BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:А.B. Borisov
DTSTART:20220414T110000Z
DTEND:20220414T120000Z
DTSTAMP:20260423T053140Z
UID:mmandim/39
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/mmandim/39/"
 >O(3)-model: Integrability. Stationary and Dynamic Magnetic Structures</a>
 \nby А.B. Borisov as part of Mathematical models and integration methods\
 n\n\nAbstract\nA three-dimensional O(3)-model for a unit vector $n(r)$ has
  numerous application\nin the field theory and in the physics of condensed
  matter. We prove that this model\nis integrable under some differential c
 onstraint\, that is\, under certain restrictions for the\ngradients of fie
 lds $Θ(r)$\, $Φ(r)$\,  parametrizing the vector $n(r)$). Under the prese
 nce of the\ndifferential constraint\, the equations of the models are redu
 ced to a one-dimensional sine-\nGordon equation determining the dependence
  of the field $Θ(r)$ on an auxiliary field $a(r)$\nand to a system of two
  equations $(∇S)(∇S) = 0$\, $\\Delta S = 0$ for a complex-valued funct
 ion\n$S(r) = a(r)+i\\cdot Φ(r)$. We show that the solution of this system
  provide all known before exact\nsolutions of models\, namely\, two-dimens
 ional magnetic instantons and three-dimensional\nstructures of hedgehog ty
 pe.\n\nWe 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 simpl
 e solutions of these equations are discussed: a new magnetic structure tha
 t represents two straight intersecting vortex threads and a "inclusion" ty
 pe structure.\n\nThe integrability of the dynamical equations the O(3)-mod
 el in four-dimensional pseudo-Euclidean space–time was investigated . We
  use a differential substitution to reduce the equations to the one-dimens
 ional sine-Gordon equation and a system of two equations for a complex-val
 ued function $S(r\, t)$ that uniquely determines a vector $n$. We prove th
 at solving the equations for this function amounts to solving a system of 
 four quasilinear equations for auxiliary fields. We obtain their exact sol
 ution in the form of an implicit function of three variables\, which then 
 determines the exact solutions of the dynamical equations with differentia
 l constraints taken into account. As examples\, we describe the dynamics o
 f a plane vortex in D = (2.1)\, a “hedgehog”-type structure\, and new 
 dynamical topological structure.\n\nReferences:\n\n1. А.Б. Борисо
 в. Трехмерные вихри в модели Гейзенберга
 \, ТМФ\, 2021\, том 208\, номер 3\, 471–480 (A.B. Borisov. Th
 ree-Dimensional Vortices in the Heisenberg Model. Theoretical and Mathemat
 ical Physics\, 208(3): 1256–1264 (2021)).\n\n2. А.Б. Борисов. 
 Об интегрируемости 𝑂(3)–модели. Уфимски
 й 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).\n\n3. А.Б. Борисов. Ди
 намика трехмерных магнитных структур в 
 модели Гейзенберга. ТМФ\, 2022\, том 210\, номе
 р 1\, страницы 115–127 (A.B. Borisov. Dynamics of Three-Dimensi
 onal Magnetic Structures in the Heisenberg Model. Theoretical and Mathemat
 ical Physics\, 210(1): 99–110 (2022)).\n
LOCATION:https://researchseminars.org/talk/mmandim/39/
END:VEVENT
END:VCALENDAR