Riemann-Hilbert problem associated with the fourth-order dispersive nonlinear Schrödinger equation in optics and magnetic mechanics

In this paper, we utilize Fokas method to investigate 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 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 the analysis and symmetry of the scattering matrix, the IBVPs of FODNLS equation is expressed as a matrix Riemann-Hilbert (RH) problem form. Then one can get the potential function solution $u(x,t)$ of the FODNLS equation by solving this matrix RH problem. In addition, we also obtained that some spectral functions admits a key global relationship.

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