In this paper, we explicitly construct an infinite number of Hopfions (static, soliton solutions with nonzero Hopf topological charges) within the recently proposed ͑3 1 1͒-dimensional, integrable, and relativistically invariant field theory. Two integers label the family of Hopfions we have found. Their product is equal to the Hopf charge which provides a lower bound to the soliton's finite energy. The Hopfions are explicitly constructed in terms of the toroidal coordinates and shown to have a form of linked closed vortices. . The emerging stringlike structures are quite intriguing and may find applications in various physical models of condensed matter physics and gauge field theory. It is therefore of direct physical interest to find a field theoretical model for which it is possible to write down in a closed form explicit soliton solutions with nonzero Hopf index (Hopfians). This will advance an understanding of stringlike soliton configurations and their properties and open a way to incorporate them into various models relevant for physical applications.In Ref.[3], we have introduced the three-dimensional field model which falls into a class of higher dimensional integrable models from the point of view of the generalized zero-curvature approach [4]. The question posed in [3] was whether this form of integrability is linked to the existence of soliton solutions as is expected from the study of two-dimensional integrable models. Our analysis of the model in [3] has indeed revealed one nontrivial soliton solution described by a standard Hopf map of unit Hopf index. To fully establish a connection between integrability and soliton solutions would require finding other topological solitons with arbitrary topological charges. This is accomplished in this Letter. The equations of motion of the model are solved in toroidal coordinates and the space of solutions is found to be represented by a family of maps ‫ޒ‬ 3 ! ‫ޒ‬ 2 labeled by two integers. The integers count the number of times the map winds around two independent angular directions.The model under consideration is described by the Lagrangian density

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A general formalism is proposed to study infinite-dimensional Noether symmetries in arbitrary field theories on group coadjoint orbits as well as in their gauged versions (coset geometric models). The basic tools are generalized group composition laws valid for any geometric action. As a main application, we present a general scheme for constructing the "hidden" Kac-Moody currents.

We use Hirota's method formulated as a recursive scheme to construct complete set of soliton solutions for the affine Toda field theory based on an arbitrary Lie algebra. Our solutions include a new class of solitons connected with two different type of degeneracies encountered in the Hirota's perturbation approach.We also derive an universal mass formula for all Hirota's solutions to the Affine Toda model valid for all underlying Lie groups. Embedding of the Affine Toda model in the Conformal Affine Toda model plays a crucial role in this analysis.

We analyze the integrability properties of models defined on the symmetric space SU(2)/U(1) in 3+1 dimensions, using a recently proposed approach for integrable theories in any dimension. We point out the key ingredients for a theory to possess an infinite number of local conservation laws, and discuss classes of models with such property. We propose a 3+1-dimensional, relativistic invariant field theory possessing a toroidal soliton solution carrying a unit of topological charge given by the Hopf map. Construction of the action is guided by the requirement that the energy of static configuration should be scale invariant. The solution is constructed exactly. The model possesses an infinite number of local conserved currents. The method is also applied to the Skyrme-Faddeev model, and integrable submodels are proposed.Comment: LaTeX, 14 pg