this Lax pair – to the mKdV Lax pair . In both of these Lax pairs, $U$ depends linearly on $\lambda$, while $V$ contains negative powers of $\lambda$ in the case of the mCH Lax pair but only positive powers of $\lambda$ in the case of the mKdV Lax pair. This indicates that the mCH equation can be viewed as a negative flow in the mKdV hierarchy.

Tri-Hamiltonian splitting in the NLS hierarchy {#splitting}
==============================================

The tri-Hamiltonian splitting method was originally applied to the NLS hierarchy in by considering the standard bi-Hamiltonian structure of the NLS equation. However, this did not lead to a peakon equation, because the operator $\Delta=1-D_x^2$ did not appear when the split Hamiltonian operators were recombined. We will show how to overcome this obstacle by using a third Hamiltonian structure of the NLS equation, which is connected with the first higher flow in the NLS hierarchy.

NLS hierarchy
-------------

We begin from the NLS equation $$\label{nls-eqn}
\s\inum v_t + \tfrac{1}{2}|v|^2 v +v_{xx}=0$$ which has the tri-Hamiltonian structure $$\label{nls-triHam-eqn}
v_t=\s\inum(\tfrac{1}{2}|v|^2 v +v_{xx})
= \I(\delta I/\delta\bar v)
= \H(\delta H/\delta\bar v)
=\D(\delta D/\delta\bar v)$$ where $$\begin{aligned}
\I & =\s \inum , 
\label{nls-Iop}
\\
\H & =-D_x -\inum vD\inv_x\Im\bar 