Stationary structures in two-dimensional continuous Heisenberg ferromagnetic spin system. (English) Zbl 1038.35133
Summary: Stationary structures in a classical isotropic two-dimensional continuous Heisenberg ferromagnetic spin system are studied in the framework of the \((2+1)\)-dimensional Landau-Lifshitz model. It is established that in the case of \(\vec S(\vec r,t)=\vec S(\vec r-\vec v t)\) the Landau-Lifshitz equation is closely related to the Ablowitz-Ladik hierarchy. This relation is used to obtain soliton structures, which are shown to be caused by joint action of nonlinearity and spatial dispersion, contrary to the well-known one-dimensional solitons which exist due to competition of nonlinearity and temporal dispersion. We also present elliptical quasiperiodic stationary solutions of the stationary \((2+1)\)-dimensional Landau-Lifshitz equation.
MSC:
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 82D40 | Statistical mechanics of magnetic materials |
| 82B21 | Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics |
Keywords:
Landau-Lifshitz model; Landau-Lifshitz equation; Ablowitz-Ladik hierarchy; soliton; quasiperiodic stationary solutionsReferences:
| [1] | Lakshmanan, M., Phys. Lett, A61, 53-54, 1977 |
| [2] | Takhtajan, L. A., Integration of the Continuum Heisenberg Spin Chain through the Inverse Scattering Method, Phys, Lett, A64, 235-237, 1977 |
| [3] | Kosevich, A. M.; Ivanov, B. A.; Kovalev, A. S., Magnetic Solitons, Phys, Rep, 194, 117-238, 1990 |
| [4] | Dodd, R. K.; Eilbeck, J. C.; Gibbon, J. D.; Morris, H. C., Solitons and Nonlinear Wave Equations, 1984, London: Academic Press, London |
| [5] | Faddeev, L. D.; Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons, 1987, Berlin: Springer-Verlag, Berlin · Zbl 0632.58004 |
| [6] | Papanicolaou, N., Duality Rotation for 2-D Classical Ferromagnets, Phys, Lett, A84, 151-154, 1981 |
| [7] | Vekslerchik, V. E., An O(3, 1) Nonlinear σ-Model and the Ablowitz-Ladik Hierarchy, J. Phys. A: Math. Gen, 27, 6299-6313, 1994 · Zbl 0844.35130 |
| [8] | Ishimori, Y., Multi-Vortex Solutions of a Two-Dimensional Nonlinear Wave Equation, Progr. Theor, Phys, 72, 33-37, 1984 · Zbl 1074.35573 |
| [9] | Lipovskii, V. D.; Shirokov, A. V., An Example of Gauge Equivalence of Multidimensional Integrable Equaitons, Funk, Analiz, 23, 65-66, 1989 · Zbl 0716.35078 |
| [10] | Ablowitz, M. J.; Ladik, J. F., Nonlinear Differential-Difference Equations, J. Math. Phys, 16, 598-603, 1975 · Zbl 0296.34062 |
| [11] | Vekslerchik, V. E., The 2D Toda Lattice and the Ablowitz-Ladik Hierarchy, Inverse Problems, 11, 463-479, 1995 · Zbl 0822.35121 |
| [12] | Vekslerchik, V. E., The Davey-Stewartson Equation and the Ablowitz-Ladik Hierarchy, Inverse Problems, 12, 1057-1074, 1996 · Zbl 0862.35113 |
| [13] | Ablowitz, M. J.; Ladik, J. F., Nonlinear Differential-Difference Equations and Fourier Analysis, J. Math. Phys, 17, 1011-1018, 1976 · Zbl 0322.42014 |
| [14] | AblowitzM J and Segur H, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. · Zbl 0472.35002 |
| [15] | Leon J and Spire A, The Zakharov-Shabat Spectral Problem on the Semi-Line: Hilbert Formulation and Applications, J. Phys. A: Math. Gen. 34 (2001) 7359-7380; nlin.PS/0105066. · Zbl 0983.35118 |
| [16] | Gutshabash E Sh, Lipovskij V D and Nikulichev S S, Nonlinear σ-Model in a Curved Space, Gauge Equivalence, and Exact Solutions of (2 + 0)-Dimensional Integrable Equations, Theor. Math. Phys. 115 (1998), 619-638; translation from Teor. Mat. Fiz. 115 (1998), 323-348; nlin.SI/0001012. · Zbl 1113.81095 |
| [17] | Levi, D.; Benguria, R., Backlund Transformations and Nonlinear Differential-Difference Equations, Proc. Natl. Acad. Sci. US, 77, 5025-5027, 1980 · Zbl 0453.35072 |
| [18] | Levi, D., Nonlinear Differential-Difference Equations as Backlund Transformations, J. Phys. A: Math. Gen, 14, 1083-1098, 1981 · Zbl 0465.35081 |
| [19] | Shabat, A.; Yamilov, R. I., Lattice Representations of Integrable Systems, Phys, Lett, A130, 271-275, 1988 |
| [20] | Kovalev A S, Private Communication. |
| [21] | Ahmad, S.; Chowdhury, A. R., On the Quasiperiodic Solutions to the Discrete Nonlinear Schrödinger Equation, J. Phys. A: Math. Gen, 20, 293-303, 1987 · Zbl 0663.35083 |
| [22] | Bogolyubov, N. N., A Discrete Periodic Problem for a Modified Nonlinear Korteweg-de Vries Equation, Teor, Math. Fiz, 50, 118-126, 1982 · Zbl 0494.34011 |
| [23] | Miller, P. D.; Ercolani, N. M.; Krichever, I. M.; Levermore, C. D., Finite Genus Solutions to the Ablowitz-Ladik Equations, Comm. on Pure and Appl, Math, 48, 1369-1440, 1996 · Zbl 0869.34065 |
| [24] | Vekslerchik, V. E., Finite Genus Solutions for the Ablowitz-Ladik Fierarchy, J. Phys. A: Math. Gen, 32, 4983-4994, 1999 · Zbl 0963.37064 |
| [25] | Erdelyi, A.; Magnus, W., Oberhettinger F and Tricomi F G, 1953, Toronto, London: McGraw-Hill Book Co., New York, Toronto, London · Zbl 0052.29502 |
| [26] | Mumford, D., Tata Lectures on Theta I, II, 1984, 1983, Boston: Birkhauser, Boston · Zbl 0509.14049 |
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