Backlund transformations and Hamiltonian flows

TL;DR

This paper demonstrates that certain Backlund transformations for integrable systems can be viewed as solutions to non-autonomous Hamiltonian equations, linking discrete maps with continuous Hamiltonian flows, exemplified by the Toda lattice.

Contribution

It introduces a novel interpretation of Backlund transformations as Hamiltonian flows and identifies the interpolating Hamiltonian for discrete transformations.

Findings

Backlund transformations can be seen as solutions to non-autonomous Hamiltonian equations.

The interpolating Hamiltonian is a linear combination of spectral curve integrals.

Application to the Toda lattice illustrates the theoretical framework.

Abstract

In this work we show that, under certain conditions, parametric Backlund transformations (BTs) for a finite dimensional integrable system can be interpreted as solutions to the equations of motion defined by an associated non-autonomous Hamiltonian. The two systems share the same constants of motion. This observation lead to the identification of the Hamiltonian interpolating the iteration of the discrete map defined by the transformations, that indeed will be a linear combination of the integrals appearing in the spectral curve of the Lax matrix. An application to the Toda periodic lattice is given.