Title: Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation,
Abstract: The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian system
possessing $N$-peakon weak solutions, for all $N\geq 1$, 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
$2$-peakon weak solutions. This perplexing situation is addressed here by
finding an explicit conserved integral for $N$-peakon weak solutions for all
$N\geq 2$. When $N$ 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 $N$ 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 $N$-peakon weak solutions. Moreover, the lack of
conservation of the two Hamiltonians of the mCH equation when they are reduced
to $2$-peakon weak solutions is shown to extend to $N$-peakon weak solutions
for all $N\geq 2$. 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.