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Zbl 0916.35047
Reyes, Enrique G.
Pseudo-spherical surfaces and integrability of evolution equations. (English)
[J] J. Differ. Equations 147, No.1, 195-230 (1998); erratum ibid. 153, No.1, 223-224 (1999). ISSN 0022-0396

An evolution equation $u_t= F(x,t,u,u_x,\dots, u_{x^k})$ 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 $\omega^2_1+ \omega^2_2$ 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.\par 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 $$u_t= u^{-3} u_{xxx}+ a_2(x, u,u_x) u^2_{xx}+ a_1(x, u, u_x)u_{xx}+ a_0(x,u,u_x)$$ 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.
[Igor Barashenkov (Rondebosch)]

MSC 2000:: *35G10 Initial value problems for linear higher-order PDE
35K25 Higher order parabolic equations, general
35Q53 KdV-like equations

Keywords: kinematic integrability; formal integrability; classification of integrable equations

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Zentralblatt MATH Berlin [Germany]