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

Abstract

<jats:p>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 dispersionless PDEs in dimensions four and higher are of Monge–Ampère type. Some partial classification results of Monge–Ampère equations in four dimensions with half-flat conformal structure are also obtained.</jats:p>

Webs, Nijenhuis operators, and heavenly PDEs

“…4. Equation G 0 fits to the definition of equations of "first heavenly type" [BFKN20] and by results of that paper should be contact equivalent to one of three equations listed in [BFKN20, Table 2].…”

“…3. Equation A 0 appeared in [BFKN20] and it was shown that A 0 and A 4 (which is the same as (0.3)) are contact nonequivalent.…”

“…a function f such that the 2-form β λ := dd ( N −λId) −1 f annihilates T F λ ; here d N is the so-called Nijenhuis exterior derivative corresponding to a Nijenhuis operator N. The two-form β λ is of rank 2, i.e. satisfies the identity β λ ∧ β λ = 0, (0.6) equivalent to a second order "heavenly" PDE on the function f , which is integrable in the sense that it possesses a dispersionless Lax pair [BFKN20], i.e. vector fields X 1 (λ), X 2 (λ) depending on the second jet of f and an auxiliary parameter λ such that the Frobenius integrability condition [X 1 , X 2 ] ∈ X 1 , X 2 holds identically modulo the equation and its differential consequences 1 .…”

Deformations of dispersionless Lax systems *

“…In this paper we study dispersionless integrable systems that arise as integrability conditions for foliation F on bundle B with 1-dimensional fibers over manifold M. This framework can be adapted to the heavenly equations [29,30], the Manakov-Santini system [24] (see also [7]), the Dunajski-Tod equation [9] (see also [3]), the hyper-CR equation [6], the dispersionless Hirota equation [8] (known also as the abc-equation [31]), equations related to GL(2)-structures and web geometry [11,17,18,[20][21][22] and many others [1,2,5,10,13,15,23,25]. The quotient space T = B/F is usually referred to as the (real) twistor space, and it gives rise to the following double fibration picture M ← B → T .…”

“…All the aforementioned systems describe well known classes of geometric structures on M. In particular, as in the case of the heavenly equations, these structures encompass anti-self-dual metrics (e.g. [1,3,7,9,29]), Einstein-Weyl structures (e.g. [7,8,10,21,24]), or higher dimensional counterparts like GL(2)-structures or Veronese and Kronecker webs (see [11,17,18,20,22]).…”