On the dbar-dressing method applicable to heavenly equation

Abstract

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, $τ$-function for the hierarchy is defined. Addition formulae (generating equations) for the $τ$-function are found. It is demonstrated that $τ$-function for heavenly equation hierarchy is given by the action for $\dbar$-problem evaluated on the solution of this problem.