Fractional solutions for the inextensible Heisenberg antiferromagnetic fow and solitonic magnetic flux surfaces in the binormal direction
DOI:
https://doi.org/10.31349/RevMexFis.67.452Abstract
Motivated by recent researches in magnetic curves and their flows in different types of geometric manifolds and physical spacetime structures, we compute Lorentz force equations associated with the magnetic b-lines in the binormal direction. Evolution equations of magnetic b-lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their explicit solutions are investigated in terms of magnetic and geometric quantities via the conformable fractional derivative method. By considering arc-length and time-dependent orthogonal curvilinear coordinates, we finally determine the necessary and sufficient conditions that have to be satisfied by these quantities to define the Lorentz magnetic flux surfaces based on the inextensible Heisenberg antiferromagnetic flow model.References
Gilmore, R. (1984). Length and curvature in the geometry of thermodynamics. Physical Review A, 30(4), 1994.
Barbashov, B. M., Nesterenko, V. (1990). Introduction to the relativistic string theory. World Scientific.
De Sabbata, V., Sivaram, C. (1994). Spin and torsion in gravitation. World Scientific.
Schief, W. K., Rogers, C. (2005). The Da Rios system under a geometric constraint: the Gilbarg problem. Journal of Geometry and Physics, 54(3), 286-300.
Littlejohn, R. G. (1983). Variational principles of guiding centre motion. Journal of Plasma Physics, 29(1), 111-125.
Kleman, M. (1980). Developable domains in hexagonal liquid crystals. Journal de Physique, 41(7), 737-745.
Körpinar, T., Demırkol, R. C. (2020). Electromagnetic curves of the linearly polarized light wave along an optical fiber in a 3D Riemannian manifold with Bishop equations. Optik, 200, 163334.
Korpinar, T., Demirkol, R. C. (2018). Frictional magnetic curves in 3D Riemannian manifolds. International Journal of Geometric Methods in Modern Physics, 15(02), 1850020.
Körpınar, T., Demirkol, R. C. (2018). Gravitational magnetic curves on 3D Riemannian manifolds. International Journal of Geometric Methods in Modern Physics, 15(11), 1850184.
Kazan, A., Karadag, H. B. (2017). Magnetic pseudo null and magnetic null curves in Minkowski 3-space. In International Mathematical Forum, 123, 119-132.
Güvenç, Ş., Özgür, C. (2019). On slant magnetic curves in S-manifolds. Journal of Nonlinear Mathematical Physics, 26(4), 536-554.
Cabrerizo, J. L. (2013). Magnetic fields in 2D and 3D sphere. Journal of Nonlinear Mathematical Physics, 20(3), 440-450.
Sun, J. (2019). Singularity properties of killing magnetic curves in Minkowski 3-space. International Journal of Geometric Methods in Modern Physics, 16(08), 1950123.
Körpınar, T., Demirkol, R. C., Körpınar, Z., Asil, V. (2020). Maxwellian evolution equations along the uniform optical fiber in Minkowski space. Revista Mexicana de Física, 66(4), 431-439.
Körpınar, T., Demirkol, R. C., Körpınar, Z., Asil, V. (2020). Maxwellian evolution equations along the uniform optical fiber in Minkowski space. Optik, 217, 164561.
Ricca, R. L. (2005). Inflexional disequilibrium of magnetic flux-tubes. Fluid Dynamics Research, 36(4-6), 319.
Ricca, R. L. (1997). Evolution and inflexional instability of twisted magnetic flux tubes. Solar Physics, 172(1-2), 241-248.
Garcia de Andrade, L.C. (2006). Non-Riemannian geometry of twisted flux tubes. Brazilian journal of physics, 36(4A), 1290-1295.
Garcia de Andrade, L.C. (2006). Riemannian geometry of twisted magnetic flux tubes in almost helical plasma flows. Physics of Plasmas, 13(2), 022309-022309.
Garcia de Andrade, L.C. (2006). Vortex filaments in MHD. Physica Scripta, 73(5), 484.
Guo, B., Ding, S. (2008). Landau-Lifshitz Equations. World Scientific.
Vieira, V. R., Horley, P. P. (2012). The Frenet--Serret representation of the Landau--Lifshitz--Gilbert equation. Journal of Physics A: Mathematical and Theoretical, 45(6), 065208.
Hasimoto, H. (1972). A soliton on a vortex filament. Journal of Fluid Mechanics, 51(3), 477-485.
Anco, S. C., Myrzakulov, R. (2010). Integrable generalizations of Schrödinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces. Journal of Geometry and Physics, 60(10), 1576-1603.
Erdoğdu, M., Özdemir, M. (2014). Geometry of Hasimoto surfaces in Minkowski 3-space. Mathematical Physics, Analysis and Geometry, 17(1-2), 169-181.
Ricca, R. L. (1992). Physical interpretation of certain invariants for vortex filament motion under LIA. Physics of Fluids A: Fluid Dynamics, 4(5), 938-944.
Balakrishnan, R., Bishop, A. R., Dandoloff, R. (1993). Anholonomy of a moving space curve and applications to classical magnetic chains. Physical Review B, 47(6), 3108.
Barros, M., Ferrández, A., Lucas, P., Merono, M. (1995). Hopf cylinders, B-scrolls and solitons of the Betchov-Da Rios equation in the 3-dimensional anti-De Sitter space. CR Acad. Sci. Paris, Série I, 321, 505-509.
Barros, M., Ferrández, A., Lucas, P., Meroño, M. A. (1999). Solutions of the Betchov-Da Rios soliton equation: a Lorentzian approach. Journal of Geometry and Physics, 31(2-3), 217-228.
Arroyo, J., Garay, Ó. J., Pámpano, Á. (2017). Binormal motion of curves with constant torsion in 3-spaces. Advances in Mathematical Physics, 2017.
Körpınar, T., Demirkol, R. C., Körpınar, Z. (2019). Soliton propagation of electromagnetic field vectors of polarized light ray traveling along with coiled optical fiber on the unit 2-sphere S². Rev. Mex. Fis. 65(6), 626-633.
Körpınar, T., Demirkol, R. C., Körpınar, Z. (2019). Soliton propagation of electromagnetic field vectors of polarized light ray traveling in a coiled optical fiber in Minkowski space with Bishop equations. The European Physical Journal D, 73(9), 203.
Körpınar, T., Demirkol, R. C., Körpınar, Z. (2019). Soliton propagation of electromagnetic field vectors of polarized light ray traveling in a coiled optical fiber in the ordinary space. International Journal of Geometric Methods in Modern Physics, 16(8), 1950117.
Balakrishnan, R., Bishop, A. R., Dandoloff, R. (1990). Geometric phase in the classical continuous antiferromagnetic Heisenberg spin chain. Physical review letters, 64(18), 2107.
Bliokh, K. Y. (2009). Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium. Journal of Optics A: Pure and Applied Optics, 11(9), 094009.
Bliokh, K. Y., Niv, A., Kleiner, V., Hasman, E. (2008). Geometrodynamics of spinning light. Nature Photonics, 2(12), 748.
Wassmann, F., Ankiewicz, A. (1998). Berry's phase analysis of polarization rotation in helicoidal fibers. Applied optics, 37(18), 3902-3911.
Balakrishnan, R. (1994). Space curve evolution, geometric phase, and solitons. Theoretical and Mathematical Physics, 99(2), 501-504.
Samuel, J., Nityananda, R. (2000). Transport along null curves. Journal of Physics A: Mathematical and General, 33(14), 2895.
Balakrishnan, R., Dandoloff, R. (2004). Classical analogues of the Schrödinger and Heisenberg pictures in quantum mechanics using the Frenet frame of a space curve: an example. European journal of physics, 25(3), 447.
Körpınar, T., Demirkol, R. C., Asil, V. (2020). Directional magnetic and electric vortex lines and their geometries. Indian Journal of Physics. Accepted. https://doi.org/10.1007/s12648-020-01885-2.
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 65-70.
Eslami, M., Rezazadeh, H. (2016). The first integral method for Wu--Zhang system with conformable time-fractional derivative. Calcolo, 53(3), 475-485.
Çenesiz, Y., Baleanu, D., Kurt, A., Tasbozan, O. (2017). New exact solutions of Burgers' type equations with conformable derivative. Waves in Random and complex Media, 27(1), 103-116.
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Copyright (c) 2021 TALAT Körpınar, Ridvan Cem Demirkol, ZELİHA KÖRPINAR, Vedat Asil
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