Analysis of the Bloch equations for the nuclear magnetization model. (English. Russian original) Zbl 1288.82059

Proc. Steklov Inst. Math. 281, Suppl. 1, S64-S81 (2013); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 18, No. 2, 123-140 (2012).
Summary: We consider a system of three ordinary first-order differential equations known in the theory of nuclear magnetism as the Bloch equations. The system contains four dimensionless parameters as coefficients. Equilibrium states and the dependence of their stability on these parameters are investigated. The possibility of the appearance of two stable equilibrium states is discovered. The equations are integrable in the absence of dissipation. For the problem with small dissipation far from equilibrium, approximate solutions are constructed by the method of averaging.


82D40 Statistical mechanics of magnetic materials
78A25 Electromagnetic theory (general)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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[1] F. Bloch, Phys. Rev. 70(7–8), 460 (1946).
[2] M. I. Kurkin and E. A. Turov, NMR in Magnetically Ordered Materials and Its Applications (Nauka, Moscow, 1990) [in Russian].
[3] A. S. Borovik-Romanov, Yu. M. Bunkov, B. S. Dumesh, M. I. Kurkin, M. P. Petrov, and V. P. Chekmarev, Physics-Uspekhi 27(4), 235 (1984).
[4] L. A. Kalyakin, O. A. Sultanov, and M. A. Shamsutdinov, Teor. Math. Physics 167(3), 762 (2011).
[5] A. G. Gurevich and G. A. Melkov, Magnetization, Oscillations and Waves (Fizmatlit, Moscow, 1994; CRC, New York, 1996).
[6] Ya. A. Monosov, Nonlinear Ferromagnetic Resonance (Nauka, Moscow, 1971) [in Russian].
[7] L. A. Kalyakin and M. A. Shamsutdinov, Proc. Steklov Inst. Math., Suppl. 2, S124 (2007).
[8] V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations (Editorial, Moscow, 2004) [in Russian]. · Zbl 0089.29502
[9] N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations (Nauka, Moscow, 1974; Gordon and Breach, New York, 1962).
[10] V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (VINITI, Moscow, 1985; Springer-Verlag, Berlin, 2006). · Zbl 0885.70001
[11] R. Z. Khas’minskii, Stability of Systems of Differential Equations under Random Perturbations of Their Parameters (Nauka, Moscow, 1969) [in Russian].
[12] V. D. Azhotkin and V. M. Babich, J. Appl. Math. Mech. 49(3), 290 (1985). · Zbl 0596.70027
[13] J. Br√ľning, S. Yu. Dobrokhotov, and M. A. Poteryakhin, Math. Notes 70(5–6), 599 (2001). · Zbl 1025.37038
[14] V. I. Arnol’d, Soviet Math. Dokl. 3, 136 (1962).
[15] A. M. Il’in, Teor. Math. Physics 118(3), 301 (1999). · Zbl 0963.34040
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