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, $\tau$-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.

11 pages

References in corpus (5)

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• Towards a theory of differential constraints of a hydrodynamic hierarchy
• Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type
• Hydrodynamic reductions of the heavenly equation
• Nonlinear Beltrami equation and tau-function for dispersionless hierarchies

Cited by in corpus (25)

• Inverse Scattering Problem for Vector Fields and the Cauchy Problem for the Heavenly Equation
• On the integrability of symplectic Monge-Ampére equations
• Dunajski generalization of the second heavenly equation: dressing method and the hierarchy
• On the solutions of the second heavenly and Pavlov equations
• Einstein--Weyl geometry, dispersionless Hirota equation and Veronese webs
• The Inverse Spectral Transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions
• Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
• `Interpolating' differential reductions of multidimensional integrable hierarchies
• Linearly degenerate hierarchies of quasiclassical SDYM type
• On the heavenly equation hierarchy and its reductions
• On a class of integrable systems of Monge-Ampère type
• Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields
• Inverse Scattering Problem for Vector Fields and the Heavenly Equation
• Dressing method based on homogeneous Fredholm equation: quasilinear PDEs in multidimensions
• Partially integrable systems in multidimensions by a variant of the dressing method. 1
• Integrable systems in 4D associated with sixfolds in Gr(4,6)
• Six-dimensional heavenly equation. Dressing scheme and the hierarchy
• Second-order PDEs in 4D with half-flat conformal structure
• Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property
• Integrability properties of symmetric 4+4-dimensional heavenly type equation
• A note on the Hyper--CR equation, and gauged $N=2$ supergravity
• The Cauchy problem for the Pavlov equation
• Doubrov-Ferapontov general heavenly equation and the hyper-Kähler hierarchy
• On some exact solutions of heavenly equations in four dimensions
• Matrix extension of multidimensional dispersionless integrable hierarchies