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.

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Cited by in corpus (8)

• A four-component Camassa-Holm type hierarchy
• The Frobenius-Virasoro algebra and Euler equations
• Double Extended Cubic Peakon Equation
• Higher dimensional integrable deformations of the modified KdV equation
• Integrable Deformation of Space Curves, Generalized Heisenberg Ferromagnet Equation and Two-Component Modified Camassa-Holm Equation
• Camassa-Holm and M-CIV equations with self-consistent sources: geometry and peakon solutions
• Zakharov-Ito equation and Generalized Heisenberg ferromagnet-type equation: equivalence and related geometric curve flows
• Generalized Heisenberg Ferromagnet type Equation and Modified Camassa-Holm Equation: Geometric Formulation, Soliton Solutions and Equivalence