Introduction

The (Schwarzian) KdV equation [1]

$$\phi_t/\phi_x+\{\phi;x\}=\lambda\eqno (1.1)$$

has the Bäcklund transformation [1]

$$\phi=(a\psi+b)/(c\psi+d)\ ,\eqno (1.2)$$

$$ad-bc=1\ ,\qquad\phi_x=\psi_x^{-1}\ ,\eqno (1.3)$$

where

$$\psi_t/\psi_x+\{\psi;x\}=\lambda\ .\eqno (1.4)$$

The expression

$$\{\phi;x\}={\partial\over\partial x}\left({\phi_{xx}\over\phi... ...ht)-{1\over 2}\left( {\phi_{xx}\over\phi_x}\right)^2\eqno (1.5)$$

is the Schwarzian derivative, which is invariant under the Möbius group (1.2) [2,3].

The effective Bäcklund transformation (BT) for (1.1) is the composition of (1.2) and (1.3). We find that [4]

$$\phi_{n+1,x}={\phi_n^2\over\phi_{n,x}}\ ,\eqno (1.6)$$

$${\phi_{n+1,t}\over\phi_{n+1,x}}+{\phi_{n,xx}\over\phi_{n,x}}=... ...)^2-4{\partial^2\over\partial x^2}\ln\phi_n+2\lambda\eqno (1.7)$$

is a BT for (1.1). The periodic fixed points of the BT are defined by Eqs. (1.6) and (1.7) with

$$n=1,2,3,4,\ldots(\hbox{mod}\ N)\ .\eqno (1.8)$$

The periodic fixed points continue to define a strong BT for (1.1). That is, the integrability conditions

$$\phi_{n+1,xt}=\phi_{n+1,tx}\eqno (1.9)$$

continue to imply that $\phi_n$ satisfy (1.1), and, by the periodicity $\hbox{mod}\ N$, the set

$$\bigl\{\phi_n,\ n=1,2,\ldots(\hbox{mod}\ N)\bigr\}\eqno (1.10)$$

are solutions of (1.1).

In a previous work [4] we have found that if

$$\xi_j=\phi_{j,x}/\phi_j\eqno (1.11)$$

then

$$\xi_{j+1,x}/\xi_{j+1}+\xi_{j,x}/\xi_j=\xi_j-\xi_{j+1}\ .\eqno (1.12)$$

Define the $N\times N$ circulant matrices [5]

$$A=\left(\matrix{1&1&0&&&\cr 0&1&1&0&&0\cr 0&0&1&1&&\cr &&&\ddots&\ddots&\cr 0&&&&1&1\cr 1&&&&&1\cr}\right)\ ,\eqno (1.13)$$

$$B=\left(\matrix{1&-1&0&&&\cr 0&1&-1&0&&0\cr 0&0&1&-1&&\cr &&&\ddots&\ddots&\cr 0&&&&1&-1\cr -1&&&&&1\cr}\right)\ .\eqno (1.14)$$

Then with

$$\widehat{\xi}=\left(\matrix{\xi_1\cr \xi_2\cr \vdots\cr \xi_n\cr}\right) \ ,\eqno (1.15)$$

$$\beta_j=\ln\xi_j\ ,\eqno (1.16)$$

$$\widehat{\xi}=\left(\matrix{\beta_1\cr \beta_2\cr \vdots\cr \beta_n\cr} \right)\ ,\eqno (1.17)$$

Eqs. (1.12) are

$$A\widehat{\beta}_{,x}=B\widehat{\xi}\ .\eqno (1.18)$$

For all $N$ the one-dimensional null space of $B$ is spanned by the $N$ vector

$$\widehat{b}_0=\left(\matrix{1\cr 1\cr \vdots\cr 1\cr}\right)\ .\eqno (1.19)$$

While for $N=2kK$

$$\vert A\vert=0\eqno (1.20)$$

and $A$ has a one-dimensional null space spanned by

$$\widehat{a}_0=\left(\matrix{-1\cr 1\cr -2\cr \vdots\cr -1\cr}\right)\ . \eqno (1.21)$$

For $N=2k+1$, $A$ is invertible and

$$A^{-1}=\textstyle{1\over 2}(I+\Omega)\ ,\eqno (1.22)$$

where

$$\Omega=\left(\matrix{0&-1&1&-1&1&\cdots&-1&1\cr 1&0&-1&1&-1&... ... 1&&&&&1&0&-1\cr -1&1&-1&1&-1&\cdots&1&0\cr}\right)\eqno (1.23)$$

is a $(2k+1,2k+1)$ antisymmetric matrix

$$\Omega^{\prime}=-\Omega\eqno (1.24)$$

with

$$\vert\Omega\vert=0\ .\eqno (1.25)$$

The one-dimensional null space of (1.23) is spanned by (1.19) and it can be shown that

$$\Omega=A^{-1}B\ .\eqno (1.26)$$

In the notation for circulant matrices [5]

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaystyl... ...\hbox{circ}[0,-1,1,-1,1,\ldots,-1,1]\ .\cr\crcr}}\,\eqno (1.27)$$

When $N$ is odd, Eqs. (1.18) can be written as Hamiltonian systems

$$\widehat{\xi}_x=\left(\matrix{\xi_1&&&0\cr &\xi_2&&\cr &&\ddo... ...dots&\cr 0&&&\xi_{_N}\cr}\right)\nabla_{\xi} H_1\ ,\eqno (1.28)$$

where

$$H_1=\sum_{j=1}^{^N}\xi_j\ .\eqno (1.29)$$

In [4] we find that (1.28) is a completely integrable, $k$-dimensional, Hamiltonian system with one Casimir

$$H_{_N}=\prod_{j=1}^{^N}\xi_j\eqno (1.30)$$

and $k$ independent integrals

$$H_{_{N-2m}}=L^m\circ H_{_N}\ ,\eqno (1.31)$$

where

$$N=2k+1\eqno (1.32)$$

and

$$L=\sum_{j=1}^{^N} {\partial^2\over\partial\xi_j\partial\xi_{j+1}}\ . \eqno (1.33)$$

The above integrals (and Casimir) are in involution with regard to the Poisson bracket

$$\{G,H\}=(\nabla_{\widehat{\xi}}G)^tM_{\widehat{\xi}}\nabla_{\widehat{\xi}} H\ , \eqno (1.34)$$

where the cosymplectic form

$$M_{\widehat{\xi}}=\left(\matrix{\xi_1&&\cr &\ddots&\cr &&\xi_{_N}\cr} \right)\ .\eqno (1.35)$$

For all $N$ the system (1.18), by contraction with (1.19), has the Casimir integral (1.29). For even $N$,

$$N=2k+2\ ,\eqno (1.36)$$

contraction of (1.18) with the null vector of $A$, (1.21), obtains the constraint condition

$$C_1=\sum_{j=1}^{^N}(-1)^j\xi_j\equiv 0\ .\eqno (1.37)$$

In Sec. II we find that, when $N=2k+2$, the system (1.18) is a $k$-dimensional completely integrable Hamiltonian system with Casimir (1.29) and $k$ independent integrals (1.31) in involution. The integrals are in involution with the constraint (1.37). That is, the constraint is preserved by the flows. Furthermore, we find that systems (1.18) are equivalent to the periodic Kac-Van Moerbeke (KM) equations [6]. In effect, the KM flow commutes with (1.18). This implies, by a known result [7], that (1.18) is equivalent to the periodic Toda lattice when $N$ is even.

In Sec. III we find that the periodic fixed points of the BT for the Boussinesq equation [8] is of the form (1.18) and (1.28) for appropriate $A$, $B$, $\Omega$. Again, the system is shown to have a Hamiltonian structure and the integrals are found by a method similar to that developed for the KdV systems. However, the Boussinesq systems are not equivalent to the KdV systems.

In Sec. IV we define, by a generalization of the KdV and Boussinesq systems, a hierarchy of Hamiltonian systems of the form (1.18). Certain integrals are found.