Riemann-Hilbert approach and double-pole solutions for the third-order flow equation of DNLS-type equation with nonzero boundary conditions

Abstract In this paper, the Riemann-Hilbert approach is applied to study a third-order flow equation of derivative nonlinear Schrödinger-type equation with nonzero boundary conditions. By utilizing the analytical, symmetric, and asymptotic properties of eigenfunctions, a generalized Riemann-Hilbert problem is formulated for the third-order flow equation of derivative nonlinear Schrödinger-type equation with nonzero boundary conditions. The formulas of N -soliton solutions for cases of single pole and double poles are given. We present some kinds of soliton solutions of these two cases according to different distributions of spectral parameters to study the dynamical behavior of them.