Integrable U ( 1 ) -invariant peakon equations from the NLS hierarchy

Highlights: • A novel NLS-type peakon equation is derived from the NLS hierarchy. • Its bi-Hamiltonian structure is obtained from tri-Hamiltonian splitting. • A modified AKNS scheme is used to derive a Lax pair in a systematic way. • Peakon solutions with an oscillatory temporal phase are investigated. • The first known example of a peakon breather is found. Two integrable $U\left(1\right)$-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 $U\left(1\right)$-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.