Integrability, conservation laws and solitons of a many-body dynamical system associated with the half-wave maps equation

We consider the half-wave maps (HWM) equation which is a continuum limit of the classical version of the Haldane-Shastry spin chain. In particular, we explore a manybody dynamical system arising from the HWM equation under the pole ansatz. The system is shown to be completely integrable by demonstrating that it exhibits a Lax pair and relevant conservation lows. Subsequently, the analytical multisoliton solutions of the HWM equation are constructed by means of the pole expansion method. The properties of the oneand two-soliton solutions are then investigated in detail as well as their pole dynamics. Last, an asymptotic analysis of the N -soliton solution reveals that no phase shifts appear after the collision of solitons. This intriguing feature is worth noting since it is the first example observed in the head-on collision of rational solitons. A number of problems remain open for the HWM equation, some of which are discussed in concluding remarks. ∗ E-mail address: matsuno@yamaguchi-u.ac.jp 1

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