Fe b 20 08 Second order quasilinear PDEs and conformal structures in projective space

We investigate second order quasilinear equations of the form fijuxixj = 0 where u is a function of n independent variables x1, ..., xn, and the coefficients fij are functions of the first order derivatives p = ux1 , ..., p n = 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 P with coordinates p, ..., p. The coefficient matrix fij defines on P n a conformal structure fij(p)dp dp . 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(p)dp dp 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. MSC: 35Q58, 37K05, 37K10, 37K25.