Second order quasilinear PDEs and conformal structures in projective space

Abstract

We investigate second order quasilinear equations of the form f_{ij} u_{x_ix_j}=0 where u is a function of n independent variables x_1, ..., x_n, and the coefficients f_{ij} are functions of the first order derivatives p^1=u_{x_1}, >..., p^n=u_{x_n} only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n+1, R), which acts by projective transformations on the space P^n with coordinates p^1, ..., p^n. The coefficient matrix f_{ij} defines on P^n a conformal structure f_{ij} dp^idp^j. In this paper we concentrate on the case n=3, although some results hold in any dimension. The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived. These conditions constitute a complicated over-determined system of PDEs for the coefficients f_{ij}, which is in involution. We prove that the moduli space of integrable equations is 20-dimensional. Based on these results, we show that any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. Reformulated in differential-geometric terms, the integrability conditions imply that the conformal structure f_{ij} dp^idp^j is conformally flat, and possesses an infinity of 3-conjugate null coordinate systems. Integrable equations provide an abundance of explicit examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms.