BÄCKLUND TRANSFORMATIONS

The KdV and Boussinesq systems are instances of the general system in component form [34]

$$\frac{\xi _{j,x}}{\xi _j} +\frac{\xi _{j+1,x}}{\xi _{j+1}} + \dots +\frac{\xi _{j+p,x}}{\xi _{j+p}} = \xi _j - \xi _{j+p}$$ (76)

where $j=1,2,\dots \pmod{N}$. The KdV systems correspond to $p=1$ and the Boussinesq to $p=2$. Let the circulant forward shift matrix [33] be

$$C=\bf c \bf i \bf r \bf c[0,1,0,0,\dots,0] .$$

In the N-vector form equations (76) are

$$A\left(\begin{array}{c} \frac{\xi _{1,x}}{\xi _1} \\ \vdots \\ \frac{\xi _{N,x}}{\xi _N} \end{array}\right) = B\hat \xi$$ (77)

with

$$\begin{array}{ll} A &= I+C+\dots +C^p \\ B &= I-C^p. \end{array}$$

The casimir integrals of (117) correspond to the null vectors of B. The null vectors of A produce the constraints.

Associated with the principal Casimir, for any N

$$H_N=\prod_{j=1}^N\xi _j$$

we find the principal integrals of (77)

$$H_{N-pm-m} = L^m\circ H_N$$ (78)

,where $m=0,1,2,\dots$ and

$$L=\sum_{j=1}^N\partial_{\xi _j}\partial_{\xi _{j+1}} \dots \partial_{\xi _{j+p}}.$$

The systems (116) have a Hamiltonian structure

$$A\left(\begin{array}{c} \frac{\xi _{1,x}}{\xi _1} \\ \vdots \... ...} & \xi _N \end{array}\right) \bigtriangledown_{\hat \xi } H_1$$ (79)

,where $H_1=\sum_{j=1}^N\xi _j.$

The higher-order equations associated with the integrals (78) are

$$A\left(\begin{array}{c} \frac{\xi _{1,x}}{\xi _1} \\ \vdots \... ...i _N \end{array}\right) \bigtriangledown_{\hat \xi }H_{N-pm-m}.$$ (80)

When A is invertible, then

$$\Omega=A^{-1}B$$

is an antisymmetric circulant matrix.

We have the systems

$$\hat \xi _{,x}=M_{\hat\xi }\bigtriangledown_{\hat\xi }H_1$$ (81)

and

$$\hat\xi _{,x}=M_{\hat\xi }\bigtriangledown_{\hat\xi }H_{N-pm-m}$$ (82)

where

$$M_{\hat\xi }=\left(\begin{array}{lll} \xi _1 & & \\ {} & \dd... ...1 & & \\ {} & \ddots & \\ {} & {} & \xi _N \end{array}\right)$$

is the co-symplectic form.