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

SIGMA 2 (2006), 044, 18 pages      nlin.SI/0512046

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

Received December 12, 2005, in final form April 12, 2006; Published online April 19, 2006; Replaced by the revised version September 29, 2006

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.

Key words: bi-Hamiltonian; soliton equation; recursion operator; symmetric space; curve flow; wave map; Schrödinger map; mKdV map.

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