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

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.

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

“…Proof of theorems 1.2, 1.3. As mentioned in the Introduction, theorem 1.2 follows from the general result of [3]. As for theorem 1.3, note that a generic four-dimensional travelling wave reduction of a multi-dimensional integrable PDE of rank 4 will automatically be non-degenerate and integrable, because a generic travelling wave reduction of a non-trivial disperionless Lax pair is itself non-trivial.…”

“…Hence the reduced PDE must be of Monge-Ampère type by theorem 1.1 demonstrated above. Indeed, by [3] the existence of a non-trivial Lax pair in four dimensions yields half-flatness of the conformal structure on any solution, so theorem 1.1 applies. Henceforth, since all travelling wave reductions of a multi-dimensional PDE are of Monge-Ampère type, by the argument similar to that in [2] we conclude that the initial equation in d dimensions should be of the Monge-Ampère type as well.…”

“…Before moving on to the next proof recall that non-triviality of dispersionless Lax pairs in four dimensions was discussed in detail in [3] (in three dimensions the non-triviality condition of [3] is somewhat more involved). In higher dimensions d > 4 we say that a dispersionless Lax pair is non-trivial if the commutativity of the generators of any modification of the Lax pair which is identical on the equation holds essentially modulo the equation.…”

“…Integral surfaces of the involutive distribution X, Y give totally null surfaces of the corresponding conformal structure [g], thus providing an alternative proof of its halfflatness. This is a particular case of the general result of [3] [20] and the novel inverse scattering approach [21,22] to PDEs with half-flat conformal structure.…”

“…It was demonstrated recently in [3] that, in four dimensions, half-flatness of [g] is equivalent to the existence of a non-trivial dispersionless Lax pair in parameter-dependent vector fields. This leads to the following result.…”