Longitudinal-transverse soliton chains analog to heisenberg ferromagnetic spin chains in (2+1) dimensional with biquadrant interactions

Abstract

AbstractIn quantum spin systems, lower energy state for a nuclear spin in an external field is spin ($$+1/2$$ + 1 / 2 ) while the higher energy state is spin ($$-1/2$$ - 1 / 2 ). A continuum analog to the discrete model led to nonlinear Schrodinger equation (NLSE) with higher order dispersion. Recently, a model equation with bilinear and biquadratic interactions in the (2 + 1) dimensional Heisenberg ferromagnetic spin chain (HFSC), was derived in the literature. This model equation is a NLSE with quartic dispersion and fifth degree nonlinearity, and it was rarely considered in the literature. The, only, work done has shown solitons solutions. This motivated us to consider this problem for inspecting the multiple characteristics of the HFSC in space-time. Here, the exact solutions of the HFSC biquadratic model equation are obtained by using the unified method (UM). In applications, it is found the UM is of low time cost in symbolic computations. So, we think that it prevails the known methods. The solutions obtained are evaluated numerically and shown in figures.These figures reveal that the solutions exhibit longitudinal-transverse (L-T) solitons chains (SCs) with presence (or absence) of tunneling, depending on the values of the parameters of high dispersivity and high nonlinearity. Also, L-T zig-zag SCs are observed, where pulses with higher amplitudes (or gaps) occur along a characteristic line. Furthermore, L-TSCs modulation along the space axes is shown. It is remarked that the contour plots show complex lattice waves, which is relevant to the spin chain system. So, the present work reveals new solitons structures induced by bilinear-biquadratic interactions of HFSC.