The Gauge Equivalence of the Zakharov Equations and (2+1)-dimensional Continuous Heisenberg Ferromagnetic Models

The gauge equivalence between the (2+1)-dimensional Zakharov equation and (2+1)-dimensional integrable continuous Heisenberg ferromagnetic model is established. Also their integrable reductions are shown explicitly. Preprint CNLP-1994-04. Alma-Ata.1994 The concepts of gauge equivalence between completely integrable equations plays important role in the theory of solitons[1,2]. In the (2+1)-dimensions such equivalence have been constructed recently for the Davey-Stewartson and Ishimori equations[3], for the some Myrzakulov and nonlinear Schrödinger type equations and so on[4-7]. In this Letter we wish find the gauge equivalent counterparts of the some (2+1)-dimensional integrable continuous Heisenberg ferromagnet models(theMyrzakulovIX equation and its integrable reductions). The Myrzakulov-IX(M-IX) equation (according to the notations of ref.[8]) looks like iSt + 1 2 [S,M1S] +A2Sx +A1Sy = 0 (1a) M2u = α 4i tr(S[Sy, Sx]) (1b) where α, b, a= consts and S = ( S3 rS − rS −S3 ) , S = S1 ± iS2 S 2 = I, r = ±1 M1 = α 2 ∂ 2 ∂y2 − 2α(b− a) ∂ ∂x∂y + (a − 2ab− b) ∂ ∂x2 ; M2 = α 2 ∂ 2 ∂y2 − α(2a+ 1) ∂ ∂x∂y + a(a+ 1) ∂ ∂x2 , A1 = 2i{(2ab+ a+ b)ux − (2b+ 1)αuy} A2 = 2i{(2ab+ a+ b)uy − α (2ab+ a + 2ab+ b)ux}. These set of equations is integrable and admits the some integrable reductions: the Myrzakulov-VIII equation as b=0 and the Ishimori equation as a = b = − 2 [8]. In general we have the two integrable cases: the M-IXA equation as α = 1 and the M-IXB equation as α = −1. Equation(1) is the (2+1)-dimensional integrable generalisation of the Heisenberg ferromagnetic model iSt = 1 2 [S, Sxx] (the isotropic LandauLifshitz equation).