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

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 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 ${\mathbb{S}}^n(1)$. Integrability to these systems is also studied.

References in corpus (4)

• Geodesic Flow and Two (Super) Component Analog of the Camassa-Holm Equation
• Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces
• Motions of Curves in the Projective Plane Inducing the Kaup-Kupershmidt Hierarchy
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