the representation theory of groups which are useful to discuss the structures of the τ-function.

A-1 Irreducible character

We consider the symmetric group S_{m}, i.e., the permutation group of m numbers. The number of elements is m!. Example The elements of S_{3} are

( \begin{matrix}1 & 2 & 3 \\ 2 & 3 & 1 \end{matrix} ) ,  ( \begin{matrix}1 & 2 & 3 \\ 3 & 1 & 2 \end{matrix} ) ,  ( \begin{matrix}1 & 2 & 3 \\ 1 & 3 & 2 \end{matrix} ) ,
( \begin{matrix}1 & 2 & 3 \\ 3 & 2 & 1 \end{matrix} ) ,  ( \begin{matrix}1 & 2 & 3 \\ 2 & 1 & 3 \end{matrix} ) ,  ( \begin{matrix}1 & 2 & 3 \\ 1 & 2 & 3 \end{matrix} ) .

The elements of S_{m} are classified into the classes (1^{α_{1}}, 2^{α_{2}}, …, m^{α_{m}}), where α_{1} + 2α_{2} + ⋯ + mα_{m} = m. Example For S_{3}, we have

(3^{1}) ∋  ( \begin{matrix}1 & 2 & 3 \\ 2 & 3 & 1 \end{matrix} ) , ( \begin{matrix}1 & 2 & 3 \\ 3 & 1 & 2 \end{matrix} ) ,
(1^{1}, 2^{1}) ∋  ( \begin{matrix}1 & 2 & 3 \\ 1 & 3 & 2 \end{matrix} ) , ( \begin{matrix}1 & 2 & 3 \\ 3 & 2 & 1 \end{matrix} ) , ( \begin{matrix}1 & 2 & 3 \\ 2 & 1 & 3 \end{matrix} ) ,
(1^{3}) ∋  ( \begin{matrix}1 & 2 & 3 \\ 1 & 2 & 3 \end{matrix} ) .

Irreducible representation of S_{m} is characterized by the partition [λ], where

[λ] = [λ_{1}, λ_{2}, …, λ_{m}]

means the set of numbers satisfying

λ_{1} + λ_{2} + ⋯ + λ_{m} = m ,
λ_{1} ≥ λ_{2} ≥ … ≥ λ_{m} ≥ 0 .

To each partition [λ] there corresponds a Young diagram.

Example For S_{3}, we have

\begin{aligned}3 + 0 + 0 & ⟷ & [3, 0, 0] & or & [3] & ⟷ & & & , \\ 2 + 1 + 0 & ⟷ & [2, 1, 0] & or & [2, 1] & ⟷ & & , \\ 1 + 1 + 1 & ⟷ & [1, 1, 1] & or & [1^{3}] & ⟷ & . \end{aligned}

The partition has the one to one correspondence to the set of numbers (ℓ_{1}, ℓ_{2}, …, ℓ_{m}) introduced in §6. Namely,

[λ] ⟷ (ℓ_{1}, ℓ_{2} − 1, …, ℓ_{m} − m + 1) ⟷ (ℓ_{1}, ℓ_{2}, …, ℓ_{m})

Example For S_{3}, we have

\begin{aligned}[3] & ⟷ & (0, 0, 3) & ⟷ & (0, 1, 5) , \\ [2, 1] & ⟷ & (0, 1, 2) & ⟷ & (0, 2, 4) , \\ [1^{3}] & ⟷ & (1, 1, 1) & ⟷ & (1, 2, 3) . \end{aligned}

If a is an element of S_{m} belonging to the class (1^{α_{1}}, 2^{α_{2}}, …, m^{α_{m}}), then we have a relation,

t_{1}^{α_{1}}t_{2}^{α_{2}} … t_{m}^{α_{m}} = \frac{1}{1^{α_{1}}2^{α_{2}} … m^{α_{m}}} ∑_{|Y |=m} χ^{Y} (a)S_{Y} (t) ,  (A.1)

where S_{Y} is defined by Eq. (6.6), χ^{Y} (a) is the irreducible character, |Y | is the size (the number of ’s) of the Young diagram, and the summation is taken over all the Young diagrams of size m. Example If a ∈ (3^{1}), then we have

t_{3}^{1} = \frac{1}{3^{1}} \{χ (a)S (t) + χ (a)S (t) + χ (a)S (t)\} .

The values of the irreducible characters are determined by Eq. (A.1). The values for all elements belonging to the same class are identical. Hence χ^{Y} (a) may be written as χ_{ρ}^{Y}, where ρ is the class to which a belongs. Example For S_{3}, we have the following chart of the irreducible characters:

class

number of elements irreducible character

χ  χ  χ
(1^{3})  1  1  2  1
(1^{1}, 2^{1})  3  1  0  −1
(3^{1})  2  1  −1  1

From the definition of the irreducible characters, we especially have

χ\underbrace{m}_{ρ} = (−1)^{m−∑_{j=1}^{m} α_{j}}χ_{ρ}^{\}m} = 1 .  (A.2)

There exists the orthogonality relation of the first kind among the irreducible characters,

\frac{1}{m!} ∑_{a∈S_{m}} χ^{Y} (a)χ^{Y′}(a) = δ_{Y Y′} ,  (A.3)

where the summation is taken over all the elements of S_{m}. Since the irreducible characters are identical for all elements belonging to the same class ρ, Eq. (A.3) may be rewritten by

\frac{1}{m!} ∑_{ρ} h_{ρ}χ_{ρ}^{Y} χ_{ρ}^{Y′} = δ_{Y Y′} ,  (A.4)

where

h_{ρ} = \frac{m!}{1^{α_{1}}2^{α_{2}} ⋯ m^{α_{m}}α_{1}!α_{2}! ⋯ α_{m}!}  (A.5)

is the number of elements belonging to the class ρ.

A-2 Schur function

Let us define F (x) and φ_{j}, j = 1, 2, …, by
F (x) = ∏_{ν=1}^{m}(1 − ϵ_{ν}x)^{−1} = 1 + φ_{1}x + φ_{2}x^{2} + · · · .  (A.6)

By equating the same powers of x in Eq. (A.6), we obtain

φ_{j}(ϵ) = φ_{j}(ϵ_{1}, ϵ_{2}, …, ϵ_{m}) = ∑_{λ_{1}+λ_{2}+···+λ_{m}=j} ϵ_{1}^{λ_{1}}ϵ_{2}^{λ_{2}} … ϵ_{m}^{λ_{m}} .  (A.7)

On the other hand, we have

F′(x)/F (x) = t_{1} + 2t_{2}x + 3t_{3}x^{2} + ⋯,  (A.8)

where

t_{ℓ} = \frac{1}{ℓ}(ϵ_{1}^{ℓ} + ϵ_{2}^{ℓ} + ⋯ + ϵ_{m}^{ℓ}) .  (A.9)

Integration of Eq. (A.8) yields

F (x) = exp ∑_{ℓ=1}^{∞} t_{ℓ}x^{ℓ} ,  (A.10)

which, by means of Eq. (3.3), reduces to

F (x) = ∑_{ℓ=0}^{∞} p_{ℓ}x^{ℓ} .  (A.11)

Then, from Eqs. (A.6) and (A.11), we find

φ_{j}(ϵ) = p_{j}(t) .  (A.12)

The Schur function is usually defined by

S̃_{[ν]}(ϵ) = \begin{matrix}| & \begin{matrix}ϵ_{1}^{ν_{m}} & ϵ_{1}^{ν_{m−1}+1} & … & ϵ_{1}^{ν_{1}+m−1} \\ ϵ_{2}^{ν_{m}} & ϵ_{2}^{ν_{m−1}+1} & … & ϵ_{2}^{ν_{1}+m−1} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ ϵ_{m}^{ν_{m}} & ϵ_{m}^{ν_{m−1}+1} & … & ϵ_{m}^{ν_{1}+m−1} \end{matrix} & | \\ | & \begin{matrix}ϵ_{1}^{0} & ϵ_{1}^{1} & … & ϵ_{1}^{m−1} \\ ϵ_{2}^{0} & ϵ_{2}^{1} & … & ϵ_{2}^{m−1} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ ϵ_{m}^{0} & ϵ_{m}^{1} & … & ϵ_{m}^{m−1} \end{matrix} & | \end{matrix} ,  (A.13)

where [ν] means the partition [ν_{1}, ν_{2}, …, ν_{m}] (See for example Ref. 17)). It is noted that the denominator is the Vandermonde’s determinant Δ(ϵ_{1}, ϵ_{2}, …, ϵ_{m}). Equation (A.13) may be rewritten as

S̃_{Y} (ϵ) = | \begin{matrix}φ_{ν_{m}} & φ_{ν_{m−1}+1} & … & φ_{ν_{1}+m−1} \\ φ_{ν_{m}−1} & φ_{ν_{m−1}} & … & φ_{ν_{1}+m−2} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ φ_{ν_{m}−m+1} & φ_{ν_{m−1}−m+2} & … & φ_{ν_{1}} \end{matrix} | ,  (A.14)

where Y is the Young diagram corresponding to the partition [ν].

By substituting Eq. (A.12) into Eq. (A.14), we find that S̃_{Y} (ϵ) is equal to S_{Y} (t) defined by Eq. (6.6). Then, by using Eq. (A.9), we see that Eq. (A.1) is equivalent to

(∑_{ℓ=1}^{m} ϵ_{ℓ})^{α_{1}}(∑_{ℓ=1}^{m} ϵ_{ℓ}^{2})^{α_{2}} ⋯ (∑_{ℓ=1}^{m} ϵ_{ℓ}^{m})^{α_{m}} = ∑_{|Y |=m} χ_{ρ}^{Y} S̃_{Y} (ϵ) .  (A.15)

A-3 Expansion of functions in S_{Y} (t)

An arbitrary analytic function f(t) may be written by

f(t) = ∑_{m=0}^{∞} ∑_{α_{1}+2α_{2}+···+mα_{m}=m} C^{(m)}(α_{1}, α_{2}, …, α_{m})t_{1}^{α_{1}}(2t_{2})^{α_{2}}(3t_{3})^{α_{3}} … (mt_{m})^{α_{m}} ,

(A.16)

where C^{(m)}(α_{1}, α_{2}, …, α_{m}) is constant. It is clear that all the elements of the set {t_{1}^{α_{1}} (2t_{2})^{α_{2}} (3t_{3})^{α_{3}} …} are linearly independent.

From Eq. (A.1), we have

t_{1}^{α_{1}}(2t_{2})^{α_{2}} … (mt_{m})^{α_{m}} = ∑_{|Y |=m} χ_{ρ}^{Y} S_{Y} (t) .  (A.17)

The orthogonality relation (A.4) gives

S_{Y} (t) = \frac{1}{m!} ∑_{ρ} h_{ρ}χ_{ρ}^{Y} t_{1}^{α_{1}}(2t_{2})^{α_{2}} … (mt_{m})^{α_{m}} .  (A.18)

Equations (A.17) and (A.18) show that there exists a linear transformation between {S_{Y} (t); |Y | = m} and {t_{1}^{α_{1}} (2t_{2})^{α_{2}} … (mt_{m})^{α_{m}}; α_{1} + 2α_{2} + ⋯ + mα_{m} = m}. Hence we find that f(t) can be uniquely expressed in S_{Y} (t) as

f(t) = ∑_{Y} S_{Y} (t)c_{Y} .  (6.10)

Substituting Eq. (A.5) into Eq. (A.18), we have

S_{Y} (t) = ∑_{ρ} χ_{ρ}^{Y} \frac{t_{1}^{α_{1}}t_{2}^{α_{2}} ⋯ t_{m}^{α_{m}}}{α_{1}!α_{2}! ⋯ α_{m}!} ,  (A.19)

which gives

S_{Y} (˜∂_{t}) = ∑_{ρ} χ_{ρ}^{Y} \frac{∂_{t_{1}}^{α_{1}}∂_{t_{2}}^{α_{2}} ⋯ ∂_{t_{m}}^{α_{m}}}{1^{α_{1}}2^{α_{2}} ⋯ m^{α_{m}}α_{1}!α_{2}! ⋯ α_{m}!} .  (A.20)

Then, using the orthogonality relation (A.4), we find

S_{Y} (˜∂_{t})S_{Y′}(t)|_{t=0} = δ_{Y Y′} .  (6.24)

Furthermore, if we apply Eq. (6.24) to Eq. (6.10), we obtain

c_{Y} = S_{Y} (˜∂_{t}) f(t)|_{t=0} .  (6.11)

References

1) C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Phys. Rev. Lett. 19(1967), 1095.

2) P. D. Lax, Comm. Pure Appl. Math. 21(1968), 467.

3) V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34(1972), 62.

4) M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Phys. Rev. Lett. 30(1973), 1262 ; ibid. 31(1973), 125.

5) V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl. 8(1974), 226.

6) R. Hirota, Phys. Rev. Lett. 27(1971), 1192 ; in Solitons, ed. by R. K. Bullough and P. J. Caudrey (Springer, Berlin 1980).

7) J. Satsuma, J. Phys. Soc. Jpn. 46(1979), 359.

8) N. C. Freeman and J. J. C. Nimmo, Phys. Lett. 95A(1983), 1 : J. J. C. Nimmo and N. C. Freeman, Phys. Lett. 95A(1983), 4 ; ibid. 96A(1983), 443 : N. C. Freeman, IMA J. Appl. Math. 32(1984), 125.

9) M. Sato, RIMS Kokyuroku 439(1981), 30.

10) E. Date, M. Kashiwara, M. Jimbo and T. Miwa, in Non-linear Integrable Systems— Classical Theory and Quantum Theory, ed. by M. Jimbo and T. Miwa (World Scientific, Singapore, 1983), 39.

11) See for example, M. Jimbo and T. Miwa, Publ. RIMS, Kyoto Univ. 19(1983), 943.

12) For example, M. Sato, Notes by M. Noumi, Sophia Kokyuroku in Math. (1984), No. 18 (in Japanese).

13) M. Sato and Y. Sato, in Nonlinear Partial Differential Equations in Applied Science ed. by H. Fujita, P. D. Lax and G. Strang (Kinokuniya / North-Holland, Tokyo, 1983) 259.

14) M. Mulase, Math. Sci. (1982), No. 228 (in Japanese).

15) R. Hirota and J. Satsuma, Phys. Lett. 85A(1981), 407 : J. Satsuma and R. Hirota, J. Phys. Soc. Jpn. 51(1982), 3390.

16) T. Miwa, Proc. Jpn. Acad. 58A(1982), 9.

17) D. E. Littlewood, The Theory of Group Characters (Clarendon, Oxford, 1950).

Figure Captions

Fig. 1 : The chain of cells to construct the Maya diagram. Fig. 2 : The Maya diagram expressing the set of numbers (2, 3, 5, 7). Fig. 3 : The Maya diagram expressing the vacuum state. Fig. 4 : The procedure to construct the Young diagram corresponding to the Maya diagram

of Fig. 2 and the resultant Young diagram.

Fig. 1 : ⋯; ⋯ −5 −4 −3 −2 −1 0 1 2 3 4 5 ⋯; ⋯

Fig. 2 : ⋯; ⋯ −5 ; ◯ −4 ; ◯ −3 −2 −1 ; ◯ 0 ; ◯ 1 2 ; ◯ 3 4 ; ◯ 5 ⋯; ⋯

Fig. 3 : ⋯; ⋯ −5 ; ◯ −4 ; ◯ −3 ; ◯ −2 ; ◯ −1 ; ◯ 0 ; ◯ 1 2 3 4 5 ⋯; ⋯

Fig. 4 :

1 2 3 4 5 →→→→→−⋯ 0↑  ↑4 3 → −1↑  ↑2 1 → −2↑ ↑0 −3↑ ↑−1 →→ −3 −2 −4↑ ↑⋮  ⟹