By Boling Guo
This can be a entire advent to Landau Lifshitz equations and Landau Lifshitz Maxwell equations, starting with the paintings by way of Yulin Zhou and Boling Guo within the early Eighties and together with lots of the paintings performed by way of this chinese language workforce led through Zhou and Guo seeing that. The ebook makes a speciality of points equivalent to the lifestyles of susceptible recommendations in multi dimensions, life and strong point of delicate recommendations in a single measurement, family with harmonic map warmth flows, partial regularity and very long time behaviors.
The ebook is a invaluable reference ebook in the event you have an interest in partial differential equations, geometric research and mathematical physics. it will probably even be used as a sophisticated 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 ideas to the method of Ferromagnetic Spin Chain.
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Extra info for Landau Lifshitz Equations
1 Spherical (Cylindrical) Symmetric Heisenberg Equations of Ferromagnetic Spin Chain Radial Symmetric Equations Consider n-dimensional Landau–Lifshitz equation St (r, t) = S × ∆S, 2 S = 1, r = (r1 , r2 , . . , rn ), S = (S x , S y , S z ), ∆= ∂2 ∂2 + · · · + . 2) 0 < r < ∞. 3) τ (x, t) = k −2 [S · (Sr × Srr )]. 2), in the natural coordinate system, exhibits in the form where eir = D ∧ ei , eit = ei ∧ Ω, i = 1, 2, 3, D = ke3 + τ e1 , Ω = ω1 e1 + ω2 e2 + ω3 e3 , krr n−1 kr 1 ω1 = − τ2 + − , k r r r ω2 = −kr − [(n − 1)/r]k, ω3 = −kτ.
26) that R1r − 2(a0 a1 + b0 b1 )x − (n − 1) = (n − 1)2 . 30) We should introduce exponential movable singular manifold except for n = 1, 2. 31) Integrability of heisenberg chain 1 (a1r + a2 ϕr ) + a1rr + 2a2r ϕr + a2 ϕrr r + 2(R1 a2 + R2 a1 + R3 a0 ) − a1 /r 3 = 0, 49 −b1t − b2 ϕt + R3 ϕr − 2(a0 a3 + b0 b3 )ϕr + R2r − 2(a0 a2 + +a21 /2 + b0 b2 + b21 /2) − 2(a1 a2 + b1 b2 )ϕr = 0. 33) that one of the functions a3 , b3 is arbitrary. Finally, it can be verified from the coefficients of (ϕ1 , ϕ1 , ϕ1 ) that a4 or b4 is arbitrary.
Substitute M1 (r, t) = M10 + m 1 (r, t), M2 (r, t) = M20 + m 2 (r, t), H (i) (r, t) = H0(i) + h(r, t). 1) where m 1 , m 2 and h are small deviations from the equilibrium values M10 , (i) M20 and H0 , and then linearize these equations. This results in a set of two linear diﬀerential equations for m 1 and m 2 . 6) where χ(k, ω) is a tensor which depends on k and ω and on the quantities char(e) acterizing the equilibrium state of the antiferromagnets. 1. Numerical results for the equilibrium states of an antiferromagnet (1).