Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 001, 19 pages      arXiv:1301.0180      https://doi.org/10.3842/SIGMA.2013.001
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

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

Changzheng Qu a, Junfeng Song b and Ruoxia Yao c
a) Center for Nonlinear Studies, Ningbo University, Ningbo, 315211, P.R. China
b) College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, P.R. China
c) School of Computer Science, Shaanxi Normal University, Xi'an, 710062, P.R. China

Received September 28, 2012, in final form December 27, 2012; Published online January 02, 2013

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 Sn(1). Integrability to these systems is also studied.

Key words: invariant curve flow; integrable system; Euclidean geometry; Möbius sphere; dual Schrödinger equation; multi-component modified Camassa-Holm equation.

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