179

数理解析研究所講究録
第 650 巻 1988 年 179-195

Algorithmic Construction of the Recursion Operatorsof Toda and Landau-Lifshitz Equation\dagger

E.Barouch, A.S. Fokas and V.G. Papageorgiou
Department of Mathematics and Computer Science
Clarkson University
Potsdam, NY 13676
Abstract
A new approach to the construction of recursion operators of completely integrable system is exhibited. It is explicitly applied to derive the hierarchy ofequations of motion of the celebrated Toda lattice as well as the well known\dagger Supported by AFOSR grant #AFOSR-87-0310.

A. Recursion Operator for the Toda Lattice
The equations of motion for the Hamiltonian System of the Toda lattice with
Hamiltonian(1)$H= \sum_{n}\{\frac{1}{2}p_{n}^{2}+e^{x_{n+1}-x_{n}}\}$

are:$x_{n,t}=p_{n}$,(2)$p_{n,t}=e^{x_{n+1}-x_{n}}-e^{x_{n}-x_{n-1}}$

and the shift operator of the Lax pair acting on the vector(3)$\psi_{n}=\{\begin{array}{l}t_{n}\epsilon_{n}\end{array}\}$is given by (Takhtadzhan and Fadeev (1979))(4)$L_{n}(\lambda)\psi_{n}=\psi_{n+1}$

where

.

(5)

$L_{n}(\lambda)=\{\begin{array}{ll}p_{n}+\lambda -e^{x_{n}}e^{-x_{n}} 0\end{array}\}$After introducing(6)$v_{n}\equiv\epsilon_{n+1}$(4) yields the following second order difference equation for$v_{n}$:(7)$\lambda v_{n}=e^{x_{n+1}-x_{n}}v_{n+1}+(-p_{n})v_{n}+v_{n-1}$

Define now,$b_{n}\equiv-p_{n}$(8)$a_{n}v_{n+1}+b_{n}v_{n}-+v_{n-1}=\lambda v_{n}$(9)$a_{n}\equiv e^{x_{n+1}-x_{n}}$then (7) is written as:
The time evolution of the auxilliary vector$\psi_{n}$is expressed in terms of$v_{n,t}=(A_{n}v_{n+1}-B_{n}v_{n})a_{n}$2$v_{n}’ s$

as

(10)

1S

and the compatibility of (9), (10) gives:$[a_{n,t}+\lambda a_{n}(A_{n+1}-A_{n})-a_{n}(b_{n+1}A_{n+1}-b_{n}A_{n})-a_{n}(a_{n+1}B_{n+1}-a_{n-1}B_{n-1})]v_{n+1}$$+[b_{n,t}+\lambda(a_{n}B_{n}-a_{n-1}B_{n-1})+b_{n}(a_{n-1}B_{n-1}-a_{n}B_{n})+a_{n-1}A_{n-1}-a_{n}A_{n+1}]v_{n}=0$(11)hence both coefficients of$v_{n}$

andshould vanish i.e.

$v_{n+1}$$a_{n,t}+\lambda a_{n}(A_{n+1}-A_{n})-a_{n}(b_{n+1}A_{n+1}-b_{n}A_{n})-a_{n}(a_{n+1}B_{n+1}-a_{n-1}B_{n-1})=0(12)$and$b_{n,t}+\lambda(a_{n}B_{n}-a_{n-1}B_{n-1})+b_{n}(a_{n-1}B_{n-1}-a_{n}B_{n})+a_{n-1}A_{n-1}-a_{n}A_{n+1}=0(13)$One may postulate

So after substitution of (15) into (12), (13) and equating coefficients ofobtains the’following equations:, $a_{n}B_{n}^{(N)}-a_{n-1}B_{n-1}^{(N)}=0$$a_{n}(A_{n+1}^{(N)}-A_{n}^{(N)})=0$(14)$B_{n}= \sum_{j=0}^{N}B_{n}^{(j)}\lambda^{j}$$A_{n}= \sum_{j=0}^{N}A_{n}^{(j)}\lambda^{j}$$\lambda^{j}$

, one
(15)

$a_{n,t}=a_{n}(b_{n+1}A_{n+1}^{(0)}-b_{n}A_{n}^{(0)})+a_{n}(a_{n+1}B_{n+1}^{(0)}-a_{n-1}B_{n-1}^{(0)})$(16)$b_{n,t}=b_{n}(a_{n}B_{n}^{(0)}-a_{n-1}B_{n-1}^{(0)})+a_{n}A_{n+1}^{(0)}-a_{n-1}A_{n-1}^{(0)}$(17)$a_{n}(A_{n+1}^{(j-1)}-A_{n}^{(j-1)})=a_{n}(b_{n+1}A_{n+1}^{(j)_{\vee}}-b_{n}A_{n}^{(j)})+a_{n}(a_{n+1}B_{n+1}^{(j)}-a_{n-1}B_{n-1}^{(j)})$(18)$a_{n}B_{n}^{(j-1)}-a_{n-1}B_{n-1}^{(j-1)}=b_{n}(a_{n}B_{n}^{(j)}-a_{n-1}B_{n-1}^{(j)})+a_{n}A_{n+1}^{(j)}-a_{n-1}A_{n-1}^{(j)}$

(19)

$Y$

.
for $j=1,$
Upon introducing the operators$\ldots,$$n$$\triangle,$$\triangle^{+}$(cf. Soliani et al., (1983))$\triangle u_{n}\equiv u_{n+1}-u_{n}$$\triangle^{+}u_{n}\equiv u_{n-1}-u_{n}$

3

(20)

.

182

one may write (17), (18) in matrix form.

$\{\begin{array}{l}a_{n,t}b_{n,t}\end{array}\}=\{\begin{array}{ll}a_{n}(\triangle-\triangle^{+})a_{n} a_{n}\triangle b_{n}-b_{n}\triangle^{+}a_{n} a_{n}\triangle-\triangle^{+}a_{n}\end{array}\}\{\begin{array}{l}B_{n}^{(0)}A_{n}^{(0)}\end{array}\}$$\equiv\Omega\{\begin{array}{l}B_{n}^{(0)}A_{n}^{(0)}\end{array}\}$(21)$\Theta\{\begin{array}{l}B_{n}^{(j-1)}A_{n}^{(j-1)}\end{array}\}=\Omega\{\begin{array}{l}B_{n}^{(j)}A_{n}^{(j)}\end{array}\}$(22)$\Theta\equiv\{\begin{array}{ll}0 a_{n}\triangle-\triangle^{+a_{n}} 0\end{array}\}$(23)

and (18), (19) expressed as:

where

Note that this operator was present by Soliani et al., (1983). The recursion
relation takes the form$\{\begin{array}{l}B_{n}^{(j-1)}A_{n}^{(j-1)}\end{array}\}=\Theta^{-1}\Omega\{\begin{array}{l}B_{n}^{(j)}A_{n}^{(j)}\end{array}\}\equiv\Psi\{\begin{array}{l}B_{n}^{(j)}A_{n}^{(j)}\end{array}\}$(24)

From (25) one obtains recursively:$\{\begin{array}{l}B_{n}^{(0)}A_{n}^{(0)}\end{array}\}=\Psi^{N}\{\begin{array}{l}B_{n}^{(N)}A_{n}^{(N)}\end{array}\}$(25)

and since a solution of (15) is$A_{n}^{(N)}=c,$4$B_{n}^{(N)}=0$

(26)

‘

where

$c$

183

is an arbitrary constant, the hierarchy of the Toda lattice is given by:$\{\begin{array}{l}a_{n,t}b_{n,t}\end{array}\}=\Omega\Psi^{N}\{\begin{array}{l}0c\end{array}\}$(27)

The first system of equations $(N=0, c=-1)$$\{\begin{array}{l}a_{n,t}b_{n,t}\end{array}\}=\{\begin{array}{l}-a_{n}\triangle b_{n}\triangle^{+a_{n}}\end{array}\}=\{\begin{array}{l}a_{n}(b_{n}-b_{n+1})a_{n-1}-a_{n}\end{array}\}$(28)

is equivalent to (2), using (8). The second system $(N=1)$ is:$\{\begin{array}{l}a_{n,t}b_{n,t}\end{array}\}=\Omega\Theta^{-1}\Omega\{\begin{array}{l}0-1\end{array}\}$(29)$\Theta^{-1}=\{\begin{array}{lll}0 a_{n}^{-1}(\triangle^{+})^{-1} \triangle^{-1}a_{n}^{-1} 0\end{array}\}$(30)

and, after noting thatwhere$( \triangle^{-1}u)_{n}\equiv-\sum_{j=n}^{+\infty}u_{j}$(31)(29) becomes:$\{\begin{array}{l}a_{n,t}b_{n,t}\end{array}\}=\{\begin{array}{ll}a_{n}(a_{n+1} a_{n}-2a_{n}+a_{n-1})-(b_{n+1}^{2}-b_{n}^{2})a_{n}-b_{n}(a_{n-1}-)-a_{n}(b_{n+1}- b_{n})+a_{n-1}b_{n-1}-a_{n}b_{n}\end{array}\}$

$t$

5

(32)

184

B. Landau-Lifshitz Equation
The Landau-Lifshitz equation (LL) is given by(1)$S_{t}=S\cross S_{xx}+S\cross JS$

where

$J$

is the diagonal matrix
$J=$

and

$S$

diag

(2)

$(J_{1}, J_{2}, J_{3})$is the classical unit spin $S=(S_{1}, S_{2}, S_{3})$ , i.e.,$S\cdot S=1$.

(3)

It is well known that (1) is completely integrable and Sklyanin (1979) andothers presented its Lax-pair. Since the LL equation is the continuum limit of
the equation of motion of the quantum non-isotropic Heisenberg Hamiltonian(the so-called XYZ), it is not surprising that the Lax pair is expressed in terms of

Jacobi elliptic functions. The algebraic structure of (1) was studied in detail byDate, Jimbo, Kashiwara and Miwa (1983) who derived its quasi-periodic solutions

as well. Furthermore, Fuchssteiner (1984) studied its master-symmetries.
Consider the equation for the auxilliary vector$\psi$

given by$\psi_{x}=-i(\sum_{j=1}^{3}S_{j}W_{j}\sigma_{j})\psi\equiv-iL\psi$

6

(4)

$|$

185

while

$L$

may be viewed as the shift operator associated with the Lax pair. The

operators$\sigma_{j}$are the Pauli spin operators given by$\sigma_{1}=(\begin{array}{ll}0 11 0\end{array}),$$\sigma_{2}=(\begin{array}{ll}0 -ii 0\end{array}),$and the Jacobi elliptic functions$W_{j}$(5)$\sigma_{3}=(\begin{array}{ll}1 00 -1\end{array})$

are given by Sklyanin as:$W_{1}= \rho\frac{1}{sn(u,k)}$(6)$W_{2}=p \frac{dn(u,k)}{sn(u,k)}$$W_{3}= \rho\frac{cn(u,.k)}{sn(u,k)}$with the modulus$k$given by(7)

$k= \{\frac{J_{2}-J_{1}}{J_{3}-J_{1}}\}^{1/2}0<k<1$

and the arbitrary normalization parameter$\rho$

as well as the parameters$\alpha,$are$\beta$defined by$W_{1}^{2}-W_{3}^{2}= \frac{1}{4}(J_{3}-J_{1})\equiv\alpha$$(8a)$$W_{2}^{2}-W_{3}^{2}= \frac{1}{4}(J_{3}-J_{2})\equiv\beta$$(8b)$

Formally, one may express the time evolution of the auxilliary vector$\psi_{t}=-iV\psi$

7$\psi$

as(9)and the structure of the operator$L$suggests that V has similar form, i.e. one

may postulate$\psi_{t}=-i\{\sum_{j=1}^{3}W_{j}V_{j}\sigma_{j}\}\psi$(10)with the compatibility condition$L_{t}-V_{x}-i[L, V]=0$(11)$\sum_{j=1}^{3}S_{j,t}W_{j}\sigma_{j}-\sum_{j=1}^{3}V_{j,x}W_{j}\sigma_{j}-ib\sum_{=1}^{3}S_{j}W_{j}\sigma_{j},\sum_{j=1}^{3}V_{j}W_{j}\sigma_{j}]=0$(12)that takes the form

Equating coefficients of$\sigma_{j}$for $j=1,2,3$ , one obtains$S_{1,t}= \frac{2W_{2}W_{3}}{W_{1}}(S_{3}V_{2}-S_{2}V_{3})+V_{1,x}$(13)

as well as other cyclic permutations.
It is convenient to introduce the parametrization$\lambda\equiv\frac{1}{2}W_{1}W_{2}W_{3}$,$\mu\equiv W_{3}^{2}$(14)

with the immediate identity$\lambda^{2}=\frac{1}{4}\mu(\mu+\alpha)(\mu+\beta)-$

8

(15)

$\sim_{-}18’/$

where$\alpha,$$\beta$have been defined by (8a), (8b). Thus, (13) and its cyclic permutations

take the form
$S_{1,t}= \frac{\mu(\mu+\beta)}{\lambda}(S_{3}V_{2}-S_{2}V_{3})+V_{1,x}$$(16a)$$S_{2,t}= \frac{(\mu+\alpha)\mu}{\lambda}(S_{1}V_{3}-S_{3}V_{1})+V_{2,x}$$(16b)$$S_{3,t}= \frac{(\mu+\beta)(\mu+\alpha)}{\lambda}(S_{2}V_{1}-S_{1}V_{2})+V_{3,x}$$(16c)$

One may formally represent the operators$V_{k}$

by the finite expansions$V_{1}= \frac{\mu(\mu+\beta)}{\lambda}\sum_{j=0}^{n}\mu^{n-j}a_{1}^{(j)}+\sum_{j=0}^{n}\mu^{n-j}b_{1}^{(j)}$$(17a)$$V_{2}= \frac{(\mu+\alpha)\mu}{\lambda}\sum_{j=0}^{n}\mu^{n-j}a_{2}^{(j)}+\sum_{j=0}^{n}\mu^{n-j}b_{2}^{(j)}$$(17b)$$V_{3}= \frac{(\mu+\beta)(\mu+\alpha)}{\lambda}\sum_{j=0}^{n}\mu^{n-j}a_{3}^{(j)}+\sum_{j=0}^{n}\mu^{n-j}b_{3}^{(j)}$$(17c)$

In other words, determination of the operators
determination of$V$$a_{k}^{(j)},$$b_{l}^{(m)}$is equivalent to a

. Upon substitution of (17) in (16a) one obtains$S_{1,t}= \frac{\mu(\mu+\beta)}{\lambda}\sum_{j=0}^{n}\mu^{n-j}a_{1,x}^{(j)}+\sum_{j=0}^{n}\mu^{n-j}b_{1,x}^{(j)}$$- \frac{\mu(\mu+\beta)}{\lambda}[S_{2}(\frac{(\mu+\alpha)(\mu+\beta)}{\lambda}\sum_{j=0}^{n}\mu^{n-j}a_{3}^{(j)}+\sum_{j=0}^{n}\mu^{n-j}b_{3}^{(j)})$$-S_{3}( \frac{\mu(\mu+\alpha)}{\lambda}\sum_{j=0}^{n}\mu^{n-j}a_{2}^{(j)}\cdot+\sum_{j=0}^{n}\mu^{n-j}b_{2}^{(j)})]$

9

(18)

188

namely$S_{1,t}= \frac{\mu(\mu+\beta)}{\lambda}\sum_{j=0}^{n}\mu^{n-j}(a_{1,x}^{(j)}-S_{2}b_{3}^{(j)}+S_{3}b_{2}^{(j)})+\sum_{j=0}^{n}\mu^{n-j}b_{1,x}^{(j)}$(19)$-4( \mu+\beta)S_{2}\sum_{j=0}^{n}\mu^{n-j}a_{3}^{(j)}+4\mu S_{3}\sum_{j=0}^{n}\mu^{n-j}a_{2}^{(j)}$or$S_{1,t}= \frac{\mu(\mu+\beta)}{\lambda}\sum_{j=0}^{n}\mu^{n-j}(a_{1,x}^{(j)}-S_{2}b_{3}^{(j)}+S_{3}b_{2}^{(j)})+\sum_{j=0}^{n}\mu$$[b_{1,x}^{(j)}-4\beta S_{2}a_{3}^{(j)}]$(20)$-4 \sum_{j=-1}^{n-1}\mu^{n-j}[S_{2}a_{3}^{(j+1)}-S_{3}a_{2}^{(j+1)}]$

Similarly, the other two equations are given by$S_{2,t}= \frac{(\mu+\alpha)\mu}{\lambda}\sum_{j=0}^{n}\mu^{n-j}(a_{2,x}^{(j)}-S_{3}b_{1}^{(j)}+S_{1}b_{3}^{(j)})+\sum_{j=0}^{n}\mu^{n-j}(b_{2,x}^{(j)}+4\alpha S_{3}a_{1}^{(j)})$$-4 \sum_{j=-1}^{n-1}\mu^{n-j}(S_{3}a_{1}^{(j+1)}-S_{1}a_{3}^{(j+1)})$$(20b)$$S_{3,t}= \frac{(\mu+\beta)(\mu+\alpha)}{\lambda}\sum_{j=0}^{n}\mu^{n-j}(a_{3,x}^{(j)}-S_{1}b_{2}^{(j)}+S_{2}b_{1}^{(j)})$$+ \sum_{j=0}^{n}\mu^{n-j}(b_{3,x}^{(j)}-4\alpha S_{1}a_{2}^{(j)}+4\beta S_{2}a_{1}^{(j)})-4\sum_{j=-1}^{n-1}\mu^{n-j}(S_{1}a_{2}^{(j+1)}-S_{2}a_{1}^{(j+1)})$Equating coefficients of$\mu^{j}$and$\lambda^{-1}\mu^{j}$$(20c)$

independently one obtains(21)$S\cross a^{(0)}=0$; $\dot{J}=0,1,$$S\cross b^{(j)}=a_{x}^{(j)}$10$\ldots,$

$n$

(22)

189

$S\cross a^{(j+1)}=\frac{1}{4}b_{x}^{(j)}-(AS)\cross a^{(j)}$; $j=0,1,$$\ldots,$$S_{t}=b_{x}^{(n)}-4(AS)\cross a^{(n)}$

$n-1$

(23)
(24)

where A is diagonal matrix given by$A=$

First, one solves (22) for

diag

(25)

$(\alpha, \beta, 0)$$b^{(j)}$$b^{(j)}=-S\cross a_{x}^{(j)}+g_{j}S$

where$g_{j}$

is a scalar function of

$x$

(26)

to be determined by requiring the solvability

condition for (23):$\{b_{x}^{(j)}-(4AS)\cross a^{(j)}\}\cdot S=0$(27)

$g_{j,x}=\{S_{x}\cross a_{x}^{(j)}+(4AS)\cross a^{(j)}\}\cdot S$(28)$g_{j}=\partial^{-1}(\{S_{x}\cross a_{x}^{(j)}+(4AS)\cross a^{(j)}\}\cdot S)$(29)

This condition gives:

$i.e$

.

where

$\partial^{-1}$indicates antiderivative with respect to .
$x$

Then
$b_{x}^{(j)}=-S\cross a_{xx}^{(j)}+[(4AS)\cross a^{(j)}\cdot S]S$

11

190

$+[\partial^{-1}(\{S_{x}\cross a_{x}^{(j)}+(4AS)\cross a^{(j)}\}\cdot S)]S_{x}$Now (23) yields:$a^{(j+1)}=-\frac{1}{4}S\cross\{b_{x}^{(j)}-(4AS)\cross a^{(j)}\}+f_{j+1}S$

where the scalar$f_{j+1}$

is to be determined by the requirement$a_{x}^{(j+1)}\cdot S=0$

for (22) to be solvable for$b^{(j)}$,. Using (31), (32) yields$f_{j+1,x}= \frac{1}{4}S_{x}\cross\{b_{x}^{(j)}-(4AS)\cross a^{(j)}\}\cdot S$

$i.e$

.
$f_{j+1}= \frac{1}{4}\partial^{-1}(S_{x}\cross\{b_{x}^{(j)}-(4AS)\cross a^{(j)}\}\cdot S)$

SO,$a^{(j+1)}=-\frac{1}{4}S\cross\{b_{x}^{(j)}-(4AS)\cross a^{(j)}\}$$+ \frac{1}{4}[\partial^{-1}(S_{x}\cross\{b_{x}^{(j)}-(4AS)\cross a^{(j)}\}\cdot S)]S$

Finally, introducing the operators:$\Theta^{-1}a\equiv S\cross a+[\partial^{-1}(S_{x}\cross a\cdot S)]S$

12

$19]_{-}$

and$\Omega a\equiv\frac{1}{4}\{-S\cross a_{xx}-(4AS)\cross a+[(4AS)\cross a\cdot S]S$(37)$+[\partial^{-1}(\{S_{x}\cross a_{x}+(4AS)\cross a\}\cdot S)]S_{x}\}$

we can write (35) and (24) as:$a^{(j+1)}=\Theta^{-1}\Omega a^{(j)}\equiv\Psi a^{(j)}$(38)$S_{t}=4\Omega a^{(n)}$

(39)

and

Next, one has to deal with the “starting points” of the recursion,$a^{(0)},$$b^{(0)}$. It

is best illustrated by an explicit derivation of the hierarchy (39) for $n=1$ :From (21), solving for$a^{(0)}$, one obtains:(40)$a^{(0)}=F_{o}S$It turns out that

$F_{o}$

is a constant in order to be able to solve (22) for$G_{o}$

is a new constant in order that (23) be solvable for$a^{(1)}$$a^{(1)}=f_{1}S+\frac{1}{4}\{G_{o}(S\cross S_{x})+F_{o}[S_{xx}-(S\cdot S_{xx})S]\}$

13

:

(41)

$b^{(0)}=-F_{o}S\cross S_{x}+G_{o}S$

where$b^{(0)}$:192$+F_{o}\{(S\cdot AS)S-AS\}$Since$a_{x}^{(1)}$

has to be normal to$S$,$f_{1,x}-F_{o} \{\frac{1}{4}(S_{x}\cdot S_{xx})-S_{x}\cdot AS)\}=0$

Since

$S$

is a unit vector, i.e. $S\cdot S=1$ , one has:$arrow S\cdot S_{xx}=S_{x}\cdot S_{x}$and$S\cdot S_{xxx}=\frac{3}{2}(S\cdot S_{xx})_{x}$so (43) yields:$f_{1}=F_{1}+ \frac{1}{8}F_{o}[(S_{xx}-4AS)\cdot S]$

where$F_{1}$

is a constant. Hence$a^{(1)}=\{F_{1}+\frac{1}{8}F_{o}[(S_{xx}-4AS)\cdot S]\}S+\frac{1}{4}G_{o}S\cross S_{x}$$+ \frac{1}{4}F_{o}(S_{xx}-(S\cdot S_{xx})S)+F_{o}[(S\cdot AS)S-AS]$and$a_{x}^{(1)}=r_{F_{1}}+\frac{1}{8}F_{o}[(S_{xx}-4AS)\cdot S]\}S_{x}$$+ \frac{F_{o}}{8}\{(S_{xx}-4AS)\cdot S_{x}+(S_{xxx}-4AS_{x})\cdot S\}S$

14

193

$+ \frac{1}{4}G_{o}S\cross S_{xx}+\frac{1}{4}F_{o}\{S_{xxx}-(S_{x}\cdot S_{xx})S-(S\cdot S_{xxx})S-(S\cdot S_{xx})S_{x}\}$$+F_{o}\{2(S_{x}\cdot AS)S+(S\cdot AS)S_{x}-AS_{x}\}$Then, solving (22) for$b^{(1)}$, one gets$b^{(1)}=g_{1}S-S\cross a_{x}^{(1)}$where$g_{1}$

(48)

(49)

has to satisfy the equation (cf. (28))$g_{1,x}= \frac{1}{4}G_{o}[S_{x}\cdot S_{xx}+4AS\cdot S_{x}]-\frac{1}{4}F_{o}[S_{x}\cross(4AS_{x})\cdot S$$+S_{xx}\cross(4AS)\cdot S-S_{x}\cross S_{xxx}\cdot S]$

$i.e$

(50)

.
$g_{1,x}= \frac{1}{8}G_{o}[S_{x}\cdot S_{x}+4AS\cdot S]_{x}$$+ \frac{1}{4}F_{o}[S_{x}\cross(S_{xx}-4AS)\cdot S]_{x}$(51)$g_{1}=G_{1}- \frac{1}{8}G_{o}[(S_{xx}-4AS)\cdot S]+\frac{1}{4}F_{o}[S_{x}\cross(S_{xx}-4AS)\cdot S]$(52)

and because of (45):One may set $n=1$ in equation (24). The resulting evolution equation contains
the arbitrary constants$G_{1},$$F_{1},$$G_{o},$$F_{o}$. By letting all but one vanish, one obtains

the hierarchy of evolution equations as:(i)$S_{t}=S_{x}$

15

(53)

194

(ii)

$S_{t}=S\cross S_{xx}+(4AS)\cross S$

which is the same as(54)$S_{t}=S\cross S_{xx}+S\cross JS$

because of $(8a,b)$ and (25).

(iii)$S_{t}=S_{xxx}+\frac{3}{2}[(S_{x}\cdot S_{x})-JS\cdot S^{\nu}+J_{3}]S_{x}+3(S_{x}\cdot S_{xx})S$(55)

This equation was obtained by Date, Jimbo, Kashiwara and Miwa (1983)(iv)$S_{t}=S\cross S_{xxxx}+S_{x}\cross S_{xxx}-\frac{1}{2}[3S\cdot S_{xx}+S\cdot JS]S\cross S_{xx}$$+[3(S_{x}\cdot S_{xx})-S_{x}\cdot JS]S\cross S_{x}-[S_{x}\cross(S_{xx}+JS)\cdot S]S_{x}$$+ \frac{1}{2}[3(S\cdot S_{xx})+(S\cdot JS)](JS)\cross S$$-[S_{x}\cross(S_{xx}+JS)\cdot S]_{x}S+(S\cross JS_{x})_{x}+S_{xx}\cross(JS)$.

Detailed account of this work, in particular the bi-Hamiltonian formulation andthe connection with the master-symmetry approach, will be published elsewhere.

16

195

References

Takhtadzhan, L.A. &Fadeev, L.D. (1979): The Quantum Method of the Inverse
problem and the Heizenberg XYZ model, Russian Math. Surveys 34:5 (1979),

11-68.
Leo, M., Leo, R.A., Soliani, G., Solombrino, L. and Mancarella, G. (1983): Sym-

metry properties and Bi-Hamiltonian structure of the Toda lattice, preprint,

Lecce, May 1983.

Sklyanin, E.K. (1979): On complete integrability of the Landau-Lifshitz equa-tion, Steklov Math. Institute LOMI preprint, E-3, (1979).
Date, E., Jimbo, M., Kashiwara, M. and Miwa, T. (1983): Landau-Lifshitz equa-tion: solitons; quasi-periodic solutions and infinite-dimensional Lie algebras.

J. Phys. $A:221-236$ .
Fuchssteiner, B. (1984): On the hierarchy of the Landau-Lifshitz equation. Phys-

ica $13D,$ $387-394$ .
Note added in the proof:

The first presentation of the Toda

recursion operator was given by H. Flaschka and D. W. McLaughlin in“Backlund Transformation“ , ed. by R.M. Miura, Lecture Notes inMathematics 515, Springer-Verlag, New york, p. 253’

7(1976).