Reductions of self-dual Einstein equations

On Reductions of Noncommutative Anti-Self-Dual Yang-Mills Equations

In this paper, we show that various noncommutative integrable equations can be derived from noncommutative anti-self-dual Yang-Mills equations in the split signature, which include noncommutative versions of Korteweg-de Vries, Non-Linear Schrödinger, N -wave, Davey-Stewartson and Kadomtsev-Petviashvili equations. U(1) part of gauge groups for the original Yang-Mills equations play crucial roles...

The dispersive self-dual Einstein equations and the Toda Lattice

The Boyer-Finley equation, or SU(∞)-Toda equation is both a reduction of the self-dual Einstein equations and the dispersionless limit of the 2d-Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both contains the Toda lattice equation and whose dispersionless limit is the familiar self-dual Einstein equation. Such a system is...

New integral equation form of integrable reductions of Einstein equations

A new development of the “monodromy transform” method for analysis of hyperbolic as well as elliptic integrable reductions of Einstein equations is presented. Compatibility conditions for some alternative representations of the fundamental solutions of associated linear systems with spectral parameter in terms of a pair of dressing (“scattering”) matrices give rise to a new set of linear (quasi...

New Examples of Einstein Self-dual Metrics

Symmetry Reductions of the Lax Pair of the Four-Dimensional Euclidean Self-Dual Yang-Mills Equations

The reduction by symmetry of the linear system of the self-dual Yang-Mills equations in four-dimensions under representatives of the conjugacy classes of subgroups of the connected part to the identity of the corresponding Euclidean group under itself is carried out. Only subgroups leading to systems of differential equations nonequivalent to conditions of zero curvature without parameter, or t...