On the Einstein-Weyl and conformal self-duality equations

Abstract

The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as `master dispersionless systems' in four and three dimensions respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note we present, in specially adapted coordinate systems, explicit forms of the corresponding equations and their Lax pairs. In particular, we demonstrate that any Lorentzian Einstein-Weyl structure is locally given by a solution to the Manakov-Santini system, and we find a system of two coupled third-order scalar PDEs for a general anti-self-dual conformal structure in neutral signature.

More references added. Final version, to appear in JMP

References in corpus (5)

• Interpolating Dispersionless Integrable System
• Integrable Background Geometries
• The Inverse Spectral Transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions
• Integrability of the Manakov--Santini hierarchy
• Quaternion-Kaehler four-manifolds and Przanowski's function

Cited by in corpus (24)

• Twistor theory at fifty: from contour integrals to twistor strings
• New integrable (3+1)-dimensional systems and contact geometry
• Integrable dispersionless PDE in 4D, their symmetry pseudogroups and deformations
• The quadric ansatz for the $mn$-dispersionless KP equation, and supersymmetric Einstein-Weyl spaces
• Veronese webs and nonlinear PDEs
• On a class of 2D integrable lattice equations
• Second-order PDEs in 3D with Einstein-Weyl conformal structure
• SDYM equations on the self-dual background
• Integrable (3+1)-dimensional system with an algebraic Lax pair
• Dispersionless (3+1)-dimensional integrable hierarchies
• Matrix extension of the Manakov-Santini system and integrable chiral model on Einstein-Weyl background
• Multidimensional integrable systems from contact geometry
• GL(2)-geometry and complex structures
• Second-order PDEs in 4D with half-flat conformal structure
• The Cayley cubic and differential equations
• Dispersionless integrable systems and the Bogomolny equations on an Einstein-Weyl geometry background
• Parallel spinors on Lorentzian Weyl spaces
• Integrable (3+1)-dimensional generalization for dispersionless Davey--Stewartson system
• Differential invariants of Einstein-Weyl structures in 3D
• Differential Invariants of Self-Dual conformal structures
• Dispersionless BKP equation, the Manakov-Santini system and Einstein-Weyl structures
• On a class of integrable Hamiltonian equations in 2+1 dimensions
• From the conformal self-duality equations to the Manakov-Santini system
• Matrix extension of multidimensional dispersionless integrable hierarchies