Dark solitonic interaction and conservation laws for a higher-order (2+1)-dimensional nonlinear Schrödinger-type equation in a Heisenberg ferromagnetic spin chain with bilinear and biquadratic interaction

In this paper, a higher-order (2+1)-dimensional nonlinear Schrödinger-type equation is investigated, which describes the nonlinear spin dynamics of the (2+1)-dimensional Heisenberg ferromagnetic spin chain with bilinear and biquadratic interaction. Lax pair and infinite-number conservation laws are constructed, which could prove the existence of the multi-soliton solutions. Via the auxiliary function, bilinear forms and dark-soliton solutions are derived. Properties and interaction patterns for the dark solitons are investigated: (i) Effects on the dark solitons arising from the bilinear interaction, biquadratic interaction and lattice parameter are discussed. (ii) Through the asymptotic analysis, elastic and inelastic interaction between the two solitons is discussed analytically and graphically, respectively. Due to the elastic interaction, amplitudes and velocities of the two dark solitons remain unchanged with the distortion of the interaction area, except for certain phase shifts. However, in virtue of the inelastic interaction, amplitudes of the dark solitons reduce to zero, without the distortion. (iii) Elastic interaction among the three dark solitons is investigated, which implies that the properties of the elastic interaction among the three are similar to that between the two, except for the more complicated distortion. Inelastic-elastic interaction is also investigated, which implies that the interaction between the inelastic region and the dark soliton is elastic. (iv) Linear stability analysis is proposed, which is used to analyze the properties of modulation instability and proves that the dark solitons are modulationally stable.