Completely exceptional $2^\textrm{nd}$ order PDEs via conformal geometry and BGG resolution

Abstract

By studying the development of shock waves out of discontinuity waves, in 1954 P. Lax discovered a class of PDEs, which he called 'completely exceptional', where such a transition does not occur after a finite time. A straightforward integration of the completely exceptionality conditions allowed Boillat to show that such PDEs are actually of Monge-Ampere type. In this paper, we first recast these conditions in terms of characteristics, and then we show that the completely exceptional PDEs, with 2 or 3 independent variables, can be described in terms of the conformal geometry of the Lagrangian Grassmannian, where they are naturally embedded. Moreover, for an arbitrary number of independent variables, we show that the space of r-th degree sections of the Lagrangian Grassmannian can be resolved via a BGG operator. In the particular case of 1st degree sections, i.e., hyperplane sections or, equivalently, Monge-Ampere equations, such operator is a close analog of the trace-free second fundamental form.