Second order quasilinear PDEs and conformal structures in projective space
Abstract
We investigate second order quasilinear equations of the form fij uxixj=0 where u is a function of n independent variables x1, ..., xn, and the coefficients fij are functions of the first order derivatives p1=ux1, >..., pn=uxn 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 Pn with coordinates p1, ..., pn. The coefficient matrix fij defines on Pn a conformal structure fij dpidpj. 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 fij, 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 fij dpidpj 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.