---
abstract: 'Two integrable $U(1)$-invariant peakon equations are derived from the NLS hierarchy through the tri-Hamiltonian splitting method. A Lax pair, a recursion operator, a bi-Hamiltonian formulation, and a hierarchy of symmetries and conservation laws are obtained for both peakon equations. These equations are also shown to arise as potential flows in the NLS hierarchy by applying the NLS recursion operator to flows generated by space translations and $U(1)$-phase rotations on a potential variable. Solutions for both equations are derived using a peakon ansatz combined with an oscillatory temporal phase. This yields the first known example of a peakon breather. Spatially periodic counterparts of these solutions are also obtained.'
author:
- |
    Stephen C. Anco${}^1$\
    \
    Fatane Mobasheramini${}^{1,2}$\
    \
    \
    \
title: |
    Integrable U(1)-invariant peakon equations\
    from the NLS hierarchy
---

[^1]

Introduction {#intro}
============

There has been considerable recent interest in equations that possess peaked solitary wave solutions, known as peakons. One of the first well-studied peakon equations is the Camassa-Holm (CH) equation [@CamHol; @CamHolHym] $u_t -u_{txx} + 3u u_x - 2 u_x u_{xx} - u u_{xxx}=0$, which arises from the theory of shallow water waves. This equation possesses peakon solutions, $u=c \exp(-|x-ct|)$, and multi-peakon solutions which are 