Компьютерная алгебра и квантовые вычисления с приложениями UDC 517.957, 530.145 On 2D and 3D Localized Solutions with Nontrivial Topology I. L. Bogolubsky, A. A. Bogolubskaya Laboratory of Information Technologies Joint Institute for Nuclear Research 6, Joliot-Curie str., Dubna, Moscow region, Russia, 141980 Localized solutions of nonlinear field models with nontrivial topological properties are discussed. <...> Existence of various systems of definitions of the topological objects, developed in this area of research historically, can potentially lead to the wrong conclusions about existence of such solutions. <...> The classification allowing to define accurately and differentiate objects with different topological properties is proposed, which prevents from inferring wrong conclusions. <...> Such solutions are divided into 2 classes: the topological solitons (TS) and topological defects (TD). <...> Solutions of both types describe the localized distributions of field energy, but they differ in topological properties. <...> We exemplify and compare stationary TSs and TDs in 2 and 3 spatial dimensions. <...> Examples of TSs are: solitons in Heisenberg magnets, Belavin–Polyakov solitons/instantons, Skyrmions, “baby-skyrmions”. <...> Examples of TDs are: sine-Gordon kinks, Nielsen–Olesen strings-vortices in the Abelian Higgs (AHM) model, ’t Hooft–Polyakov hedgehog-monopoles in the Georgi–Glashow model. <...> We note some technical problems with TDs, which are not met in the case of TSs. <...> Soliton analogs of Nielsen-Olesen TDs in the AHM have been found: they are TSs in the A3M model. <...> Key words and phrases: topological charge, solitons, defects, mapping degree, AbelianHiggs model, Yang-Mills field, Heisenberg antiferromagnet, Georgi-Glashow and WeinbergSalam models, hedgehog ansatz. 1. <...> Introduction field equations with nontrivial topological properties is the important approach to nonperturbative field theory. <...> Historically the first localized solutions with nontrivial topology were skyrmions, Investigation of localized energy distributions described by solutions of nonlinear found in [1] and used for the description of baryons. <...> However we believe that the usage of the same term “a soliton” both for TSs and TDs may turn out misleading in some cases (for an example see Sect.6, where TLs in the Standard Model are discussed). 2. <...> Definitions Both topological defects, TD <...>