Boris G. Konopelchenko
(Universita` di Lecce, Phys. Dept.)
On the Dressing method applicable to the 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.