Functionals and their applications: Selected topics by Evans G.C.

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36) It follows from simple computations that A0r r A0 + 2Aϕr r A0 = r B0r r B0 + 2Bϕr r B0 = 0. 37) r Let iλ be the integral constant, we have A0r + iλA0 = −2qϕr , B0r − iλB0 = 2Bϕr . 38) Define square characteristic functions as follows: A0 = iΨ21 , B0 = iΨ22 , ϕr = −iΨ1 Ψ2 . 40) Landau–Lifshitz Equations 50 where V1 = −iλr/2 q . 44) and the compatible condition is U1t − V1r + [U1 , V1 ] = 0. 34): λt = 2λ2 . 45) Introduce normal change: ˆ = g −1 Ψ, Ψ g(r, t) ∈ GL(2, C). 42) to give new eigenvalue equation ˆ r = Uˆr Ψ, ˆ Ψ where ˆ t = Vˆ1 Ψ, ˆ Ψ Uˆ1 = g −1 U1 g − g −1 gr , Vˆ1 = g −1 V1 g − g −1 gt , gr = U1 (λ = 0)g, gt = V1 (λ = 0)g.

79) for a uniaxial crystal. 38) n2j cos2 θk − µ(1 − n2j sin2 θk ) ρj = . 40) We note that and consequently (1) b1 (1) b2 (2) =− b2 (2) b1 = iρ1 . 36) that (j) h1 (j) h2 (j) = iρj , h3 (j) h2 = µ sin θk + (µ − 1) sin θk cos θk ) , µ sin2 θk + cos2 θk where as before the subscript j represents waves with refractive index nj . 37) for the refractive indices and find the values of ω for which the wave vector is zero. 42) it follows that k will be zero together with ω, and when ω k1 = ω µ(0) c , k2 = ω µ(0) c gM0 , 1 cos2 θk + µ(0) sin2 θk .

J(x, t) = k τ = 16 4 2 And, we also have √ √ 3x 3x −1/2 tan h (λt + δ) (λt + δ)−1/2 4 4 √ √ 3x 3x (λt + δ)−1/2 − cos (λt + δ)−1/2 4 4 √ √ 3x 3x −1/2 y S = − sech tan h (λt + δ) (λt + δ)−1/2  4 4    √ √     3x 3x  −1/2  − sin cos (λt + δ) (λt + δ)−1/2    4 4    √    3x   2 z  (λt + δ)−1/2 . 25) Inhomogeneous Heisenberg Chain Inhomogeneous Ferromagnetic Equations Consider the nonhomogeneous ferromagnetic chain equation St (x, t) = f (x)S × Sxx + fx (x)(S × Sx ). 1) Let e1 (x, t), e2 (x, t), e3 (x, t) be the tangential vector, normal vector and co-normal vector of a moving space curve and form a natural coordinate system.