F eb 2 00 4 A class of Einstein – Weyl spaces associated to an integrable system of hydrodynamic type

A class of Einstein–Weyl spaces associated to an integrable system of hydrodynamic type. Abstract HyperCR Einstein–Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can be in principle constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions yield new Einstein–Weyl structures. Let us consider a pair of quasi-linear PDEs u t + w y + uw x − wu x = 0, u y + w x = 0, (1.1) for two real functions u = u(x, y, t), w = w(x, y, t). This system of equation has recently attracted a lot of attention in the integrable systems literature [18, 9, 10, 17]. In [3] it arouse in a different context, as a symmetry reduction of the heavenly equation. The system (1.1) shares many properties with two more prominent dispersionless integrable equations: the dispersionless Kadomtsev–Petviashvili equation (dKP), and the SU (∞) Toda equation , but it is simpler in some ways. (1.2) does not contain derivatives with respect to the spectral parameter λ (the Lax pairs for SU (∞) Toda, and dKP contain such terms). • Consider a one-form e(λ) = dx − udy + wdt + λ(dy − udt) + λ 2 dt. The system (1.1) is equivalent to the Frobenius integrability condition e(λ) ∧ de(λ) = 0, (1.3) where d keeps λ constant. This formulation is dual to the Lax representation (1.2), because the distribution spanned by L and M can be defined as the kernel of e(λ). The analogous dual formulations of dKP and SU (∞) Toda involve distributions defined by two–forms [16, 21, 5], and are considerably more complicated. One of the aims of this paper is to provide a twistor description of (1.1) given by the following