Gauge Equivalence between Two-dimensional Heisenberg Ferromagnets with Single-site Anisotropy and Zakharov Equations

Gauge equivalence between the two-dimensional continuous classical Heisenberg ferromagnets(CCHF) of spin 1 2 -the M-I equation with singleside anisotropy and the Zakharov equation(ZE) is established for the easy axis case. The anisotropic CCHF is shown to be gauge equivalent to the isotropic CCHF. Preprint CNLP-1994-05. Alma-Ata.1994 In the study of 1+1 dimensional ferromagnets, a well known Lakshmanan and gauge equivalence take place between the continuous classical Heisenberg ferromagnets of spin 1 2 and the nonlinear Schrodinger equations (e.g., Lakshmanan 1977, Zakharov and Takhtajan 1979, Nakamura and Sasaga 1982, Kundu and Pashaev 1983, Kotlyarov 1984). At the same time, there exit some integrable analogues of the CCHF in 2+1 dimensions(Ishimori 1982, Myrzakulov 1987). One integrable (2+1)-dimensional extension of the CCHF is the following Myrzakulov-I(M-I) equation with one-ion anisotropy St = (S ∧ Sy + uS)x + vS ∧ n, (1a) ux = −S · (Sx ∧ Sy), (1b) vx = △(Sy · n) (1c) were the spin field S = (S1, S2, S3) with the magnetude normalized to unity, u and v are scalar functions, n = (0, 0, 1), and △ < 0 and △ > 0 correspond respectively to the system with an easy plane and to that with an easy axis. Note that if the symmetry ∂x = ∂y is imposed then the M-I equation (1) reduces to the well known Landau-Lifshitz equation with single-site anisotropy St = S ∧ (Sxx +△(S · n)n). (2) The aim of this letter is the construction the (2+1)-dimensional nonlinear Schrodinger equation which is gauge equivalent to the M-I equation (or two-dimensional CCHF)(1) with the easy-axis anisotropy (△ > 0). Besides the gauge equivalence between anisotropic (△ 6 = 0) and isotropic (△ = 0) CCNF (1) is established. The Lax representation of the M-I equation (1) may be given by (Myrzakulov 1987) ψx = L1ψ, ψt = 2λψy +M1ψ (3) where L1 = iλS+μ[σ3, S], M1 = 2λA+2iμ[A, σ3] + 4iμ {σ3, V }σ3 (4)