Integration of the equations of the Heisenberg model (2D) and the chiral SU(2) models by differential geometry methods

Abstract: In the report, to integrate the two-dimensional Heisenberg equation and the three-dimensional chiral SU(2) model, the differential-geometric method of integration is used, the essence of which is as follows. First, we perform the hodograph transformation, i.e. change the role of dependent and independent coordinates. Unlike the standard hodograph transformation, we do not just introduce derivatives of the old coordinates with respect to new ones, but define through these derivatives new fields associated with the components of the metric tensor that appears when the hodograph transformation is performed. Since the original independent coordinates were Euclidean, the curvature tensor in terms of the introduced metric must vanish. Ultimately, we obtain a self-consistent system of equations for calculating the components of the metric tensor. In this case, the equations guaranteeing the curvature tensor to vanish turn out to be the main ones, and the system of nonlinear equations of the models is their reduction. The solutions of the constructed equations make it possible to write the solutions of the original models in the form of implicit functions. It is important that the differential-geometric method of model integration, based on the embedding of a non-linear partial differential equation in a certain differential relation in Euclidean space, makes it possible to analyze a wide variety of spatial structures, the study of which by other methods is extremely difficult. The solutions found in the chiral SU(2) model describe three-dimensional configurations containing, in particular, spatial vortices, sources, non-localized textures, and structures with a mapping degree equal to one, similar to topological solitons. In the Heisenberg model we find a vortex strip (a limited vortex region in a plane). Many of the obtained solutions depend on arbitrary functions.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods