Multi-Hamiltonian Lax Dynamics in (1+1)-Dimensions

Abstract

In the previous chapters we presented a variety of multi-Hamiltonian integrable dynamical systems of finite and infinite dimensions and their common geometric and algebraic features. What we still lack, is a systematic method for the construction of multi-Hamiltonian infinite dimensional systems. This chapter will present an approach to a unified description of the integrable equations, based on the use of a simple and powerful algebraic tool, the so called R —structure (or R —matrix) [62], [175], [176]. This approach is formulated in a rather abstract algebraic way, but as an advantage one gets a simple and effective method for analysis of the multi-Hamiltonian structure of integrable systems. The crucial point of this approach is the observation that the Lax equation (1.2) can be treated as an abstract dynamical system from which the ‘physical’ dynamical systems are obtained by introducing suitable charts. Hence, the phase space for most of these equations can be regarded as given by the set of Lax operators taking values from some Lie algebra. This abstract representation of integrable dynamics is referred to as the Lax dynamics. The multi-Hamiltonian construction of the integrable equations becomes quite transparent when using the terminology of R —structures. Based on results of Gelfand and Dikii [91], Adler [4]used a Lie algebraic setting to describe integrable partial differential equations via their Lax representations. As an important consequence it turned out that integrable systems of a different nature (discrete lattice systems or differential equations) may be constructed in a similar manner using Lie algebra techniques. The celebrated Adler-Gelfand-Dikii (AGD) scheme starts with a dual Lie algebra as a natural phase space for integrable equations. The Lie-Poisson bracket associated with the Lie algebra structure provides a natural Hamiltonian structure, and the invariant functions provide a natural set of functions in involution on the algebra. In order to obtain a nontrivial integrable dynamics for these functions, only a few additional structures have to be provided, which are again of a purely Lie algebraic nature. As the simplest example, a decomposition of the original algebra into proper subalgebras gives rise to a hierarchy of integrable Hamiltonian equations. It turn out that this construction may be regarded as a special case of a yet larger picture. Following Drinfeld’s ideas [62]Semenov showed that the notion of classical R —structures leads to an algebraic construction of integrable Systems generalizing the AGD scheme [175]. In [117],[150]it was shown that there are in fact three natural Poisson brackets associated with such classical R —structures. They lead to an abstract tri-Hamiltonian formulation of the Lax equations describing the nonlinear integrable systems. Applications of this general construction to the particular algebra of pseudo-differential operators or algebra of shift operators leads to a compact formulation of the multi-Hamiltonian structures for certain classes of integrable hierarchies.