Second-order PDEs in four dimensions with half-flat conformal structure

  title={Second-order PDEs in four dimensions with half-flat conformal structure},
  author={S. Berjawi and Eugene V. Ferapontov and Boris S. Kruglikov and V. S. Novikov},
  journal={Proceedings of the Royal Society A},
We study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable… 
1 Citations


Integrability via Geometry: Dispersionless Differential Equations in Three and Four Dimensions
We prove that the existence of a dispersionless Lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation (PDE) is equivalent to the canonical
On the Einstein-Weyl and conformal self-duality equations
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
Integrability of Dispersionless Hirota-Type Equations and the Symplectic Monge–Ampère Property
We prove that integrability of a dispersionless Hirota-type equation implies the symplectic Monge–Ampère property in any dimension $\geq 4$. In 4D, this yields a complete classification of
Integrable Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations
We study integrable non-degenerate Monge–Ampère equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining those
Dispersionless integrable systems in 3D and Einstein-Weyl geometry
For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if
Self-dual gravity as a two-dimensional theory and conservation laws
Starting from the Ashtekar Hamiltonian variables for general relativity, the self-dual Einstein equations (SDE) may be rewritten as evolution equations for three divergence-free vector fields given
Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian
We investigate integrable second-order equations of the formwhich typically arise as the Hirota-type relations for various (2 + 1)-dimensional dispersionless hierarchies. Familiar examples include
On an integrable multi-dimensionally consistent 2n + 2n-dimensional heavenly-type equation
The ‘universal’ character of this novel equation of vanishing Pfaffian type is demonstrated by retrieving and generalizing to higher dimensions a great variety of well-known integrable equations such as the dispersionless Kadomtsev–Petviashvili and Hirota equations and various avatars of the heavenly equation governing self-dual Einstein spaces.