A remark on the Hankel determinant formula for solutions of the Toda equation

0.2 MB·English

by  Kenji Kajiwara

A Remark on the Hankel Determinant Formula for
Solutions of the Toda Equation
7 Kenji Kajiwara1, Marta Mazzocco2 and Yasuhiro Ohta 3
0
0 1 GraduateSchool ofMathematics, KyushuUniversity,6-10-1Hakozaki,
2 Fukuoka812-8581, Japan
2 School ofMathematics,TheUniversityofManchester, SackvilleStreet,

n Manchester M601QD,UnitedKingdom
a 3 DepartmentofMathematics,KobeUniversity,Rokko,Kobe657-8501,Japan
J

5 Abstract. We consider the Hankel determinant formula of the τ functions of
1
the Toda equation. We present a relationship between the determinant formula

andtheauxiliarylinearproblem,whichischaracterizedbyacompactformulafor
] theτ functionsintheframeworkoftheKPtheory. Similarphenomenathathave
I
S been observed for the Painlev´e II and IV equations are recovered. The case of
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[ AMSclassificationschemenumbers: 37K10,37K30,34M55,34M25,34E05

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Hankel determinant formula for the Toda equation 2
1. Introduction
The Toda equation[24]
d2y
n =eyn−1−yn −eyn−yn+1, (1.1)
dt2
where n∈Z, is one of the most important integrable systems. It can be expressed in
various forms such as
dV dI
n =V (I −I ), n =V −V , (1.2)
dt n n n+1 dt n−1 n
dα dβ
n =α (β −β ), n =2(α2 −α2 ), (1.3)
dt n n+1 n dt n n−1
where the dependent variables are related to y as
n
Vn =eyn−yn+1, In = dyn, αn = 1eyn−2yn+1, βn =−1dyn.(1.4)
dt 2 2 dt
The Toda equation can be reduced to the bilinear equation
τ′′τ −(τ′)2 =τ τ , (1.5)
n n n n+1 n−1
by the dependent variable transformation
τ τ τ d τ
y =log n−1, V = n+1 n−1, I = log n−1. (1.6)
n τ n τ2 n dt τ
n n n
In general, the determinant structure of the τ function (dependent variable of
bilinear equation) is the characteristic property of integrable systems. For example,
the Casoratideterminantformula of the N-soliton solutionof the Toda equation (see,
for example, [25, 4])
τn =et22 det(fn(i+)j−1)i,j=1,...,N, fn(k) =pnk epkt+ηk0 +p−kn ep1kt+ξk0, (1.7)
wherep ,η andξ (k =1,...,N)areconstants,isadirectconsequenceoftheSato
k k0 k0
theory;the solutionspace ofsolitonequationsis the universalGrassmanmanifold, on
which infinite dimensional Lie algebras are acting [9, 15, 25].
If we consider the Toda equation on semi-infinite or finite lattice, the soliton
solutions do not exist but another determinantalsolution arises. For the semi-infinite
case we impose the boundary condition as
τ =0, τ =1, V =0, n≥0. (1.8)
−1 0 0
Then τ admits the Hankel determinant formula[13, 5, 6]
n
τ =det(a ) , a =τ , a =a′ , n∈Z . (1.9)
n i+j−2 i,j=1,...,n 0 1 i i−1 ≥0
The important feature of this determinant formula is that the lattice site n appears
as the determinant size, while for the soliton solutions the determinant size describes
the number of solitons. This type of determinant formula is actually a special case of
thedeterminantforulafortheinfinitelattice[12]. Howeverthemeaningoftheformula
has not been yet fully understood.
The purpose of this article is to establish a characterization of the Hankel
determinant formula of the Toda equation; entries of the matrices in the determinant
formula are closely related to the solution of auxiliary linear problem. Moreover,this
relationshipcanbe describedby a compactformula inthe frameworkof the theory of
KP hierarchy.
In section 2, we discuss the Hankel determinant formula of the infinite Toda
equation and present the relationship between the determinant formula and auxiliary
linear problem. In section 3, we apply the results to the Painlev´e II equation. We
consider the case of finite lattice in section 4.
Hankel determinant formula for the Toda equation 3
2. Hankel determinant formula of the solution of the Toda equation
2.1. Determinant formula and auxiliary linear problem
The Hankel determinant formula for τ satisfying the infinite Toda equation (1.5) is
n
given by as follows:
Proposition 2.1 [12] For fixed k ∈Z, we have:
det(a(k) ) n>0,
τ i+j−2 i,j=1,...,n
k+n = 1 n=0, (2.1)
τ 
k  det(b(i+k)j−2)i,j=1,...,|n| n<0,
 τ i−2 τ
a(k) =a(k)′+ k−1 a(k)a(k) , a(k) = k+1,
i i−1 τ l i−2−l 0 τ
 k k
l=0
 b(k) =b(k)′+ τk+1 iX−2b(k)b(k) , b(k) = τk−1. (2.2)
i i−1 τ l i−2−l 0 τ
k k
We shall now relate the determinant fXol=rm0 ula to the auxiliary linear problem of the
Toda equation (1.2) given by
V Ψ +I Ψ +Ψ =λΨ ,
n−1 n−1 n n n+1 n
dΨ (2.3)
 n =V Ψ ,
 dt n−1 n−1
or

L Ψ =λΨ ,
n n n
dΨ (2.4)
 n =B Ψ ,
 dt n n
where

L =V e−∂n +I +e∂n, B =V e−∂n. (2.5)
n n−1 n n n−1
The adjoint linear problem associated with the linear problem (2.3) is given by
Ψ∗ +I Ψ∗ +V Ψ∗ =λΨ∗,
n−1 n n n n+1 n
dΨ∗ (2.6)
 n =−V Ψ∗ ,
 dt n n+1
or

L∗Ψ∗ =λΨ∗,
n n n
dΨ∗ (2.7)
 − n =B∗Ψ∗,
 dt n n
where

L∗ =V e∂n +I +e−∂n, B∗ =V e∂n. (2.8)
n n n n n
The compatibility condition for each problem
dL dL∗
n =[B ,L ], n =[−B∗,L∗] (2.9)
dt n n dt n n
yields the Toda equation (1.2), respectively.
One of our main results is that the entries of the determinants in the Hankel
determinant formula arise as the coefficients of asymptoric expansions at λ = ∞ of
Hankel determinant formula for the Toda equation 4
the ratio of solutions of the linear and adjoint linear problems. To state the result
more precisely, we define
Ψ (t,λ) Ψ∗ (t,λ)
Ξ (t,λ)= k , Ω (t,λ)= k+1 . (2.10)
k Ψ (t,λ) k Ψ∗(t,λ)
k+1 k
Theorem 2.2 (i) The ratios Ξ (t,λ) and Ω (t,λ) admits two kinds of asymptotic
k k
expansions as functions of λ as λ→∞
Ξ(−1)(t,λ)=u λ−1+u λ−2+···, (2.11)
k −1 −2
Ξ(1)(t,λ) =v λ+v +v λ−1+···, (2.12)
k 1 0 −1
and
Ω(−1)(t,λ)=u λ−1+u λ−2+···, (2.13)
k −1 −2
Ω(1)(t,λ) =v λ+v +v λ−1+···, (2.14)
k 1 0 −1
respectively.
(ii) The above asymptotic expantions are related to the Hankel determinants entries
a(k) and b(k) as follows:
i i
∞
1 τ
Ξ(−1)(t,λ)= k b(k) λ−i, (2.15)
k λτ i
k−1
i=0
X
′
Ξ(1)(t,λ)= τk2 λ− τkτ+k1 − 1 τk ∞ a(k+1) (−λ)−i , (2.16)
k τk+1τk−1  (cid:16)τkτ+k1(cid:17) λτk+1 Xi=0 i 
 
and  
∞
1 τ
Ω(−1)(t,λ)= k a(k) (−λ)−i, (2.17)
k λτ i
k+1
i=0
X
′
Ω(1)(t,λ)= τk2 λ− τkτ−k1 − 1 τk ∞ b(k−1) λ−i . (2.18)
k τ τ  (cid:16)τk−1(cid:17) λτ i 
k+1 k−1 τk k−1 Xi=0
 
 
(iii) Ξ(±1) and Ω(±1) are related as follows:
k k
τ2 τ2
Ω(1)(t,λ) Ξ(−1)(t,λ)= k , Ω(−1)(t,λ) Ξ(1)(t,λ)= k . (2.19)
k k τ τ k k τ τ
k+1 k−1 k+1 k−1
Brief sketch of the proof of Theorem 2.2. One canproveTheorem2.2 by direct
calculation. From the linear problem (2.3) and (2.10), we see that Ξ (t,λ) satisfies
k
the Riccati equation
∂Ξ
k =−V Ξ2 +(λ−I )Ξ −1. (2.20)
∂t k k k k
PluggingseriesexpansionΞ =λρ ∞ h λ−i into(2.20)andconsideringthebalance
k i=0 i
of leading terms, we find that ρ must be ρ = 1,−1, which proves (2.11) and (2.12) .
P
Moreover, it is possible to verify (2.15) and (2.16) by deriving recursion relations of
coefficients for each case and comparing them with (2.2). Similarly, from the Riccati
equation for Ω
∂Ω
k =V Ω2 +(−λ+I )Ω +1, (2.21)
∂t k k k+1 k
Hankel determinant formula for the Toda equation 5
one can prove the statements for Ω. For (iii), putting Xk = τk+τ1kτ2k−1Ξk(−11) = VkΞ1k(−1),
plugging this expressioninto the Riccati equation(2.21) and using (1.2), we find that
X satisfies (2.20). Since the expansionof Ξ(−1) starts from λ−1, the leading order of
k k
X is λandthus X =Ω(1). The secondequationof(2.19)canbe provedina similar
k k
manner. (cid:3)
2.2. KP theory
Theresultsintheprevioussectioncanbecharacterizedbyacompactformulainterms
of the language of the KP theory[9, 15, 19].
We introduce infinitely many independent variables x = (x ,x ,x ,...), x = t,
1 2 3 1
and let τ (x) be the τ function of the one-dimensional Toda lattice hierarchy[25, 9]
n
and the first modified KP hierarchy[9]. Namely, τ , nZ, satisfy the following bilinear
n
equations
1 1
D p D˜ τ ·τ =p D˜ τ ·τ , j =0,1,2..., (2.22)
x1 j+1 2 n n j 2 n+1 n−1
(cid:18) (cid:19) (cid:18) (cid:19)
1 1 1
D p D˜ −p D˜ +p − D˜ τ ·τ =0, j =0,1,2..., (2.23)
x1 j 2 j+1 2 j+1 2 n+1 n
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
where p (x), p (x), ··· are the elementary Schur functions
0 1
∞ ∞
p (x)κn =exp x κi, (2.24)
n i
n=0 i=1
X X
and
1 1
D˜ =(D , D ,..., D ,...), (2.25)
x1 2 x2 n xn
D (i=1,2,...)being theHirota’sD-operator. Thenwehavethefollowingformula:
xi
Proposition 2.3 For fixed k ∈Z, we have
det(a(k) ) n>0,
τ i+j−2 i,j=1,...,n
k+n = 1 n=0, (2.26)
τ 
k  det(b(i+k)j−2)i,j=1,...,|n| n<0,
where 
τ τ
a(k) =p (∂˜) k+1, b(k) =(−1)ip (−∂˜) k−1, (2.27)
i i τ i i τ
k k
and
1 1
∂˜=(∂ , ∂ ,..., ∂ ,...). (2.28)
x1 2 x2 n xn
Remark 2.4 It might be interesting to remark here that a(k) = τk+1 and b(k) = τk−1
0 τk 0 τk
satisfy the nonlinear Schro¨dinger hierarchy. In fact, Equations (2.2) and (2.27) with
i=2 imply for a=a(k) and b=b(k)
0 0
a =a +2a2b, b =−(b +2a2b). (2.29)
x2 x1x1 x2 x1x1
Similarly, for i=3 we have
a =a +6aba , b =b +6abb . (2.30)
x3 x1x1x1 x1 x3 x1x1x1 x1
Hankel determinant formula for the Toda equation 6
Before proceeding to the proof, we note that the auxiliary linear problem (2.3) and
its adjoint problem (2.6) are recovered from the bilinear equations (2.22) and (2.23).
In fact, supposethat τ depends ona discrete independent variablel andsatisfiesthe
n
discrete modified KP equation
1
D τ (l+1)·τ (l)=− τ (l+1)τ (l), (2.31)
x1 n n λ n+1 n−1
1
( D +1)τ (l+1)·τ (l)−τ (l+1)τ (l)=0, (2.32)
λ x1 n+1 n n n+1
thenonecanshowthatEquations(2.22)and(2.23)areequivalentto(2.31)and(2.32),
respectively, through the Miwa transformation[14, 9]
l ∂ 1 ∂ 1 ∂ 1 ∂
x = or =− + +···+ +···. (2.33)
n n(−λ)n ∂l λ∂x 2λ2∂x j(−λ)j ∂x
1 2 j
Putting
τ (l+1)
Ψ∗ =λ−n n , (2.34)
n τ (l)
n
τ (l)τ (l) d τ (l)
V = n+1 n−1 , I = log n−1 , (2.35)
n τ (l)2 n dt τ (l)
n n
and noticing t=x , the bilinear equations (2.31) and (2.32) are rewritten as
1
Ψ∗′ =−V Ψ∗ ,
n n+1 n+1
(2.36)
Ψ∗ +I Ψ∗ +V Ψ∗ =λΨ∗ ,
n n+1 n+1 n+2 n+2 n+1
which are equivalent to the adjoint linear problem (2.6). Similarly, shifting l →l−1
in (2.31) and (2.32) and putting
τ (l−1)
Ψ =λn n , (2.37)
n+1 τ (l)
n
we obtain
Ψ′ =V Ψ ,
n+1 n n
(2.38)
V Ψ +I Ψ +Ψ =λΨ ,
n n n+1 n+1 n+2 n+1
which is also equivalent to the linear problem (2.3).
Proof of Proposition 2.3. From (2.37), (2.33) and (2.24) we have
Ψ (t,λ) 1τ (l−1)τ (l) 1 τ (l) τ (l)
k = k−1 k = k e−∂∂l k−1
Ψ (t,λ) λτ (l−1)τ (l) λτ (l) τ (l)
k+1 k k−1 k−1 k
∞
1 τ (l) − P 1 ∂ τ (l) 1 τ (l) ∞ τ (l)
= k e j=1j(−λ)j ∂xj k−1 = k pn(−∂˜) k−1 (−λ)−n.
λτ (l) τ (l) λτ (l) τ (l)
k−1 k k−1 k
n=0
X
Therefore Equation (2.15) in Theorem 2.2 implies
τ (l)
b(k) =(−1)np (−∂˜) k . (2.39)
n n τ (l)
k−1
Similarly we have from (2.34), (2.33) and (2.24)
Ψ∗ (t,λ) 1τ (l+1) τ (l) 1 τ (l) τ (l)
k+1 = k+1 k = k e∂∂l k+1
Ψ∗(t,λ) λ τ (l+1) τ (l) λτ (l) τ (l)
k k k+1 k+1 k
∞
1 τ (l) P 1 ∂ τ (l) 1 τ (l) ∞ τ (l)
= k ej=1j(−λ)j ∂xj k+1 = k pn(∂˜) k+1 (−λ)−n.
λτ (l) τ (l) λτ (l) τ (l)
k+1 k k+1 k
n=0
X
Hankel determinant formula for the Toda equation 7
Therefore comparing with (2.17) we obtain
τ (l)
a(k) =p (∂˜) k+1 , (2.40)
n n τ (l)
k
which proves Proposition 2.3. (cid:3)
3. Painlev´e equations
3.1. Local Lax pair
Originally the relations between determinant formula of the solutions and auxiliary
linear problem have been derived for the Painlev´e II and IV equations [10, 11]. In
the particular case of the rationalsolutions of the Painlev´eII and IV equations these
results give the relation between determinant formula and the Airy function found in
[7, 3]. It may be natural to regard those relationships as originating from the Toda
equation,sincethesequenceofτ functionsgeneratedbytheB¨acklundtransformations
of Painlev´e equations are described by the Toda equation [20, 21, 22, 23, 8, 12]. In
this section, we show that the results for the Painlev´e II equation can be recovered
fromtheresultsinthesection2. Thekeyingredientofthecorrespondenceisthelocal
Laxpair,whichistheauxiliarylinearproblemforTodaequationformulatedbyapair
of 2×2 matrices[1]:
L φ =φ , L (t,λ)= −In+λ −e−yn , (3.1)
n n n+1 n eyn 0
(cid:18) (cid:19)
edφn =B φ , Be (t,λ)= −21 0 λ+ 0 e−yn , (3.2)
dt n n n 0 1 −eyn−1 0
(cid:18) 2 (cid:19) (cid:18) (cid:19)
e e φ(1) τ
φ = n , y =log n−1. (3.3)
n φ(n2) ! n τn
Similarly, the adjoint linear problem is given by
L∗φ∗ =φ∗ , L∗(t,λ)= −In+λ eyn , (3.4)
n n n−1 n −e−yn 0
(cid:18) (cid:19)
edφ∗n =B∗φ∗, Be∗(t,λ)= 21 0 λ+ 0 eyn , (3.5)
dt n n n 0 −1 −e−yn+1 0
(cid:18) 2 (cid:19) (cid:18) (cid:19)
e e φ∗(1)
φ∗ = n . (3.6)
n φ∗n(2) !
Compatibility condition for each problem
dL dL∗
n =B L −L B , n =−B∗ L∗ +L∗B∗, (3.7)
dt n+1 n n n dt n−1 n n n
gives Toda equatieon (1.2), respectively. Byecomparing (3.1) and (3.2) with (2.3)
e e e e e e e e
similarly by comparing(3.4) and(3.5) with (2.6), one sees thatthere is a relationship
between the solutions of the linear problems:
τ
φ(n1) =e−12λt Ψn, φ(n2) =e−12λtτn−2 Ψn−1, (3.8)
n−1
τ
φ∗n(1) =e12λt Ψ∗n, φ∗n(2) =−e21λt nτ+1 Ψ∗n+1. (3.9)
n
Hankel determinant formula for the Toda equation 8
3.2. Painlev´e II equation
In this section we consider the Painlev´e II equation (P )
II
d2u 1
=2u3−4tu+4 α+ . (3.10)
dt2 2
(cid:18) (cid:19)
We denote (3.10) as P [α] when it is necessary to specify the parameter α explicitly.
II
Suppose that τ and τ satisfy the bilinear equations
0 1
(D2−2t) τ ·τ =0, (3.11)
t 1 0
1
(D3−2tD −4(α+ )) τ ·τ =0, (3.12)
t t 2 1 0
then it is easily verified that
d τ
u= log 1 (3.13)
dt τ
0
satisfies P [α] (3.10). If we generate the sequence τ (n∈Z) by the Toda equation
II n
1
D2 τ ·τ =τ τ , (3.14)
2 t n n n+1 n−1
then it is shown that τ satisfy
n
(D2−2t) τ ·τ =0, (3.15)
t n+1 n
1
(D3−2tD −4(α+ +n)) τ ·τ =0, (3.16)
t t 2 n+1 n
and that
d τ
u= log n+1 (3.17)
dt τ
n
satisfies P [α+n]. In this sense, the Toda equation (3.14) describes the B¨acklund
II
transformation of P (see, for example, [18]). Therefore one can apply Proposition
II
2.1 to obtain the determinant formula: for fixed k ∈Z we have
det(a(k) ) n>0,
τ i+j−2 i,j=1,...,n
k+n = 1 n=0, (3.18)
τ 
k  det(b(i+k)j−2)i,j=1,...,|n| n<0,
where

i−2
τ τ
a(k) =a(k)′+ k−1 a(k)a(k) , a(k) = k+1,
i i−1 τ l i−2−l 0 τ
 k k
l=0
 b(k) =b(k)′+ τk+1 iX−2b(k)b(k) , b(k) = τk−1. (3.19)
i i−1 τ l i−2−l 0 τ
k k
Now consider theauxiliary linear problXle=m0 for PII[α] (3.10)[8]:
∂Y 1 0 0 −1τ1
∂λ =AY, A= 04 −1 λ2+ 1τ−1 20τ0 λ
(cid:18) 4 (cid:19) (cid:18) 2 τ0 (cid:19)
′
−z+t 1 τ1
+ 2 2 τ0 , (3.20)
 ′ (cid:16) (cid:17) 
1 τ−1 z+t
 2 τ0 2 
∂Y −1 0  0(cid:16) (cid:17)τ1 
∂t =BY, B = 02 1 λ+ −τ−1 τ00 , (3.21)
(cid:18) 2 (cid:19) (cid:18) τ0 (cid:19)
Hankel determinant formula for the Toda equation 9
Y τ τ
Y = 1 , z =− 1 −1. (3.22)
Y τ2
(cid:18) 2 (cid:19) 0
Comparing (3.21) with (3.2), we immediately find that
B =B , Y =φ . (3.23)
1 1
We note that it is possible to regard (3.20) as the equation defining λ-flow which is
f
consistent with evolution in t. Also, the linear equation (3.1) describes the B¨acklund
transformation. Similarly, we have the adjoint problem
∂Y∗ 1 0 0 1τ−1
=A∗Y∗, A∗ = 4 λ2+ 2 τ0 λ
∂λ (cid:18) 0 −41 (cid:19) −12ττ01 0 !
′
−z+t 1 τ−1
+ 2 2 τ0 , (3.24)
 ′ (cid:16) (cid:17) 
1 τ1 z+t
 2 τ0 2 
∂Y∗ 1 0  (cid:16)0 (cid:17)τ−1 
∂t =B∗Y∗, B∗ = 02 −1 λ+ −τ1 τ00 , (3.25)
(cid:18) 2 (cid:19) (cid:18) τ0 (cid:19)
Y τ τ
Y∗ = 1 , z =− 1 −1, (3.26)
Y τ2
(cid:18) 2 (cid:19) 0
where we have a correspondence
B∗ =B∗, Y∗ =φ∗. (3.27)
0 1
Therefore, if we apply Theorem 2.2 noticing (3.8) and (3.9), we have the following:
e
Proposition 3.1 We put
Y Y∗
Λ(t,λ)= 2, Π(t,λ)= 2 . (3.28)
Y Y∗
1 1
(i) The ratios Λ and Π admit two kinds of asymptotic expansions as functions of λ
as λ→∞
Λ(−1)(t,λ)=u λ−1+u λ−2+···, (3.29)
−1 −2
Λ(1)(t,λ) =v λ+v +v λ−1+···, (3.30)
1 0 −1
and
Π(−1)(t,λ)=u λ−1+u λ−2+···, (3.31)
−1 −2
Π(1)(t,λ) =v λ+v +v λ−1+···, (3.32)
1 0 −1
respectively.
(ii) The above asymptotic expansions are related to the Hankel determinants entries
a(k) and b(k) as follows:
i i
∞
1
Λ(−1)(t,λ)= b(0) λ−i, (3.33)
λ i
i=0
X
′
Λ(1)(t,λ)= τ0 λ− ττ01 − 1τ0 ∞ a(1) (−λ)−i , (3.34)
τ  (cid:16)τ0(cid:17) λτ i 
1 τ1 1 Xi=0
 
 
Hankel determinant formula for the Toda equation 10
and
∞
1
Π(−1)(t,λ)= a(0) (−λ)−i, (3.35)
(−λ) i
i=0
X
′
τ−1 ∞
Π(1)(t,λ)=− τ0 λ− τ0 − 1 τ0 b(−1) λ−i . (3.36)
τ  (cid:16)τ−1(cid:17) λτ i 
−1 τ0 −1 Xi=0
 
(iii) Λ(±1) and Π(±1) are related as follows: 
Π(1)(t,λ) Λ(−1)(t,λ)=1, Π(−1)(t,λ) Λ(1)(t,λ)=1. (3.37)
Proposition 3.1 is equivalent to the results presented in [7, 10]. In other words, the
relations between determinant formula for the solution of P and auxiliary linear
II
problems originate from the structure of the Toda equation. We also note that one
can recover the results for the Painlev´e IV equation[3, 11] in similar manner.
4. Toda equation on finite lattice
4.1. Determinant formula
Let us consider the Toda equation on the finite lattice. Namely, we impose the
boundary condition
V =0, V =0,
0 N
y =−∞, y =∞, (4.1)
0 N+1
α =0, α =0,
0 N
on the Toda equation (1.1), (1.2) and (1.3), respectively. In order to realize this
conditionon the levelofthe τ funtion, we proceedas follows: in the bilinear equation
(1.5), imposing the boundary condition on the left edge of lattice
τ =0, τ 6=0, (4.2)
−1 0
it immediately follows τ = 0 and one can restrict the Toda equation on the semi-
−2
infinite lattice n≥0 . In this case, the determinant formula reduces to
τ τ
k =det(a(0) ) (n≥1), a(0) =a(0)′, a(0) = 1, (4.3)
τ i+j−2 i,j=1,···,k i+1 i 0 τ
0 0
which is equivalent to (1.9). Moreover,imposing the boundary condition on the right
edge of lattice
τ 6=0, τ =0, (4.4)
N N+1
then we have the finite Toda equation
τ′′τ −(τ′)2 =τ τ , n=0,···,N, τ =τ =0. (4.5)
n n n n+1 n−1 −1 N+1
It is easily verified that the boundary condition (4.4) is satisfied by putting
N
a(0) = c eµit, (4.6)
0 i
i=1
X
where c and µ (i=1,...,N) are arbitrary constants.
i i
It is sometimes convenient to consider the finite Toda equation in the form of
(1.3). One reason for this is that the auxiliary linear problem associated with (1.3)
dΦ
n
α Φ +β Φ +α Φ =µΦ , =−α Φ +α Φ , (4.7)
n−1 n−1 n n n n+1 n dt n−1 n−1 n n+1