O(3)-model: Integrability. Stationary and Dynamic Magnetic Structures

Abstract: A three-dimensional O(3)-model for a unit vector $n(r)$ has numerous application in the field theory and in the physics of condensed matter. We prove that this model is integrable under some differential constraint, that is, under certain restrictions for the gradients of fields $Θ(r)$, $Φ(r)$, parametrizing the vector $n(r)$). Under the presence of the differential constraint, the equations of the models are reduced to a one-dimensional sine- Gordon equation determining the dependence of the field $Θ(r)$ on an auxiliary field $a(r)$ and to a system of two equations $(∇S)(∇S) = 0$, $\Delta S = 0$ for a complex-valued function $S(r) = a(r)+i\cdot Φ(r)$. We show that the solution of this system provide all known before exact solutions of models, namely, two-dimensional magnetic instantons and three-dimensional structures of hedgehog type.

We show that the found in this way exact solution of the system for the field $S(r)$ leads one to exact solution of equations of O(3)–model in the form of an arbitrary implicit function of two variables. Two simple solutions of these equations are discussed: a new magnetic structure that represents two straight intersecting vortex threads and a "inclusion" type structure.

The integrability of the dynamical equations the O(3)-model in four-dimensional pseudo-Euclidean space–time was investigated . 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 $S(r, t)$ that uniquely determines a vector $n$. 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 three variables, which then determines the exact solutions of the dynamical equations with differential constraints taken into account. As examples, we describe the dynamics of a plane vortex in D = (2.1), a “hedgehog”-type structure, and new dynamical topological structure.

References:

1. А.Б. Борисов. Трехмерные вихри в модели Гейзенберга, ТМФ, 2021, том 208, номер 3, 471–480 (A.B. Borisov. Three-Dimensional Vortices in the Heisenberg Model. Theoretical and Mathematical Physics, 208(3): 1256–1264 (2021)).

2. А.Б. Борисов. Об интегрируемости 𝑂(3)–модели. Уфимский математический журнал. Том 13. № 2 (2021). С. 6-10 (А.B. Borisov, On integrability of O(3)–model, Ufimsk. Mat. Zh., 2021, Volume 13, Issue 2, 6–10).

3. А.Б. Борисов. Динамика трехмерных магнитных структур в модели Гейзенберга. ТМФ, 2022, том 210, номер 1, страницы 115–127 (A.B. Borisov. Dynamics of Three-Dimensional Magnetic Structures in the Heisenberg Model. Theoretical and Mathematical Physics, 210(1): 99–110 (2022)).

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

Audience: researchers in the topic

Mathematical models and integration methods