Localized waves and interaction solution for the fifth-order nonlinear Schrödinger equation from Heisenberg ferromagnetism

Abstract Under investigation in this paper, we consider an inhomogeneous fifth-order nonlinear Schrödinger equation from Heisenberg ferromagnetism, which describes the dynamics of a site-dependent Hisenberg ferromagnetic spin chain. Based on the Lax pair, N th-fold Darboux transformation is constructed. Three types of the located waves, including soliton, breather and interaction solutions, are obtained via Darboux transformation. Firstly, a bound-state soliton is investigated. Secondly, the nonlinear dynamics of two-breather are studied. Next, we demonstrate the interaction solutions between rogue waves and one-breather. Furthermore, the influences of coefficients for the shape and position of background waves are discussed. In addition, we analyze the interactions between the above three types of the located waves graphically. Finally, we introduce the fundamental principles of the physics-informed neural network, obtain the pulse propagation process of the data-driven solution for the equation via the algorithm and we proceed to train the parameter of equation in the subsequent stage.