2024/02/19 by W. K. Schief, Schief, W. K., Udo Hertrich-Jeromin +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Waves and Solitons #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2402.11779
openalex publication_date 2024/02/19 · openalex created_date 2024/02/21 · openalex updated_date 2026/07/28
It is shown that a canonical geometric setting of the integrable TED equation is a Kahlerian tangent bundle of an affine manifold. The remarkable multi-dimensional consistency of this 4+4-dimensional dispersionless partial differential equation arises naturally in this context. In a particular 4-dimensional reduction, the affine manifolds turn out to be self-dual Einstein spaces of neutral signature governed by Plebanski's first heavenly equation. In another reduction, the affine manifolds are Hessian, governed by compatible general heavenly equations. The Legendre invariance of the latter gives rise to a (dual) Hessian structure. Foliations of affine manifolds in terms of self-dual Einstein spaces are also shown to arise in connection with a natural 5-dimensional reduction.