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

[摘要] 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 Schrödinger 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 Möbius sphere and n -dimensional sphere S n (1). Integrability to these systems is also studied.

[关键词] invariant curve flow;integrable system;Euclidean geometry;M¨obius sphere;dual Schr¨odinger equation;multi-component modified Camassa–Holm equation