A class of quasilinear second order partial differential equations which describe spherical or pseudospherical surfaces

TL;DR

This paper classifies a broad class of second order PDEs that describe spherical or pseudospherical surfaces, linking them to geometric structures and zero curvature representations, and provides explicit examples including well-known equations.

Contribution

It offers a complete explicit classification of second order PDEs describing spherical or pseudospherical surfaces based on arbitrary functions, expanding understanding of their geometric and integrability properties.

Findings

Includes well-known equations like the short pulse and constant astigmatism equations.

Provides explicit forms and classifications of PDEs for spherical and pseudospherical surfaces.

Demonstrates the connection between these PDEs and zero curvature representations.

Abstract

Second order partial differential equations which describe spherical surfaces (ss) or pseudospherical surfaces (pss) are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature $K = 1$ or $K = -1$, respectively, and they can be seen as the compatibility condition of an associated su(2)-valued or sl(2, R)-valued linear problem, also referred to as a zero curvature representation. Under certain assumptions we give a complete and explicit classiffcation of equations of the form $z_{tt} = A(z, z_x , z_t) z_{xx} + B(z, z_x , z_t ) z_{xt} + C(z, z_x , z_t)$ describing pss or ss, in terms of some arbitrary differentiable functions. Several examples of such equations are provided by choosing the arbitrary functions. In particular, well known equations which describe pseudospherical surfaces, such as the short pulse and the constant astigmatism equations, as well as their generalizations and their spherical analogues are included in the paper.