SECOND-ORDER QUASILINEAR PDEs AND CONFORMAL STRUCTURES IN PROJECTIVE SPACE

Abstract

We investigate second-order quasilinear equations of the form f ij u x i x j = 0, where u is a function of n independent variables x 1 , …, x n , and the coefficients f ij depend on 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 (p)dp i dp j . The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived, implying that the moduli space of integrable equations is 20-dimensional. Any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. The integrability conditions imply that the conformal structure f ij (p) dp i dp j is conformally flat, and possesses infinitely many three-conjugate null coordinate systems parametrized by three arbitrary functions of one variable. Integrable equations provide examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms.