Riemann-Hilbert approach and soliton solutions for the higher-order dispersive nonlinear Schrödinger equation with nonzero boundary conditions

Abstract

In this work, the higher-order dispersive nonlinear Schrödinger equation with non-zero boundary conditions at infinity is investigated including the simple and double zeros of the scattering coefficients. We introduce a appropriate Riemann surface and uniformization variable in order to deal with the double-valued functions occurring in the process of direct scattering. Then, the direct scattering problem is analyzed involving the analyticity, symmetries and asymptotic behaviors. Moreover, for the cases of simple and double poles, we study the discrete spectrum and residual conditions, trace foumulae and theta conditions and the inverse scattering problem which is solved via the Riemann-Hilbert method. Finally, for the both cases, we construct the soliton and breather solutions under the condition of reflection-less potentials. Some interesting phenomena of the soliton and breather solutions are analyzed graphically by considering the influences of each parameters.

35 pages, 8 figures

References in corpus (3)

• Inverse scattering transform and soliton solutions for the focusing Kundu-Eckhaus equation with nonvanishing boundary conditions
• Riemann-Hilbert approach to the modified nonlinear Schröinger equation with non-vanishing asymptotic boundary conditions
• Riemann-Hilbert approach for the NLSLab equation with nonzero boundary conditions

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• Riemann-Hilbert problem for the sextic nonlinear Schrödinger equation with non-zero boundary conditions