| The integrability of nonlinear evolution equations is an important research topic in soliton theory.The Heisenberg ferromagnet model is a significant integrable sys-tem and plays a vital role in physics and mathematics.It reveals that ferromagnetic material the nature of magnetization,and the relationship between magnetization and temperature change is presented.The Heisenberg supermagnet model can be regard-ed as a super extension of Heisenberg ferromagnet model,which plays a key role in superstring and topological fields.In this paper,by introducing the general auxiliary matrix variables,some(2+1)-dimensional higher order integrable Heisenberg supermagnet models are constructed under two constraints.Meanwhile,we establish their corresponding gauge equivalent counterparts,i.e.the(2+1)-dimensional super and fermionic nonlinear Schr?dinger equations,respectively.Moreover,new solutions of the supersymmetric integrable systems are also derived by means of the Bšacklund transformations.In recent years,the multi-component integrable systems have attracted the at-tention of many mathematical physicists.It is showed that the multi-component KP hierarchy and the multi-component Toda system have received a lot of attention for their widely application on representation theory,random matrix model,noninter-secting Brownian motions.By virtue of taking values in a commutative subalgebra Z_nof Lie algebra gl(n,C),we construct the Z_n-Heisenberg ferromagnet model which contains many Heisenberg ferromagnet type equations.In terms of the gauge transfor-mation,the gauge equivalent counterpart of the Z_n-Heisenberg ferromagnet model has been presented.Based on the motion curve of Euclidean space,the corresponding geo-metrical equivalence between the Z_n-Heisenberg ferromagnet model and Z_n-nonlinear Schr?dinger equation has also been established.Furthermore,we also discuss the Z_n-generalized inhomogeneous Heisenberg ferromagnet model and investigate their structure and integrability. |
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