Hamiltonian differential operators and contact geometry (Q1097558)

The problem of classification of Hamiltonian differential operators is investigated. An operator (1) \(L=\sum^{N}_{i=0}a_ i(x,u,u_ x,...)(\frac{d}{dx})^ i\) is called the Hamiltonian if the formula \[ \{I,J\}=\int (\delta I/\delta u)L(\delta J/\delta u)dx \] defines a Poisson bracket on the space of all functionals \(H=\int h(x,u,u_ x,...)dx\). The purpose of the paper is to get some Darboux-type results to classify Hamiltonian operators of the first and the third order under contact Bäcklund transformations. The notion of a Hamiltonian pair of operators, i.e. \(L_ 1,L_ 2\) of the form (1), such that \(\lambda L_ 1+\mu L_ 2\) is a Hamiltonian operator for any \(\lambda\),\(\mu\in {\mathbb{R}}\), is very important in the theory of integrability of evolution equations. The author studies the problem of classification of such pairs, consisting of \(L_ 1\) having first order and \(L_ 2\) having third order.