Whitham modulation theory and periodic solutions for the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain

Abstract

In this paper, we investigate the periodic solutions and Whitham modulation theory for the fifth-order nonlinear Schrödinger equation, which can describe the one-dimensional anisotropic Heisenberg ferromagnetic spin chain. First, we introduce the principle of the finite-gap integration method. Then, we discuss the single-phase periodic solutions and their degenerate forms in two limit cases. In addition, we analyze the influence of higher-order term parameters on the propagation of periodic solutions and solitons. Further, we derive the single-phase Whitham modulation equation for the fifth-order nonlinear Schrödinger equation. Moreover, we systematically derive the two-phase periodic solutions and the corresponding Whitham modulation equations.