Classification and Soliton for a Generalized Fourth-Order Dispersive Nonlinear Schrödinger Equation in a Heisenberg Spin Chain

The main objective of this paper is to study bifurcations and solitons for a generalized fourth-order dispersive nonlinear Schrödinger equation in a Heisenberg spin chain via the theory of differential dynamical systems. By introducing the complex traveling wave transformation, we obtain the singular traveling wave system and the Hamiltonian function. By applying the classical analysis method, we investigate the bifurcations thoroughly and obtain the phase portraits of the bifurcations in four cases. Based on the phase portraits, we obtain a large number of analytic parametric representations for bounded solitons, including periodic wave solutions, solitary wave solutions, kink (anti-kink) wave solutions, compacton solutions, and a special class of bounded soliton solutions, corresponding to stable and unstable manifolds of the saddle point. These analytic parametric representations enhance the understanding of dynamic properties and characteristics of soliton solutions.