Pseudo-spherical Surfaces and Integrability of Evolution Equations

Abstract

A complete classification of evolution equationsut=F(x,t,u,ux,...,ux) which describe pseudo-spherical surfaces, is given, thus providing a systematic procedure to determine a one-parameter family of linear problems for which the given equation is the integrability condition. It is shown that for every second-order equation which admits a formal symmetry of infinite rank (formalintegrability) such a family exists (kinematicintegrability). It is also shown that this result cannot be extended as proven to third-order formally integrable equations. This fact notwithstanding, a special case is proven, and moreover, several equations of interest, including the Harry-Dym, cylindrical KdV, and a family of equations solved by inverse scattering by Calogero and Degasperis, are shown to be kinematically integrable. Conservation laws of equations describing pseudo-spherical surfaces are studied, and several examples are given. © 1998 Academic Press.