Long-time asymptotic behaviors of generalized higher-order nonlinear Schrödinger equation

We discuss the Cauchy problem for a generalized higher-order nonlinear Schrödinger (gH-NLS) equation, which governs the dynamics of one-dimensional anisotropic Heisenberg ferromagnetic spin chain. Based on the inverse scattering transform, the basic Riemann-Hilbert problem for the gH-NLS equation is formulated to reconstruct its solution. Furthermore, we analyze the long-time asymptotic behaviors of solution to the rapidly vanishing initial value problem for the gH-NLS equation by applying the nonlinear steepest-descent method. This process is complicated by the inclusion of higher-order dispersion terms in this model, which introduce additional stationary phase points. This study further extends the application of this research to higher-order integrable equations.