By Boling Guo
This can be a accomplished creation to Landau Lifshitz equations and Landau Lifshitz Maxwell equations, starting with the paintings through Yulin Zhou and Boling Guo within the early Eighties and together with many of the paintings performed through this chinese language crew led by way of Zhou and Guo on the grounds that. The e-book specializes in facets comparable to the lifestyles of vulnerable strategies in multi dimensions, lifestyles and distinctiveness of tender options in a single size, family with harmonic map warmth flows, partial regularity and very long time behaviors.
The booklet is a precious reference publication if you have an interest in partial differential equations, geometric research and mathematical physics. it can even be used as a complicated textbook via graduate scholars in those fields.
Contents: Spin Waves and Equations of Ferromagnetic Spin Chain; Integrability of Heisenberg Chain; One-Dimensional Landau Lifshitz Equations; Landau Lifshitz Equations and Harmonic Maps; Landau Lifshitz Maxwell Equations; very long time habit of strategies to the approach of Ferromagnetic Spin Chain.
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Extra resources for Landau-Lifshitz equations
17), we have l 0 (Zxx · f (x, t, Z))dx = Zx (l, t) · f (l, t, Z(l, t)) − Zx (0, t) · f (0, t, Z(0, t)) − l 0 (Zx · Dx f )dx. It follows from assumption (1) that Qτ (Zx · Dx f )dxdt = Qτ + ≤ Qτ b+ + (Zx · fx (x, t, Z)Zx )dxdt (Zx · fx (x, t, Z))dxdt 1 2 1 2 l Zx (·, t) 0 Qτ 2 L2 (0,l) dt |fx (x, t, Z)|2 dxdt. It follows from assumption (2) and interpolation formula that 1 2 Qτ and |fx |2 dxdt ≤ c(x, t) l 0 2 L∞ (QT ) |Z(x, t)|6 dx ≤ C1 Z(·, t) Qτ 4 L2 (0,l) |Z|6 dxdt + d · Z(·, t) 2 L2 (QT ) 2 H01 (0,l) .
1. Numerical results for the equilibrium states of an antiferromagnet (1). 2. Computational results for the equilibrium states of an antiferromagnet (2). 9) (e) 2δ(αij − αij )ki kj + (H1 /M0 )2 ± gH0 , in which χ0 = δ −1 . 11) (e) = gM0 2δ(αij − αij )ki kj + (H1 /M0 )2 + (H0 /M0 )2 , = gM0 2δ(αij − αij )ki kj + (H1 /M0 )2 . (e) H2 and H2 = 2δM0 (the z-axis lies along the anisotropy It is assumed that H0 (e) axis and the x-axis lies along H0 ). Consider now an antiferromagnet with magnetic anisotropy of the “easy plane” (e) (e) H2 , the type.
23) C(k, ω) = detµij (k, ω), and ∆ij (k, ω) are the minors of the determinant |µij (k, ω)|. This dispersion relation will in general deﬁne not one but several frequencies for a given wave vector k. For a ferromagnet there are three such frequencies, whereas for an antiferromagnet there are four. Diﬀerent frequencies corresponding to the same k deﬁne diﬀerent branches of the oscillations. 25) for the spin waves. Since the spin wave frequency ωS (k) is of the order of gM0, it may be said that spin waves correspond to wave vector k satisfying the inequality k gM0 c λ c .
Landau-Lifshitz equations by Boling Guo