@OlvRos; @FokOlvRos] and [@ChoQu], some new aspects will be developed, including an explicit recursion formula for all of the Hamiltonians in the mCH hierarchy, and a derivation of the mCH recursion operator directly from the AKNS zero-curvature equation [@AblKauNewSeg].

In section \[splitting\], the tri-Hamiltonian splitting method will be applied to the Hamiltonian structures in the NLS hierarchy. Rather than use the standard bi-Hamiltonian structure of the NLS equation itself, whose tri-Hamiltonian splitting is known to lead to a somewhat trivial equation [@OlvRos], a third Hamiltonian structure of the NLS equation [@MarSanWan; @AncMyr] will be used instead. This Hamiltonian structure is connected with a higher flow in the NLS hierarchy, given by the Hirota equation [@Hir1973], which is a complex, $U(1)$-invariant generalization of the mKdV equation.

In section \[derivation\], the Hamiltonian operators derived from the tri-Hamiltonian splitting method will be used to construct the two $U(1)$-invariant peakon equations along with their bi-Hamiltonian formulation. These two peakon equations will also be shown to arise as potential flows in the NLS hierarchy by applying the NLS recursion operator to the flows generated by $x$-translations and $U(1)$-phase rotations on a potential variable. One of the peakon equations has an NLS-type form which does not 