Riemann–Hilbert Problem Associated with the Fourth-Order Dispersive Nonlinear Schrödinger Equation in Optics and Magnetic Mechanics

Abstract

In this paper, by using Fokas method, we study the initial-boundary value problems (IBVPs) of the fourth-order dispersive nonlinear Schrödinger (FODNLS) equation on the half-line, which can simulate the nonlinear transmission and interaction of ultrashort pulses in the high-speed optical fiber transmission system, and can also describe the nonlinear spin excitation phenomenon of one-dimensional Heisenberg ferromagnetic chain with eight poles and dipole interaction. By discussing the eigenfunctions of Lax pair of FODNLS equation and analyzing symmetry of the scattering matrix, we get a matrix Riemann–Hilbert (RH) problem from for the IBVPs of FODNLS equation. Moreover, we get the potential function solution u(x, t) of the FODNLS equation by solving this matrix RH problem. In addition, we also obtain that some spectral functions satisfy an important global relation.