be inverted to give $Q^{(n)}= (2n+1) D\inv_xv^{-1} D_x h^{(n)}$. Next, replacing $Q^{(n)}= {\R^*}Q^{(n-1)} =(2n-1) (D_xv^{-1} D_x h^{(n-1)}+v h^{(n-1)})$ in expression yields $(2n+1)h^{(n)} = (2n-1)D\inv_x\big(vD_x(v+D_x v^{-1}D_x) h^{(n-1)}\big)$ modulo a total $x$-derivative, and thus $$\label{mkdv-H-expr}
h^{(n)} = \tfrac{2n-1}{2n+1}D\inv_x\big(vD_xv(1+(v^{-1}D_x)^2)h^{(n-1)}\big) .$$ This provides an explicit recursion formula for all of the Hamiltonians in the mKdV hierarchy, starting from the Hamiltonian density $h^{(0)}= \tfrac{1}{2}v^2$.

Tri-Hamiltonian splitting method
--------------------------------

The basis for the method of tri-Hamiltonian splitting [@OlvRos] is the observation that if a linear operator $\D$ is Hamiltonian then so is the scaled operator $\D_{(\lambda)}$ defined by scaling $v\to \lambda v$ (and similarly scaling all $x$-derivatives of $v$) in $\D$. When $\D_{(\lambda)}$ consists of two terms with different powers of $\lambda$, each term will define a Hamiltonian operator, and the resulting two operators can be shown to be a compatible Hamiltonian pair.

Under scaling, the first mKdV Hamiltonian operator is invariant, $\H_{(\lambda)} = -D_x = \H$, while the second mKdV Hamiltonian operator becomes $\E_{(\lambda)} = -D_x^3 -\lambda^2 D_xvD\inv_x vD_x$, which yields two operators $$\label{mkdv-splitEops}
\E_1=-D_x^3,
\quad
\E_2=-D_xvD\inv_x vD_x .$$ These operators are a compatible Hamiltonian pair. Furthermore, since the first operator is invariant, all three operators and can be shown to be mutually compatible. Hence, 