*(English)*Zbl 0916.35047

An evolution equation ${u}_{t}=F(x,t,u,{u}_{x},\cdots ,{u}_{{x}^{k}})$ is said to describe pseudospherical surfaces if it is a necessary and sufficient condition for the existence of functions ${f}_{\alpha \beta}$, $\alpha =1,2,3$; $\beta =1,2$, depending on $x$, $t$, $u$ and its derivatives, such that one-forms ${\omega}_{\alpha}={f}_{{\alpha}_{1}}dx+{f}_{{\alpha}_{2}}dt$ satisfy the structure equations of a surface of constant Gaussian curvature equal to $-1$, with metric ${\omega}_{1}^{2}+{\omega}_{2}^{2}$ and connection one-form ${\omega}_{3}$. On the other hand, an equation is said to be formally integrable if it has a formal symmetry of infinite rank.

The author shows that every second-order equation ${u}_{t}=F(x,t,u,{u}_{x},{u}_{xx})$ which is formally integrable, describes a one-parameter family of pseudospherical surfaces. To this end, he finds explicitly the pseudospherical structures associated with each of the four second-order equations appearing in the exhaustive list of formally integrable equations available in literature. It is shown that this result cannot be extended to third-order formally integrable equations. This fact notwithstanding, a special case of the equation of the form

is considered, and every formally integrable equation of this form is proved to describe a one-parameter family of pseudospherical surfaces. Finally, several popular nonlinear equations, including the Harry-Dym, cylindrical KdV and the Calogero-Degasperis family are shown to describe pseudospherical surfaces.