A family of integrable nonlinear equations of hyperbolic type

Abstract

A new system of integrable nonlinear equations of hyperbolic type, obtained by a two-dimensional reduction of the anti-self-dual Yang–Mills equations, is presented. It represents a generalization of the Ernst–Weyl equation of General Relativity related to colliding neutrino and gravitational waves, as well as of the fourth order equation of Schwarzian type related to the KdV hierarchies, which was introduced by Nijhoff, Hone, and Joshi recently. An auto-Bäcklund transformation of the new system is constructed, leading to a superposition principle remarkably similar to the one connecting four solutions of the KdV equation. At the level of the Ernst–Weyl equation, this Bäcklund transformation and the associated superposition principle yield directly a generalization of the single and double Harrison transformations of the Ernst equation, respectively. The very method of construction also allows for revealing, in an essentially algorithmic fashion, other integrability features of the main subsystems, such as their reduction to the Painlevé transcendents.