Lax pair, interactions and conversions of the nonlinear waves for a (2+1)-dimensional nonlinear Schrödinger equation in a Heisenberg ferromagnetic spin chain

Spin waves, the collective excitations of the electron spin systems in the ferromagnetic metals, are used in the telecommunication systems and radars. Under investigation is a (2+1)-dimensional nonlinear Schrödinger equation which describes the spin dynamics of a Heisenberg ferromagnetic spin chain. We construct its Lax pair which is different from the published ones. With respect to the coherent magnetism amplitude for the bosonic operators at the spin lattice sites, we derive the n-th-order breather solutions with n as a positive integer. We convert, under certain conversion conditions, the breathers to the lumps, rogue waves and two types of periodic waves which are named as the periodic-I and periodic-II waves in this paper. The breathers and periodic-I waves are affected by the uniaxial crystal field anisotropy parameter A as well as the bilinear exchange interactions. Periods of the periodic-II waves are affected by A and the lattice parameter. Via the second-order breather solutions, interactions of the two breathers, of the two periodic waves and of a breather and a periodic-I wave are graphically discussed. Through the theoretical and graphical analyses, it is found that the lumps and rogue waves are the long-wave limits of the breathers and periodic waves, respectively.