NUCLEAR Nuclear Physics B393 (1993) 65—78 P HY S I CS B North-Holland _________________ Skyrmion model in 2 + 1 dimensions with soliton bound states Bernard Piette and Wojtek J. Zakrzewski Department of Mathematical Sciences, Unirersity of Durham, Durha,n DI-Il 3LE, UK Received 7 September 1992 Accepted for publication 20 October 1992 We consider a class of skyrmion models in 2+ I dimensions which possess bound stable solitons. We show that these models have one-soliton solutions as well as static solutions corresponding to their bound states. We study the scattering and stability properties of these solutions, compute their energies and estimate their binding energies. 1. Introduction Over the last few years, many physical processes have been described by nonlinear partial differential equations. These include areas like solid-state physics, hydrodynamics and classical field theories. Integrable models, in particular, have been popular because of their properties and the existence of methods to construct all their solutions. The integrable equations possess stable solutions (solitons) which keep their shape as they move or interact with each other. Most of the studied equations depend only on two space (or space-time) variables, and not much work has been done yet on higher-dimensional models. Integrable equations in 1 + I dimensions usually exhibit lump-like solutions which propagate at a constant speed. When two lumps are sent towards each other, they collide, modify their shape, and eventually emerge as if nothing has happened apart from a phase shift. In one space dimension the collision can only be head-on as there is no impact parameter. In higher dimensions, solitons look like extended structures with localised energy density, which can move in any direction. Not only can these objects perform a head on collision, but they can scatter at a nonzero impact parameter. In field theories, the models we are interested in are Lorentz invariant. Unfortunately, so far, nobody has been able to construct a model in 2 + 1 dimensions which is both Lorentz invariant and integrable. In our previous papers [1—3],we studied a class of (2 + 1)-dimensional minkowskian models which ap- peared to be not integrable but had many features in common with integrable 0550-3213/93/$06.00 © 1993 — Elsevier Science Publishers B.V. All rights reserved