Riemann-Hilbert approach for the Kundu equation with non-vanishing boundary conditions: Simple and double poles

Abstract

The inverse scattering transform for the Kundu equation with non-vanishing boundary conditions at infinity is studied via the Riemann-Hilbert approach. A detailed spectral analysis for analyticity, symmetry properties, and asymptotic behavior of the modified Jost eigenfunctions and scattering matrix is discussed. And an appropriate matrix Riemann-Hilbert problem is formulated. A closed system is solved for the explicit N-soliton solutions to the Kundu equation in the case of the reflectionless potential. Besides, the trace formula for the analytic scattering coefficients and the so-called theta condition for the phase difference between the boundary values are derived. A consistent framework is constructed to obtain solutions associated with double zeros to the analytic scattering coefficients. The solitons in the simple and double poles case of the Chen-Lee-Liu equation as a reduced example are studied, including any combination of dark, bright and breather solitons, and the dynamical properties of these solutions are displayed and analyzed.