the Hamiltonians can be obtained by the general scaling method [@Anc03] shown in the appendix. This yields the formula $$\label{mkdv-peakon-H-Q-expr}
\wtil H^{(n)} = \int_{-\infty}^{\infty} \tilde h^{(n)}\,dx,
\quad
\tilde h^{(n)}= \tfrac{1}{2(n+1)} u \wtil Q^{(n)}$$ modulo boundary terms. Note this formula reproduces the Hamiltonian for $n=0$.

Finally, the $n=1$ flow $$\label{mch-flow}
\wtil P^{(1)} = \wtil\R({-v_x}) = - \tfrac{1}{2} ((u^2-u_x^2)v)_x$$ yields the mCH equation $$\label{mch-eqn}
v_t = - \tfrac{1}{2} ((u^2-u_x^2)v)_x 
= \wtil\H(\delta \wtil E/\delta v)=\wtil\E(\delta \wtil H/\delta v)$$ along with its bi-Hamiltonian structure, where $\wtil H=\wtil H^{(0)}$ is the Hamiltonian and $\wtil E=\wtil H^{(1)}$ is the Hamiltonian given by $$\label{mkdv-peakon-E}
\wtil E = \int_{-\infty}^{\infty} \tfrac{1}{8}( u(u^2-u_x^2)v ) \,dx
= \int_{-\infty}^{\infty} \tfrac{1}{24}( 3u^4 +6 u^2 u_x^2 -u_x^4) \,dx$$ from formula with $n=1$, after integration by parts. Thus, we will refer to the integrable hierarchy of higher flows $$\label{mch-hierarchy}
\wtil P^{(n)} = \wtil\R^n({-v_x}),
\quad
n=1,2,\ldots$$ as the mCH hierarchy.

The flows in the mCH hierarchy correspond to symmetry operators $\X^{(n)}=\wtil P^{(n)}\partial/\partial_v$, $n=1,2,\ldots$, while the Hamiltonians $\wtil H^{(n)}$ for these flows correspond to conservation laws $\frac{d}{dt}\wtil H^{(n)}=0$, $n=0,1,2,\ldots$, all of which are admitted by each equation $v_t= \wtil P^{(n)}$, $n=1,2,\ldots$, in the hierarchy.

A useful remark is that the recursion formula for Hamiltonian gradients can be 