Integrability via Geometry: Dispersionless Differential Equations in Three and Four Dimensions
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Communications in
Mathematical Physics
Integrability via Geometry: Dispersionless Differential Equations in Three and Four Dimensions David M. J. Calderbank1 , Boris Kruglikov2 1 Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
E-mail: [email protected]
2 Institute of Mathematics and Statistics, UiT the Arctic University of Norway, 90-37 Tromsø, Norway.
E-mail: [email protected] Received: 18 February 2020 / Accepted: 18 September 2020 © The Author(s) 2020
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. 1. Introduction and Main Results The 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]. Recently, such a relation has been established for several classes of second order equations in 3D and one class in 4D [17]. Namely the following equivalences have been established: Integrability via hydrodynamic reductions : dJ tt JJJ tt JJ t t JJ tt JJ t t JJ t t JJ tt JJ t t JJ t t JJ ztt $ Dispersionless Lax pair Integrable background ks +3 with spectral parameter geometry
D. M. J. Calderbank, B. Kruglikov
Hydrodynamic integrability in 2D (also written “1 + 1 dimension”) was introduced in [9] and elaborated in [35]. Integrability via hydrodynamic reductions in d 3 dimensions was developed in [16]. This method, although constructive, is not universal, as it applies only to translation invariant equations (invariantly, this requires the existence of a d-dimensional abelian contact symmetry group). Thus the upper part of the above diagram, at least at present, does not extend to the general class of second order PDEs. On the other hand, the two other ingredients of the diagram are universal. The main aim of this paper is to prove the bottom equivalence for large class of 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 (these geometries are “backgrounds” for integrable gauge theories [2]). Consider a second order PDE E : F(x, u, ∂u, ∂ 2 u) = 0
(1)
for a scalar function u of an independent variable x on a connected manif
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