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Hopf solitons in modern mathematical physics

Hopf solitons in modern mathematical physics

Hopfions

Hopf index

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Hopfions in:

Skyrme-Faddeev model

Hydrodynamics

Electromagnetic fields

Magnetohydrodynamics

Bose-Einstein condensate

^{3}He

ferromagnets:

Heisenberg

Uniaxial

Incommensurate

Next neighbour

Hopf index

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Hopfions in:

Skyrme-Faddeev model

Hydrodynamics

Electromagnetic fields

Magnetohydrodynamics

Bose-Einstein condensate

ferromagnets:

Heisenberg

Uniaxial

Incommensurate

Next neighbour

There are many magnetic crystals without an inversion center, where exchange-relativistic interactions lead to the formation of long-period magnetic structures whose periods are incommensurate with the crystal-chemical period. In [V.G. Bar'yakhtar and E.P. Stefanovsky Fiz. Tverd. Tela 11, 1946 (1969) copy; Sov. Phys. Solid State 11, 1566 (1970)], [P. Bak and M.H. Jensen "Theory of helical magnetic structures and phase transitions in MnSi and FeGe", J. Phys. C 13 L881 (1980) ref] was proposed the energy functional (1.1) (1.2) In [A. Bogdanov "New localized solutions of the nonlinear field equations", JETP Lett. 62, 247 (1995) ref] was proved that Hobart-Derrick theorem does not prohibit the existence of three-dimensional localized solutions for the energy functional {1.1}. The main difference between this model from "Heisenberg model" and "Uniaxial model" that it admits the possibility of a completely static three-dimensional solitons.

In [A.B. Borisov, F.N. Rybakov "Three-dimensional static vortex solitons in incommensurate magnetic crystals", Low Temp. Phys. 36, 766 (2010) ref; arXiv:1108.4330v1], the authors found numerically static solitons with total Hopf index equal to zero, but the structure consists of pairs "hopfion-antihopfion". Found structures are unstable to some perturbations.

Later authors have reported [A B. Borisov, F.N. Rybakov "Three-dimensional solitons in incommensurate ferromagnets", Int. Workshop "Spin Chirality and Dzyaloshinskii-Moriya Interaction" DMI 2011, St.-Petersburg (2011), Progr. and Abstr., p.40 , ref] on the results of calculations of toroidal hopfion. But it also unstable to some perturbations.

- Any analytical soliton solution in 3D
- Existence of an absolutely stable static solitons. Necessary to perform the calculations for various types of anisotropies V(n).
- Accounting for the effects of magnetic dipole-dipole interaction

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