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This article is cited in 34 scientific papers (total in 34 papers)
Hamiltonian Flows of Curves in $G/SO(N)$ and Vector Soliton Equations of mKdV and Sine-Gordon Type
Stephen C. Anco Department of Mathematics, Brock University, Canada
Abstract:
The bi-Hamiltonian structure of the two known vector generalizations of the mKdV hierarchy of soliton equations is derived in a geometrical fashion from flows of non-stretching curves in Riemannian symmetric spaces $G/SO(N)$. These spaces are exhausted by the Lie groups $G=SO(N+1),SU(N)$. The derivation of the bi-Hamiltonian structure uses a parallel frame and connection along the curve, tied to a zero curvature Maurer–Cartan form on $G$, and this yields the mKdV recursion operators in a geometric vectorial form. The kernel of these recursion operators is shown to yield two hyperbolic vector generalizations of the sine-Gordon equation. The corresponding geometric curve flows in the hierarchies are described in an explicit form, given by wave map equations and mKdV analogs of Schrödinger map equations.
Keywords:
bi-Hamiltonian; soliton equation; recursion operator; symmetric space; curve flow; wave map; Schrödinger map; mKdV map.
Received: December 12, 2005; in final form April 12, 2006; Published online April 19, 2006
Citation:
Stephen C. Anco, “Hamiltonian Flows of Curves in $G/SO(N)$ and Vector Soliton Equations of mKdV and Sine-Gordon Type”, SIGMA, 2 (2006), 044, 18 pp.
Linking options:
https://www.mathnet.ru/eng/sigma72 https://www.mathnet.ru/eng/sigma/v2/p44
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Abstract page: | 422 | Full-text PDF : | 64 | References: | 43 |
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