Riemann-Hilbert Approach and Soliton Solutions of the Higher-Order Dispersive Nonlinear Schrödinger Equations with Single and Double Poles

Abstract

The higher-order dispersive nonlinear Schrödinger equation with the zero boundary conditions at the infinity is studied by the Riemann-Hilbert approach. We consider the direct scattering problem, corresponding eigenfunctions, scattering matrix and establish some of their properties. These results are used in the construction of an associated Riemann-Hilbert problem. Assuming that the scattering coefficients possess single or double poles, we derive the problem solutions. Finally, we present graphical examples of 1-, 2- and 3-soliton solutions and discuss their propagation.