A note on isometric immersions and differential equations which describe pseudospherical surfaces

TL;DR

This paper introduces new second-order nonlinear PDEs describing pseudospherical surfaces that admit local isometric immersions in three-dimensional space, including examples like the short pulse equation, expanding the class of known equations with this property.

Contribution

It presents families of pss equations with isometric immersions, including explicit examples like the short pulse equation, challenging the prior belief that only sine-Gordon had this property.

Findings

New families of pss equations with isometric immersions

Explicit examples including the short pulse equation

Expansion of known equations with this geometric property

Abstract

In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equations occupy a particularly special place amongst pss equations, since a series of recent papers [2, 7, 11, 12, 13], on several classes of pss equations, seemed to suggest that only the sine-Gordon equation had the above property. Explicit examples are given, which include the short pulse equation and some generalizations.