by applying a gauge transformation that consists of the mKdV scaling symmetry $v\to\lambda v$, $x\to\lambda^{-1}x$, $t\to\lambda^{-3}t$, combined with the scaling $U\to\lambda U$, $V\to\lambda^3 V$. This is easily seen to produce the standard mKdV matrix Lax pair $$\label{mkdv-laxpair}
U = \tfrac{1}{2} 
\begin{pmatrix}
\lambda & v \\ -v & -\lambda
\end{pmatrix}, 
\quad
V = \tfrac{1}{2} 
\begin{pmatrix}
-\tfrac{1}{2}\lambda v^2 - \lambda^3 & -v_{xx} -\tfrac{1}{2}v^3 -\lambda^2 v -\lambda v_x
\\
v_{xx} +\tfrac{1}{2}v^3 +\lambda^2 v -\lambda v_x & \tfrac{1}{2}\lambda v^2 + \lambda^3
\end{pmatrix}$$ up to a rescaling of the spectral parameter $\lambda$.

We now use the relation between the mKdV recursion operator and the recursion operator of the mCH equation to re-write the flow equation coming from the zero-curvature equation. This gives $$\label{mkdv-lax-pair-peakon-flow}
v_t = (\wtil\R-1)\Delta P = (\wtil\R-1)\wtil P$$ with $$P = \Delta\inv\wtil P .$$ Since $\wtil\R(-v_x) = \wtil P^{(1)}$ produces the mCH flow when $D\inv_x(0)=0$, we consider $\wtil P = -v_x$ and, correspondingly, $P=-u_x$. Then the zero-curvature equations – give $K=-\tfrac{1}{2}(u^2-u_x^2)+ c$ and $\omega = -\tfrac{1}{2}(u^2-u_x^2)v -u_{xx}+ cv$, where $c=D\inv_x(0)$ is an arbitrary constant. Hence, the flow equation becomes $$v_t = (\wtil\R-1)(-v_x) = (c+1)v_x -\tfrac{1}{2}((u^2-u_x^2)v)_x$$ which is the mCH equation up to a convective term $(c+1)v_x$. We put $c=-1$ to make this term 