On Differential Equations Describing Pseudo-Spherical Surfaces

We give a complete classification of the evolution equations ∂u/∂t = F(u, ∂u/∂x, ..., ∂ku/∂xk) which describe pseudo-spherical surfaces, without any a priori assumptions on the presence of a spectral parameter. We also prove a local existence theorem to the effect that given two differential equations describing pseudo-spherical surfaces (not necessarily evolutionary), there exists, under a technical assumption, a smooth mapping transforming any suitably generic solution of one equation into a solution of the other