subsequent expressions. This equation has the bi-Hamiltonian structure $$\label{mkdv-biHam-eqn}
v_t=\H(\delta H/\delta v)=\E(\delta E/\delta v)$$ where $$\begin{aligned}
\H & =-D_x , 
\label{mkdv-Hop}
\\
\E & =-D_x^3-D_xvD\inv_x vD_x
\label{mkdv-Eop}\end{aligned}$$ are compatible Hamiltonian operators, and where $$\begin{aligned}
H & =\int_{-\infty}^{\infty} \tfrac{1}{8}v^4 -\tfrac{1}{2}v_x^2\,dx , 
\\
E & =\int_{-\infty}^{\infty} \tfrac{1}{2}v^2\,dx\end{aligned}$$ are the corresponding Hamiltonians.

Recall [@Olv-book], a linear operator $\D$ is a Hamiltonian operator iff its associated bracket $$\label{bracket}
\{H,E\}_\D = \int_{-\infty}^{\infty} (\delta H/\delta v)\D(\delta E/\delta v)\, dx$$ is skew and satisfies the Jacobi identity, for all functionals $H$ and $E$. Two Hamiltonian operators are called compatible if every linear combination of them is a Hamiltonian operator. Also recall, the composition of a Hamiltonian operator and the inverse of a compatible Hamiltonian operator is a recursion operator that obeys a hereditary property [@FucFok]. In particular, a linear operator $\R$ is hereditary if it satisfies $$\label{hereditaryprop}
\lieder_{\X_{\R\eta}}\R = \R(\lieder_{\X_{\eta}}\R)$$ holding for all differential functions $\eta(x,v,v_x,\ldots)$, where $\lieder_{\X_f}$ denotes the Lie derivative with respect to a vector field $\X_f= f(x,v,v_x,\ldots)\partial_v$. An equivalent formulation of the hereditary property is given by [@Olv-book] $$\label{hereditaryop}
\pr\X_{\R\eta}\R +[\R,(\R\eta){}^\prime]= \R(\pr\X_{\eta}\R+[\R,\eta{}^\prime])$$ where $f^\prime$ denotes the Frechet derivative of a differential function $f(x,v,v_x,\ldots)$ and $\pr\X_f$ denotes the prolongation of $\X_f$ to the coordinate space 