in the two equations [@ChoQu]. Thirdly, the Hamiltonian structures of the CH equation can be derived from those of the KdV equation by a tri-Hamiltonian splitting method [@OlvRos].

The KdV equation is well-known to be related to the modified KdV (mKdV) equation $v_t + v^2 v_x +v_{xxx}=0$ by a Miura transformation. This equation is also a well-known integrable system, and it is related to a peakon equation $u_t -u_{txx} + ((u^2- u_x^2)(u -u_{xx}))_x =0$ called the modified CH (mCH) equation (also known as the Fokas-Olver-Rosenau-Qiao (FORQ) equation), which is an integrable system. In particular, the mCH equation arises from the theory of surface water waves [@Fok95b] and its integrability was derived by applying the tri-Hamiltonian splitting method to the Hamiltonian structures of the mKdV equation [@OlvRos; @Fok95a; @FokOlvRos]. It can also be obtained as a potential flow by using the mKdV recursion operator. A derivation based on spectral methods is given in where the Lax pair and single peakon solutions of the mCH equation were first obtained. Other recent work on the mCH equation appears in .

The main purpose of the present paper is to derive complex, $U(1)$-invariant peakon equations from the integrable hierarchy that contains the nonlinear Schrodinger 