Static solitons with nonzero Hopf number

Abstract

We investigate a generalized nonlinear O(3) model in three space dimensions where the fields are maps from R 3 ഫ͕ϱ͖ to S 2 . Such maps are classified by a homotopy invariant called the Hopf number which takes integer values. The model exhibits soliton solutions of closed vortex type which have a lower topological bound on their energies. We numerically compute the fields for topological charge 1 and 2 and discuss their shapes and binding energies. The effect of an additional potential term is considered and an approximation is given for the spectrum of slowly rotating solitons. ͓S0556-2821͑97͒00520-1͔

Solitons, links and knots

“…[5]) go much further than resolution of the differences between refs. [8,10]. We present candidate minimum energy configurations for charges one to eight which suggest that exotic linked and knotted solitons may also exist at higher charges.…”

“…They make no assumption as to the spatial symmetry of the solution or its structure at the origin and hence provide an obvious improvement on the methods employed in refs. [8,10]. The only extra ingredient required is the imposition of a boundary condition, which can be done simply at the edge of the box, mimicking spatial infinity.…”

“…[8,10] In each case we investigated an exactly axially symmetric solution and also the inclusion of non-symmetric perturbations given by (3.10).…”

“…In refs. [8,10] configurations of charge one and two were produced by employing a toroidal or cylindrical ansatz. Here, we describe how a wide range of fields can be obtained for any Hopf charge without the need to refer to a particular coordinate system which is tailored to a given configuration.…”

“…There have been two preliminary numerical investigations [8,10] of solitons in this model with charges one and two. Both concentrated mainly on axially symmetric configurations, using toroidal or cylindrical coordinates so that the numerical problem is effectively reduced to one in two spatial dimensions.…”