Abstract
We present and solve a soliton equation which we call the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation. This equation, which depends on a parameter δ > 0, describes the time evolution of two coupled spin densities propagating on the real line, and in the limit δ → ∞ it reduces to two decoupled half-wave maps (HWM) equations of opposite chirality. We show that the ncIHF equation is related to the A-type hyperbolic spin Calogero-Moser (CM) system in two distinct ways: (i) it is obtained as a particular continuum limit of an Inozemtsev-type spin chain related to this CM system, (ii) it has multi-soliton solutions obtained by a spin-pole ansatz and with parameters satisfying the equations of motion of a complexified version of this CM system. The integrability of the ncIHF equation is shown by constructing a Lax pair. We also propose a periodic variant of the ncIHF equation related to the A-type elliptic spin CM system.
Highlights
The relationship between Calogero-Moser-Sutherland (CMS1) systems and quantum field theory is a fruitful and symbiotic one yielding new insights into both domains
We show that the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation is related to the A-type hyperbolic spin Calogero-Moser (CM) system in two distinct ways: (i) it is obtained as a particular continuum limit of an Inozemtsev-type spin chain related to this CM system, (ii) it has multi-soliton solutions obtained by a spin-pole ansatz and with parameters satisfying the equations of motion of a complexified version of this CM system
We derive a Hamiltonian formulation for the ncIHF equation, show how (1.1) is formally related, via (1.8), to the well-known Heisenberg ferromagnet equation, collect some identities we use throughout the paper, provide some technical details related to conservation laws, derive a generalization of the Cotlar identity needed in the proof our Lax pair results, and compute the energy of one-soliton solutions
Summary
The relationship between Calogero-Moser-Sutherland (CMS1) systems and quantum field theory is a fruitful and symbiotic one yielding new insights into both domains. HWM equation are known in the periodic case [20–24]; while we introduce a periodic generalization of the ncIHF equation related to the elliptic spin CM model below, the focus of the present paper is on the real-line case. We derive a Hamiltonian formulation for the ncIHF equation (appendix A), show how (1.1) is formally related, via (1.8), to the well-known Heisenberg ferromagnet equation (appendix B), collect some identities we use throughout the paper (appendix C), provide some technical details related to conservation laws (appendix D), derive a generalization of the Cotlar identity needed in the proof our Lax pair results (appendix E), and compute the energy of one-soliton solutions (appendix F)
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