On a class of systems of hyperbolic equations describing pseudo-spherical or spherical surfaces

2021/12/09 by Filipe Kelmer, Kelmer, Filipe, Keti Tenenblat +1 · 3 citations
Mathematics · #35L51 #53B20 #58J60 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #Differential Geometry (math.DG) #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2112.05040

openalex publication_date 2021/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider systems of partial differential equations of the form \ u_xt=F(u,u_x,v,v_x),
v_xt=G(u,u_x,v,v_x), . describing pseudospherical (pss) or spherical surfaces (ss), meaning that, their generic solutions u(x,t) v(x,t) provide metrics, with coordinates (x,t), on open subsets of the plane, with constant curvature K=-1 or K=1. These systems can be described as the integrability conditions of \mathfrakg-valued linear problems, with \mathfrakg=\mathfraksl(2,ℝ) or \mathfrakg=\mathfraksu(2), when K=-1, K=1, respectively. We obtain characterization and also classification results. Applications of the theory provide new examples and new families of systems of differential equations, which contain generalizations of a Pohlmeyer-Lund-Regge type system and of the Konno-Oono coupled dispersionless system.

Cited by

• On a class of third order differential equations describing pseudospherical or spherical surfaces
• Special vector fields on Riemannian manifolds of constant negative sectional curvature and conservation laws
• Local isometric immersions of pseudospherical surfaces described by a class of third order partial differential equations

Related

• On Third-Order Evolution Systems Describing Pseudo-Spherical or Spherical Surfaces
• A class of third order partial differential equations describing spherical or pseudospherical surfaces
• On a class of third order differential equations describing pseudospherical or spherical surfaces
• Correspondence theorems for hierarchies of equations of pseudo-spherical type
• Local Isometric immersions of pseudo-spherical surfaces and evolution equations