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Multi-Component Integrable Systems and Invariant Curve Flows in Certain GeometriesKeywords: invariant curve flow , integrable system , Euclidean geometry , M bius sphere , dual Schr dinger equation , multi-component modified Camassa-Holm equation 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 dinger 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 bius sphere and n-dimensional sphere S^n(1). Integrability to these systems is also studied.
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