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Pseudo-spherical surfaces and integrability of evolution equations. (English) Zbl 0916.35047

An evolution equation ${u}_{t}=F\left(x,t,u,{u}_{x},\cdots ,{u}_{{x}^{k}}\right)$ 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 ${u}_{t}=F\left(x,t,u,{u}_{x},{u}_{xx}\right)$ 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

${u}_{t}={u}^{-3}{u}_{xxx}+{a}_{2}\left(x,u,{u}_{x}\right){u}_{xx}^{2}+{a}_{1}\left(x,u,{u}_{x}\right){u}_{xx}+{a}_{0}\left(x,u,{u}_{x}\right)$

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.

##### MSC:
 35G10 Initial value problems for linear higher-order PDE 35K25 Higher order parabolic equations, general 35Q53 KdV-like (Korteweg-de Vries) equations