Bäcklund Transformations in Discrete Variational Principles for Lie-Poisson Equations

Abstract

We consider a dynamical system on the dual of a Lie algebra. On the dual of that algebra there is another linear Poisson structure. This system is integrable for one of the Poisson structures because it admits a suitable Lax representation. The discrete variational principle is applied to the problem given by the non-usual linear Poisson structure to obtain Lie-Poisson integrators which preserve all the Casimir functions of the system. In the 19th century Backlund transformations were introduced in the area of partial differential equations as transformations that map solutions to solutions. It is known that Backlund transformations satisfy some specific properties such as commutativity. We geometrically define Backlund transformations associated with the obtained Lie-Poisson integrators under some invariance assumptions. The existence of an invariant scalar product that identifies the Lie algebra and the dual of the Lie algebra allows to establish the connection with the results proved in Suris’ book on the Lie algebra. We will make clear the constructions by looking at the Toda Lattice example.