Pseudo-potentials and local isometric immersion for a generalized Camassa-Holm equation describing pseudospherical surfaces

TL;DR

This paper investigates a generalized Camassa-Holm equation related to pseudospherical surfaces, demonstrating its geometric integrability, deriving conservation laws, and exploring local isometric immersion into 3D space.

Contribution

It introduces a geometric approach to analyze the integrability and immersion properties of a generalized Camassa-Holm equation.

Findings

The equation is shown to be geometrically integrable.

An infinite hierarchy of conservation laws is derived.

The study addresses local isometric immersion into Euclidean space.

Abstract

A generalized Camassa-Holm equation, which describes pseudospherical surfaces, is considered. Using geometric methods, it is demonstrated that the equation is geometrically integrable. Additionally, an infinite hierarchy of conservation laws is derived. Furthermore, the paper delves into the investigation of the problem of locally isometric immersion into three-dimensional Euclidean space based on the equation.