The soliton solutions for the higher-order nonlinear Schrödinger equation with nonzero boundary conditions: Riemann-Hilbert method

This paper explores the Riemann-Hilbert method for deriving exact N-soliton solutions of the sixth-order nonlinear Schr & ouml;dinger (6th-NLS) equation with nonzero boundary condition. The analytical process comprises three fundamental steps. First, transformations are used to simplify the nonzero boundaries. Next, the inverse scattering method establishes a crucial link between the solutions of the 6th-NLS equation and the corresponding Riemann-Hilbert problem. Finally, this Riemann-Hilbert problem is systematically solved. Additionally, selected parameter values in the solutions generate graphical representations, vividly illustrating the solutions to the 6th-NLS equation.