DIS

Abstract

An unied theory of the construction of the bi-Hamiltonian nonlinear evolution hierarchies such as eld, lattice and q-discrete soliton hierarchies, will be presented. I will give a brief review of the concept of time scales, including definitions of
-derivative and
-integral. A construction of the bi-Hamiltonian structures for integrable systems on regular time scales will be presented. The main result consists on the denition of the trace functional on an algebra of -pseudo-dierential operators, valid on an arbitrary regular time scale. I will illustrate the theory by
-dierential counterparts of AKNS and Kaup-Broer hierarchies. The talk will be based on the article: arXiv:0810.0766. (This is joint work with Maciej Blaszak and Burcu Silindir.)