SECOND-ORDER QUASILINEAR PDEs AND CONFORMAL STRUCTURES IN PROJECTIVE SPACE

Abstract

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 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(p)dpidpj. 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 fij(p) dpidpj 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.

The four-dimensional Martínez Alonso–Shabat equation: Differential coverings and recursion operators

“…which involves a parameter λ and can be considered as a four-dimensional generalization of the ABC equation (6) and reduces to the latter if z = t. Moreover, there exists a 4D generalization of (9): for µ = λ the system…”

Linearly degenerate partial differential equations and quadratic line complexes

“…Calculating the Cotton tensor (whose vanishing is responsible for conformal flatness in three dimensions) we obtain a complete list of quadratic complexes with the flat conformal structure. Recall that the flatness of f ij dp i dp j is a necessary condition for integrability of the corresponding PDE [7]. We observe that the requirement of conformal flatness imposes further constraints on the parameters appearing in cases 1-11 of Theorem 2, which are characterised by certain coincidences among eigenvalues of the corresponding Jordan normal forms of QΩ −1 (some Segre types do not possess conformally flat specialisations at all).…”

“…It was pointed out in [7] that flatness of the conformal structure (2) is the necessary condition for integrability (this simplifies the classification of integrable equations, indeed, the corresponding complexes must be contained in the list of Theorem 3). The main result reads as follows: …”

“…In what follows we label conformally flat subcases by their 'refined' Segre symbols, e.g., the symbol [ (11)(11)(11)] denotes the subcase of [111111] with three pairs of coinciding eigenvalues, the symbol [(111)(111)] denotes the subcase with two triples of coinciding eigenvalues, etc, see [22]. Although the subject is classical, the following result is new: Since conformal flatness is the necessary condition for integrability, a complete list of linearly degenerate integrable PDEs can be obtained by going through the list of Theorem 3 and either calculating the integrability conditions as derived in [7], or verifying the existence of a Lax pair. Another possibility is to utilise the recent result of [15] …”

“…This correspondence between quasilinear wave equations and conformal structures in projective space was proposed and thoroughly investigated in [7].…”