Integrability via Geometry: Dispersionless Differential Equations in Three and Four Dimensions

David M J Calderbank,Boris Kruglikov

doi:10.1007/s00220-020-03913-y

Abstract

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 conformal structure defined by the symbol being Einstein–Weyl on any solution in 3D, and self-dual on any solution in 4D. The first main ingredient in the proof is a characteristic property for dispersionless Lax pairs. The second is the projective behaviour of the Lax pair with respect to the spectral parameter. Both are established for nondegenerate determined systems of PDEs of any order. Thus our main result applies more generally to any such PDE system whose characteristic variety is a quadric hypersurface.

Highlights

Introduction and Main ResultsThe integrability of dispersionless partial differential equations is well known to admit a geometric interpretation
Twistor theory [26,29] gives a framework to visualize this for several types of integrable systems, as demonstrated by many examples [2,10,11, 14, 19, 30, 37]
The main aim of this paper is to prove the bottom equivalence for large class of partial differential equation (PDE) systems, including general second order PDEs, in 3D and 4D, where “integrable background geometry” means that a canonical conformal structure on solutions of the equation is Einstein–Weyl in 3D and self-dual in 4D

Summary

Introduction and Main Results

The integrability of dispersionless partial differential equations is well known to admit a geometric interpretation. The EW/SD property provides a canonical characteristic Lax pair, which, if the PDE on u has order , depends on at most + 1 derivatives of u ( if the PDE is quasilinear), and satisfies a ‘normality’ condition off shell which is useful in computations. None of these properties were assumed a priori. This condition means that the equation E appears nontrivially in the symbol of the integrability condition for (i.e., at highest order) From this we deduce the projective property of the Lax pair, and prove Theorem 2. We discuss pseudopotentials and their relation to contact coverings, the twistor interpretation of this relationship, and potential generalizations of the theory

Lax Pairs

The Characteristic Condition for Dispersionless Lax Pairs

Projective Dependence on the Spectral Parameter

Applications and Generalizations