We consider evolution equations, mainly of type ut = F(u, ux,..., ∂ku/∂xk), which describe pseudo‐spherical surfaces. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. Moreover, we investigate how the geometrical properties of surfaces provide analytic information for such equations.

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.

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The generalized wave equation and generalized sine-Gordon equations are known to be natural multidimensional differential geometric generalizations of the classical two-dimensional versions. In this paper we associate a system of linear differential equations with these equations and show how the direct and inverse problems can be solved for appropriately decaying data on suitable lines. An initial-boundary-value problem is solved for these equations.

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A method to derive conservation laws for evolution equations that describe pseudospherical surfaces is introduced based on a geometrical property of these surfaces. A new third-order evolution equation is obtained as a first example for a nongeneric case in the classification given by Chern and Tenenblat [Stud. Appl. Math. 74, 1 (1986)].

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