Quantitative analysis on the bifurcations and exact travelling wave solutions of a generalized fourth-order dispersive nonlinear Schrödinger equation in Heisenberg spin chain

In this paper, we analyzed the bifurcations and travelling wave solutions of a general fourth-order dispersive nonlinear Schrödinger equation arising in one dimension Heisenberg spin chain with twist interaction. First of all, based on the bifurcation theory of planar dynamical system, we obtained six types of phase portraits in the ( x , y ) -plane under different parametric conditions and revealed qualitatively the existence of the bright solitary wave, dark solitary wave, periodic wave, periodic breaking wave and unbounded wave solutions. Furthermore, we established the quantitative correspondence between the phase orbits and the energy level h . Finally, we derived the travelling wave solutions corresponding to the phase trajectories, which are found to be entirely consistent with the qualitative analysis on the types of solutions. Our results can be more easily applied to model the nonlinear wave excitations in the Heisenberg ferromagnetic spin chains with twist interactions.