Dynamics of the 3D magnetic structures for the Heisenberg model

We study the integrability of the dynamical Heisenberg equations (the $O(3)$ model in four-dimensional pseudo-Euclidean space–time), which are widely used in field theory and condensed matter physics. We use a differential substitution to reduce the equations to the one-dimensional sine-Gordon equation and a system of two equations for a complex-valued function $S(\mathbf{r},t)$ that uniquely determines a vector $\mathbf{n}$. We prove that solving the equations for this function amounts to solving a system of four quasilinear equations for auxiliary fields. We obtain their exact solution in the form of an implicit function of two variables, which then determines the exact solutions of the dynamical Heisenberg equations with differential constraints taken into account. As examples, we describe the dynamics of a plane vortex in $\mathbb{R}^2$, a “hedgehog”-type structure, and new dynamical topological structures in $\mathbb{R}^3$.