Spin generalizations of the Benjamin–Ono equation

Abstract

We present new soliton equations related to the A-type spin Calogero–Moser (CM) systems introduced by Gibbons and Hermsen. These equations are spin generalizations of the Benjamin–Ono (BO) equation and the recently introduced non-chiral intermediate long-wave (ncILW) equation. We obtain multi-soliton solutions of these spin generalizations of the BO equation and the ncILW equation via a spin-pole ansatz where the spin-pole dynamics is governed by the spin CM system in the rational and hyperbolic cases, respectively. We also propose physics applications of the new equations, and we introduce a spin generalization of the standard intermediate long-wave equation which interpolates between the matrix Korteweg-de Vries equation, the Heisenberg ferromagnet equation, and the spin BO equation.

On the non-chiral intermediate long wave equation

“…Note that κ(z) reduces to a constant in the limits ℓ → ∞ and/or δ → ∞ (as can been seen by evaluating κ(z) in (1.2) using the corresponding degenerations of the functions ℘ 2 (z) and ζ 2 (z) presented below in (2.2) and (2.1), respectively). This is one reason why the cases treated in [1] are significantly easier than the elliptic case treated in the present paper. We also note that ζ 2 (z) = ζ 1 (z) + γ 0 z with the constant…”

“…This generalization is non-trivial in several regards; in particular, our solutions include a dynamical background term which, as we show, provides a non-trivial generalization even in the hyperbolic limit when the spatial period becomes infinite. We also present corresponding generalizations of known solutions to the spin Benjamin-Ono (sBO) equation introduced in [1] and, in this way, obtain the full correspondence between sCM models and soliton equations conjectured in [1].…”

“…In a recent paper [1], we introduced and solved new soliton equations related to the A-type spin Calogero-Moser (sCM) systems of Gibbons and Hermsen [2] (see also [3]). One of these equations, the spin non-chiral intermediate long wave (sncILW) equation, was shown to have multi-soliton solutions with dynamics described by the hyperbolic sCM system.…”

“…While soliton equations with this feature have been studied for a long time (the classical examples are the Benjamin-Ono [5,6] and intermediate long wave [7,8] equations), there has recently been considerable interest in constructing and analyzing novel nonlocal soliton equations; see, for instance, [9,10,11,12,13,14,15,16]. In this context, in addition to resolving a conjecture posed in [1], this paper serves to exhibit the sncILW equation with periodic boundary conditions as an interesting system worthy of further study.…”

“…Throughout this paper, we denote by ζ(z) and ℘(z) the usual Weierstrass ζ-and ℘-functions of a complex variable z with half-periods (ℓ, iδ), with ℓ > 0 and δ > 0 fixed parameters and i := √ −1 (for the convenience of the reader, we give the definitions of these functions in Appendix A. 1). We find it convenient to use the following variants of ζ(z),…”