Nonlinear localized waves in the (2+1)-dimensional Heisenberg ferromagnetic spin chain system on the periodic waves background

We consider a (2+1)-dimensional Heisenberg ferromagnetic spin chain system (HFSCS) and present a general solutions of its Lax pair by the variable separation method and the constant change method. As an application of the general solutions of the Lax pair, three different families of the breathers and solitons for the (2+1)-dimensional HFSCS on single and double periodic waves background are derived by the Darboux transformation approach. Combining the Darboux transformation and the nonlinearization of Lax pair, the rogue waves on two kinds periodic background are expressed by the Jacobian elliptic functions dn and cn. We show that linear superposition of various nonlinear wave solutions of the (2+1)-dimensional HFSCS results into several kinds of nonlinear localized waves structures. Moreover, the results represented in this paper enrich the dynamics of the higher dimensional nonlinear wave equations.