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

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 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.

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/ Minor changes made (typos corrected and more discussion added about parallel frames and vector SG equations)

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• Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I
• The Sasa--Satsuma (complex mKdV II) and the complex sine-Gordon II equation revisited: recursion operators, nonlocal symmetries, and more

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