
SIGMA 2 (2006), 044, 18 pages nlin.SI/0512046
https://doi.org/10.3842/SIGMA.2006.044
Hamiltonian Flows of Curves in G/SO(N) and Vector Soliton Equations of mKdV and SineGordon 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
Abstract
The biHamiltonian structure of the two known vector generalizations
of the mKdV hierarchy of soliton equations is derived in a
geometrical fashion from flows of nonstretching 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 biHamiltonian structure uses
a parallel frame and connection along the curve,
tied to a zero curvature MaurerCartan 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 sineGordon 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:
biHamiltonian; soliton equation; recursion operator; symmetric space; curve flow; wave map; Schrödinger map; mKdV map.
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