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

An evolution equation u t =F(x,t,u,u x ,,u x k ) is said to describe pseudospherical surfaces if it is a necessary and sufficient condition for the existence of functions f αβ , α=1,2,3; β=1,2, depending on x, t, u and its derivatives, such that one-forms ω α =f α 1 dx+f α 2 dt satisfy the structure equations of a surface of constant Gaussian curvature equal to -1, with metric ω 1 2 +ω 2 2 and connection one-form ω 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

u t =u -3 u xxx +a 2 (x,u,u x )u xx 2 +a 1 (x,u,u x )u xx +a 0 (x,u,u x )

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:
35G10Initial value problems for linear higher-order PDE
35K25Higher order parabolic equations, general
35Q53KdV-like (Korteweg-de Vries) equations