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

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.

💡 Research Summary

This paper presents a systematic construction of two integrable, U(1)-invariant complex peakon equations derived from the nonlinear Schrödinger (NLS) hierarchy by employing the tri‑Hamiltonian splitting method. The authors first revisit the well‑known derivation of the modified Camassa–Holm (mCH) equation from the modified Korteweg‑de Vries (mKdV) hierarchy, emphasizing the role of compatible Hamiltonian operators H, E₁, E₂ and their combination into a new pair (Ĥ, Ė). By introducing a potential variable u through the relation v = Δu with Δ = 1 − ∂ₓ², they obtain a hereditary recursion operator
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