Multi-solitons of the half-wave maps equation and Calogero-Moser spin-pole dynamics

Abstract

We consider the half-wave maps (HWM) equation which provides a continuum description of the classical Haldane-Shastry spin chain on the real line. We present exact multi-soliton solutions of this equation. Our solutions describe solitary spin excitations that can move with different velocities and interact in a non-trivial way. We make an ansatz for the solution allowing for an arbitrary number of solitons, each described by a pole in the complex plane and a complex spin variable, and we show that the HWM equation is satisfied if these poles and spins evolve according to the dynamics of an exactly solvable spin Calogero-Moser (CM) system with certain constraints on initial conditions. We also find first order equations providing a Bäcklund transformation of this spin CM system, generalize our results to the periodic HWM equation, and provide plots that visualize our soliton solutions.

arXiv comment32 pages, 5 figures

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• Spin generalizations of the Benjamin-Ono equation
• Integrability, conservation laws and solitons of a many-body dynamical system associated with the half-wave maps equation
• The non-chiral intermediate Heisenberg ferromagnet equation
• How coordinate Bethe ansatz works for Inozemtsev model
• On uniqueness for half-wave maps in dimension d >= 3
• Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data
• Periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation described by elliptic spin Calogero-Moser dynamics