Analysis of a Mathematical Model for Nuclear Spins in an Antiferromagnet

    Received 14 December 2017; accepted 29 January 2018

    2018, Vol. 14, no. 2, pp.  217-234

    Author(s): Kalyakin L. A.

    This paper is concerned with a system of three nonlinear differential equations, which is a mathematical model for a system of nuclear spins in an antiferromagnet. The model has arisen in recent physical studies and differs from the well-known and well-understood Landau – Lifshitz and Bloch models in the manner of incorporating dissipation effects. It is established that the system under consideration is related to the Landau – Lifshitz system by the passage to the limit only on one invariant sphere. The initial equations contain three dimensionless parameters. Equilibrium points and their stability are examined depending on these parameters. The position of the bifurcation surface is found in the parameter space. It is proved that the corresponding equilibrium is of saddle-node type. Exact statements are illustrated by results of numerical experiments.
    Keywords: nonlinear equations, equilibrium, stability, bifurcation
    Citation: Kalyakin L. A., Analysis of a Mathematical Model for Nuclear Spins in an Antiferromagnet, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp.  217-234
    DOI:10.20537/nd180206


    Download File
    PDF, 608.44 Kb

    References
    [1] Haberman, R. and Ho, E. K., “Boundary of the Basin of Attraction for Weakly Damped Primary Resonance”, Trans. ASME J. Appl. Mech., 62:4 (1995), 941–946  crossref  mathscinet  zmath
    [2] Itin, A. P., Neishtadt, A. I., and Vasiliev, A. A., “Capture into Resonance in Dynamics of a Charged Partice in Magnetic Field and Electrostatic Wave”, Phys. D, 141:3–4 (2000), 281–296  crossref  mathscinet  zmath
    [3] Kiselev, O. M. and Glebov, S. G., “An Asymptotic Solution Slowly Crossing the Separatrix near a Saddle-Centre Bifurcation Point”, Nonlinearity, 16:1 (2003), 327–362  crossref  mathscinet  zmath  adsnasa
    [4] Bautin, N. N. and Leontovich, E. A., Methods and Ways of the Qualitative Analysis of Dynamical Systems in a Plane, 2nd ed., Nauka, Moscow, 1990, 496 pp. (Russian)  mathscinet
    [5] Borich, M. A., Bunkov, M. Yu., Kurkin, M. I., and Tankeyev, A. P., “Nuclear Magnetic Relaxation Induced by the Relaxation of Electron Spins”, JETP Lett., 105:1 (2017), 21–25  mathnet  crossref  adsnasa; Pis'ma v Zh. Èksper. Teoret. Fiz., 105:1 (2017), 23–27 (Russian)
    [6] Gurevich, A. G. and Melkov, G. A., Magnetization Oscillations and Waves, CRC, New York, 1996, 464 pp.
    [7] Kalyakin, L. A., “Analysis of the Bloch Equations for the Nuclear Magnetization Model”, Proc. Steklov Inst. Math., 281:1 (2013), S64–S81  mathscinet  zmath
    [8] Kalyakin, L. A., Sultanov, O. A., and Shamsutdinov, M. A., “Asymptotic Analysis of a Model of Nuclear Magnetic Autoresonance”, Theoret. and Math. Phys., 167:3 (2011), 762–771  mathnet  crossref  mathscinet  adsnasa; Teoret. Mat. Fiz., 167:3 (2011), 420–431 (Russian)  crossref
    [9] Kalyakin, L. A. and Shamsutdinov, M. A., “Adiabatic Approximations for Landau – Lifshitz Equations”, Proc. Steklov Inst. Math., 259:2 (2007), S124–S140  mathnet  crossref  zmath
    [10] Monosov, Ya. A., Nonlinear Ferromagnetic Resonance, Nauka, Moscow, 1971, 376 pp. (Russian)
    [11] Nemytskii, V. V. and Stepanov, V. V., Qualitative Theory of Differential Equations, Princeton Math. Ser., 22, Princeton Univ. Press, Princeton, N.J., 1960, viii+523 pp.  mathscinet



    Creative Commons License
    This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License