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

@article{Berntson2020MultisolitonsOT,
title={Multi-solitons of the half-wave maps equation and Calogero–Moser spin–pole dynamics},
author={Bjorn K Berntson and Rob Klabbers and Edwin Langmann},
journal={Journal of Physics A},
year={2020},
volume={53},
pages={505702}
}
• Published 30 June 2020
• Physics, Mathematics
• Journal of Physics A
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…
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## References

SHOWING 1-10 OF 41 REFERENCES
Waves and solitons in the continuum limit of the Calogero-Sutherland model.
The generic wave is shown to correspond to a two-band state in the quantum description of the system, while the limiting cases of solitons and phonons correspond to particle and hole excitations.
Emergence of the Calogero family of models in external potentials: duality, solitons and hydrodynamics
• Physics, Mathematics
• 2017
We present a first-order formulation of the Calogero model in external potentials in terms of a generating function, which simplifies the derivation of its dual form. Solitons naturally appear in
A Short Primer on the Half-Wave Maps Equation
We review the current state of results about the half-wave maps equation on the domain $\mathbb{R}^d$ with target $\mathbb{S}^2$. In particular, we focus on the energy-critical case $d=1$, where we
Crum transformation and rational solutions of the non-focusing nonlinear Schrödinger equation
A Crum transformation for the linear problem of the non-focusing nonlinear Schrodinger (NLS) equation is used to construct the sequence of rational solutions, given in terms of tau-functions
Numerical Solution of Some Nonlocal, Nonlinear Dispersive Wave Equations
• Mathematics, Computer Science
J. Nonlinear Sci.
• 2000
A spectral method is used to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, which there is at present no analytic proof.
Algebraic internal wave solitons and the integrable Calogero–Moser–Sutherland N‐body problem
• Physics
• 1979
The Benjamin–Ono equation that describes nonlinear internal waves in a stratified fluid is solved by a pole expansion method. The dynamics of poles which characterize solitons is shown to be
Integrable hydrodynamics of Calogero-Sutherland model: Bidirectional Benjamin-Ono equation
• Mathematics, Physics
• 2008
We develop a hydrodynamic description of the classical Calogero-Sutherland liquid: a Calogero-Sutherland model with an infinite number of particles and a non-vanishing density of particles. The
Generalized hydrodynamics of the classical Toda system
• B. Doyon
• Physics, Mathematics
Journal of Mathematical Physics
• 2019
We obtain the exact generalized hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and