Generalized Hirota bilinear identity and integrable q-difference and lattice hierarchies.

Hirota bilinear identity for Cauchy-Baker-Akhieser (CBA) kernel is introduced as a basic tool to construct integrable hierarchies containing lattice and q-difference times. Determinant formula for the action of meromorphic function on CBA kernel is derived. This formula gives opportunity to construct generic solutions for integrable lattice equations.Comment: 6 pages, LaTeX, the text of the talk at NLS-94, Chernogolovka, Russia, July 94

On the dbar-dressing method applicable to heavenly equation

The \dbar-dressing scheme based on local nonlinear vector \dbar-problem is developed. It is applicable to multidimensional nonlinear equations for vector fields, and, after Hamiltonian reduction, to heavenly equation. Hamiltonian reduction is described explicitely in terms of the \dbar-data. An analogue of Hirota bilinear identity for heavenly equation hierarchy is introduced, $\tau$-function for the hierarchy is defined. Addition formulae (generating equations) for the $\tau$-function are found. It is demonstrated that $\tau$-function for heavenly equation hierarchy is given by the action for \dbar-problem evaluated on the solution of this problem.Comment: 11 page