124 research outputs found
Hamiltonian Flows of Curves in symmetric spaces G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type
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.Comment: 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
Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation
The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian system
possessing -peakon weak solutions, for all , in the setting of an
integral formulation which is used in analysis for studying local
well-posedness, global existence, and wave breaking for non-peakon solutions.
Unlike the original Camassa-Holm equation, the two Hamiltonians of the mCH
equation do not reduce to conserved integrals (constants of motion) for
-peakon weak solutions. This perplexing situation is addressed here by
finding an explicit conserved integral for -peakon weak solutions for all
. When is even, the conserved integral is shown to provide a
Hamiltonian structure with the use of a natural Poisson bracket that arises
from reduction of one of the Hamiltonian structures of the mCH equation. But
when is odd, the Hamiltonian equations of motion arising from the conserved
integral using this Poisson bracket are found to differ from the dynamical
equations for the mCH -peakon weak solutions. Moreover, the lack of
conservation of the two Hamiltonians of the mCH equation when they are reduced
to -peakon weak solutions is shown to extend to -peakon weak solutions
for all . The connection between this loss of integrability structure
and related work by Chang and Szmigielski on the Lax pair for the mCH equation
is discussed.Comment: Minor errata in Eqns. (32) to (34) and Lemma 1 have been fixe
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