Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries

TL;DR

This paper introduces multi-component generalizations of the Camassa-Holm equation, explores their geometric formulations in various geometries, and investigates their integrability, expanding the understanding of integrable systems in differential geometry.

Contribution

It presents new multi-component integrable equations derived from invariant curve flows in different geometries, with geometric and integrability analyses.

Findings

New multi-component Camassa-Holm type equations introduced.

Geometric formulations in Euclidean, M"obius sphere, and n-sphere geometries provided.

Integrability properties of these systems studied.

Abstract

In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schr\"odinger equation, the complex Camassa-Holm equation and the multi-component modified Camassa-Holm equation are provided. It is shown that these equations arise from non-streching invariant curve flows respectively in the three-dimensional Euclidean geometry, the two-dimensional M\"obius sphere and $n$-dimensional sphere ${\mathbb S}^n(1)$. Integrability to these systems is also studied.