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

TL;DR

This paper studies the integrability and soliton solutions of the half-wave maps equation, revealing a completely integrable many-body system with unique collision properties, including no phase shifts after soliton collisions.

Contribution

It demonstrates the integrability of the associated many-body system, constructs multisoliton solutions, and analyzes their collision behavior, highlighting novel features of the half-wave maps equation.

Findings

The system exhibits a Lax pair and conservation laws.

Explicit multisoliton solutions are constructed.

Solitons collide without phase shifts, a novel phenomenon.

Abstract

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 many-body 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 one- and 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.