New PDF release: Landau Lifshitz Equations (Frontiers of Research With the

New PDF release: Landau Lifshitz Equations (Frontiers of Research With the

By Boling Guo

This can be a complete creation to Landau Lifshitz equations and Landau Lifshitz Maxwell equations, starting with the paintings by means of Yulin Zhou and Boling Guo within the early Nineteen Eighties and together with many of the paintings performed by means of this chinese language team led via Zhou and Guo in view that. The booklet makes a speciality of features reminiscent of the life of susceptible options in multi dimensions, lifestyles and distinctiveness of delicate suggestions in a single measurement, relatives with harmonic map warmth flows, partial regularity and very long time behaviors. The booklet is a useful reference booklet if you happen to have an interest in partial differential equations, geometric research and mathematical physics. it might even be used as a sophisticated 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 recommendations to the method of Ferromagnetic Spin Chain.

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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.

51) and has potential q (r), and q (r, t) = q(r, t) + f (Γ, λ). 34). Take, for example, Γ and q as follows: Γ = 1/Γ∗ , q + q = −4βrΓ/(1 + |Γ|2 ). 51) with potential q (r, t) where λ(t) = α(t) + iβ(t), β > 0. 53) as Γ = −[2βγ + 4β 2 r 2 − |q + q|1/2 ]/(q ∗ + q ∗ ). 34): qr + qr = (q + q )/r 1 + (q + q)(4β 2 r 2 − |q + q|2 )1/2 − iαr(q + q), 2 iq qt + iqt = 4α(q + q ) + iqr + − αrq (4β 2 r 2 − |q + q|2 )1/2 r iq |q|2 |q + q|2 + i(q + q) r dr − ir 2 q(α2 − β 2 ) 4 r 0 i 2 r 2 2 2 − ir q(α + β ) − (q q + q|2 q).

E. we can assume that m(r, t) and h(r, t) satisfy the equations    rot h(r, t) = 0,   div h(r, t) = 4π div m(r, t). 32) Transforming to the Fourier components in these equations, we obtain    [k, h(k, ω)]  k = 0, · h(k, ω) = 4π k · m(k, ω). 33) From the first equation it follows that the magnetic field h(k, w) is parallel to the wave vector k : h(k, ω) = −ik · ϕ(k, ω), where ϕ(k, ω) is the Fourier component of the magnetic potential. 33) in the form (k 2 + 4πki kj χij (k, ω))ϕ(k, ω) = 0 and hence k 2 + 4πkikj χij (k, ω) = 0.

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