Hamiltonian Theory of Bäcklund Transformations

Abstract

We consider theory of Bäcklund transformations of dynamical system

$$iqt = \delta H/\delta p,ipt = - \delta H/\delta q$$

(1)

with polynomial hamiltonia.n density H

$$H = {p_x}{q_x} + {q_x}^2A(p) + {q_x}B(p) + C(p)$$

(2.1)

where A(p),B(p), C(p) are any quadratic in p polynomials. This theory establishes a direct between dynamical systems (1), (2) and fully discrete lattice equations described in Section 2. We will show these equations can be considered as a discrete model of the corresponding dynamical system (1). In Sections 3, 4 we discuss some applications of this relationship between (1) and its discrete counterpart.