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

TL;DR

This paper derives two integrable U(1)-invariant peakon equations from the NLS hierarchy, providing their Lax pairs, recursion operators, and explicit peakon solutions including the first known peakon breather.

Contribution

It introduces new integrable peakon equations linked to the NLS hierarchy and constructs their analytical solutions, including a novel peakon breather.

Findings

Derived two integrable U(1)-invariant peakon equations from NLS hierarchy.

Constructed explicit peakon and periodic solutions, including the first peakon breather.

Established bi-Hamiltonian structure and hierarchy of symmetries for these equations.

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.