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

In this paper, we investigate the periodic solutions and Whitham modulation theory for the fifthorder 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 Schr¨odinger equation. Moreover, we systematically derive the two-phase periodic solutions and the corresponding Whitham modulation equations.