On equations of Lund-Regge type

Abstract

We introduce a new class of partial differential equations admitting a geometric interpretation, the class of equations of Lund-Regge type. These equations describe surfaces immersed in a three dimensional euclidean sphere, admit conservation laws, and they are the integrability condition of $3\times 3$ overdetermined $so(3,R)$-valued linear systems. As examples, we present equations describing minimal surfaces, equations describing spherical surfaces, a generalization of the integrable Konno-Oono coupled system, and an elliptic Lund-Regge equation that generalizes the sinh-Poisson equation of plasma physics. We also present a structural result on second order equations of Lund-Regge type.