Towards a Multi-Hamiltonian Theory of (2+1)-Dimensional Field Systems

Abstract

The last chapter of this book is devoted to the algebraic theory of (2 + 1)-dimensional field systems. The reason why the algebraic theory of integrable field systems in multidimensions is remarkably different from that in one space dimension was an important observation made in 1985 by Konopelchenko and Zakharov [195]. Actually, they proved that for (n + 1)-dimensional field theory for n > 1 there are no recursion operators expressible by pseudo-differential matrix operators as for the (1 + 1) case. As a consequence, there are no second Poisson structures in the same form. Hence, one can try to find them only in the form of bilocal objects, i.e. in a kernel representation, for example. It makes the algebraic theory of integrable field (lattice) systems in multidimensions more complex when compared to these in one space dimension.