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.
Highlights
For a long time, finding the solutions of integrable equations has been a very important research topic in theory and application
Due to the inverse scattering transform (IST) method is suitable for the limitations of the initial value conditions at infinity, it is almost only used to study the pure initial value problem of integrable equations
In 1997, Fokas proposed a unified transformation method from the initial value problem to the initial-boundary value problems (IBVPs) based on the IST method idea
Summary
For a long time, finding the solutions of integrable equations has been a very important research topic in theory and application. In 1997, Fokas proposed a unified transformation method from the initial value problem to the IBVPs based on the IST method idea. This method can be used to investigate IBVPs of partial differential equation [6]. The long-time asymptotic behavior of the solution of the integrable systems were discussed by the nonlinear steepest descent method proposed by Deift and Zhou [29], see for instance the cases of the three-component coupled mKdV system [30] and the three-component coupled NLS system [31].
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