Isometric immersions and differential equations describing pseudospherical surfaces

Abstract

We provide families of second order nonlinear partial differential equations describing pseudospherical surfaces (pss equations) with the property of having local isometric immersions in E3 with principal curvatures depending on a jet of finite order of their solutions. These equations occupy a particularly special place amongst pss equations since a series of recent papers on several classes of pss equations seemed to suggest that only the sine-Gordon equation had the above property. Explicit examples of such equations are given, which include the short pulse equation and some generalizations.