Using the notion of a differential equation which describes an 1j-pseudospherical surface (1j-p.s.s.), we give a characterization of the equations of type U xt = F(u, u x "'" aku/ax k ), k ~ 2, with this property. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. The equations of type u xt = F(u, uJ were characterized by Rabelo and Tenenblat in another paper. The theory is applied to several equations, some of which were not known to describe 1j-p.S.S.

In is known that the equations [u t -g(u)ux]x = ± g'(u) describe pseudo-spherical surfaces, i.e. that these equations are the integrability conditions for the structural equations of such surfaces, provided g satisfies gil + p.g = (). In this paper we obtain self-Backlund transformations for these equations by a geometric method, and show how the inverse scattering method generates global solutions.

The equations uxt=F(u,ux), which describe η-pseudospherical surfaces, are characterized. In particular, when F does not depend on ux, the sine–Gordon, sinh–Gordon, and Liouville equations are essentially obtained. Moreover, it is shown that an equation uxt=F(u) has a self-Bäclund transformation if and only if it describes an η-pseudospherical surface.

The equations of type ut = uxxx + G(u,ux,uxx) which describe η-pseudospherical surfaces are characterized. A new class of such equations is obtained. This class together with the Korteweg–de Vries (KdV) equation, the modified Korteweg–de Vries (MKdV) equation, and the linear equation completes the classification.