Fe b 20 08 Second order quasilinear PDEs and conformal structures in projective space
نویسنده
چکیده
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.
منابع مشابه
9 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 depend on 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 coordina...
متن کاملSecond 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 depend on 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 coordina...
متن کامل`-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras
We construct, for any given ` = 12 + N0, the second-order, linear PDEs which are invariant under the centrally extended Conformal Galilei Algebra. At the given `, two invariant equations in one time and ` + 12 space coordinates are obtained. The first equation possesses a continuum spectrum and generalizes the free Schrödinger equation (recovered for ` = 12) in 1 + 1 dimension. The second equat...
متن کاملFe b 20 08 Boundary Conformal Field Theory ∗
Boundary conformal field theory (BCFT) is simply the study of conformal field theory (CFT) in domains with a boundary. It gains its significance because, in some ways, it is mathematically simpler: the algebraic and geometric structures of CFT appear in a more straightforward manner; and because it has important applications: in string theory in the physics of open strings and D-branes, and in ...
متن کاملar X iv : 0 80 2 . 28 20 v 1 [ m at h - ph ] 2 0 Fe b 20 08 Lagrangian and Hamiltonian two - scale reduction ∗
Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a...
متن کامل