Dark solitons interaction for a (2+1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain

Abstract

Under investigation in this paper is a (2 + 1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Via the symbolic computation and Hirota method, the bilinear forms, dark one-, two- and three-soliton solutions are derived. Propagation and interaction for the dark solitons are illustrated graphically: Amplitude and shape of the dark one soliton keep invariant during the propagation, which imply the transport of the energy is stable in the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain. Through the asymptotic analysis, elastic and inelastic interactions between the dark two solitons are discussed. For the elastic interaction, oblique, head-on and overtaking interactions between the dark two solitons are displayed, where the amplitudes and shapes remain unchanged after interaction except for certain phase shifts. However, in the area of the inelastic interaction, amplitudes of the dark two solitons vanish after interaction. For the elastic interaction among the dark three solitons, oblique, overtaking and interaction among the dark two parallel solitons and a single one are presented, whose characteristics are similar to those of the dark two solitons. Inelastic-elastic interaction is investigated as well. Linear stability analysis is used to analyze modulation instability and prove the dark solitons are stable.