Hamiltonian Flows of Curves in symmetric spaces G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type

Details Hamiltonian Flows of Curves in symmetric spaces G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type Hamiltonian Flows of Curves in symmetric spaces G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type

2005-12-19

arxiv.org/abs/nlin/0512046v8

SIGMA 2 (2006), 044, 18 pages

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/ Mino

Scientific paper

10.3842/SIGMA.2006.044

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 curves, tied to a zero curvature Maurer-Cartan form on G, and this yields the vector mKdV recursion operators in a geometric O(N-1)-invariant 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 Schrodinger map equations.

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