Dispersionless integrable PDEs via geometry

David Calderbank, University of Bath. Part of the algebra, geometry and integrable systems colloquium.

A determined nonlinear PDE system on a manifold M is integrable by a dispersionless Lax pair if it arises as the integrability condition for a rank 2 distribution on a rank 1 bundle over M. By establishing that any such Lax pair is characteristic on solutions, we show that when the characteristic variety is a quadric (e.g. for nondegenerate second order scalar PDEs), the PDE is dispersionless integrable if and only if the induced conformal structure is self-dual (for n=4) or Einstein-Weyl (for n=3) on any solution. These results unify and extend work of E. Ferapontov et al. (Joint work with Boris Kruglikov).

David Calderbank, University of Bath