The non-chiral intermediate Heisenberg ferromagnet equation

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.

Spin generalizations of the Benjamin-Ono equation

“…Our results in the present paper, together with results in the literature on the standard (chiral) ILW equation [30] and on the non-chiral intermediate Heisenberg ferromagnet equation [32], suggest that the following is also an integrable generalization of the sBO equation,…”

“…As shown by two of us in collaboration with Klabbers [32], there exists a non-chiral variant of an intermediate generalization of the Heisenberg ferromagnetic equation which generalizes the HWM equation and which has soliton solutions given by a spin-pole ansatz governed by the hyperbolic sCM model, in generalization of a result in [16] for the HWM equation mentioned above; this non-chiral intermediate Heisenberg ferromagnetic equation is given by…”

“…for two R 3 -valued functions m = m(x, t) and n = n(x, t) satisfying m 2 = n 2 = 1 [32], and we checked that it is obtained from the sncILW equation (1.13) in a similar way as the HWM equation is obtained from the sBO equation (1.1) (as explained in the paragraph containing (1.5)), after changing the sign of V (the latter is merely a convention).…”

Periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation described by elliptic spin Calogero-Moser dynamics

“…The non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation is a generalization of the HWM equation related to the elliptic spin CM system. Together with Langmann, we introduced the ncIHF equation in [21] as a continuum limit of a classical Inozemtsev spin chain [22]; the latter is simultaneously an elliptic generalization of the Haldane-Shastry spin chain and a limiting case of the elliptic spin CM system. It is important to note that the ncIHF equation comes in two related variants: (i) an equation with periodic boundary conditions and (ii) an equation posed on the real line, which may be obtained as an infinite-period limit of the first.…”

“…The ncIHF equation on the real line [21] is obtained in the ℓ → ∞ limit; it is given by (1.1) with the ℓ → ∞ limit of the operators (1.2) obtained using (1.4). We refer to [24] for further details on the relationship between the periodic and real-line versions of (1.2).…”