Landau-Lifshitz equations by Boling Guo

By Boling Guo

It is a finished advent 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 via this chinese language staff led through Zhou and Guo given that. The e-book makes a speciality of elements equivalent to the lifestyles of susceptible options in multi dimensions, life and distinctiveness of tender strategies in a single measurement, family members with harmonic map warmth flows, partial regularity and very long time behaviors.

The booklet is a useful reference booklet if you have an interest in partial differential equations, geometric research and mathematical physics. it might even be used as a complicated textbook by way of 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 recommendations to the method of Ferromagnetic Spin Chain.

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6) where S0 is a constant vector. 6) with S, we have by noting |S| = 1 that: S · S0 = c. 7) Let A be the unit vector along the direction S0 and construct the other two vectors B, C such that they are orthogonal to A and orthogonal with each other in the righthand system. 8) where α is the constant angle between S and S0 , ϕ(u) is the angle between the projection of vector S onto (B, C)-plane and vector B. 4), we get c ϕ (u) = . 9) cos α This implies that ϕ(u) = kx − ωt, where k = c/cos α, ω = k 2 cos α.

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 . 26) gM0 , in addition to the spin wave there are also two proper electromagnetic When k c waves characterized by the dispersion relation ck ω=√ . 27) Our problem now is to determine the properties of the branches of electromagnetic gM0 . oscillations for k c 31 Spin Waves and Equations of Ferromagnetic Spin Chain gM0 Since √1α , it follows that in this region of wave vectors αk 1 and, consec quently, the spatial dispersion of the high-frequency magnetic susceptibility tensor is unimportant.

7) Let A be the unit vector along the direction S0 and construct the other two vectors B, C such that they are orthogonal to A and orthogonal with each other in the righthand system. 8) where α is the constant angle between S and S0 , ϕ(u) is the angle between the projection of vector S onto (B, C)-plane and vector B. 4), we get c ϕ (u) = . 9) cos α This implies that ϕ(u) = kx − ωt, where k = c/cos α, ω = k 2 cos α. 2). 10)   w = cos θ(x, t) where S(x, t) = (u(x, t), v(x, t), w(x, t)). 1) that  dθ   = −2θx ϕx cos θ − ϕxx sin θ, dt   ϕt sin θ = θxx − (ϕx )2 sin θ cos θ.

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