Double and triple pole solutions for the Gerdjikov–Ivanov type of derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions

In this work, the double and triple pole soliton solutions for the Gerdjikov–Ivanov type of the derivative nonlinear Schrödinger equation with zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs) are studied via the Riemann–Hilbert (RH) method. With spectral problem analysis, we first obtain the Jost function and scattering matrix under ZBCs and NZBCs. Then, according to the analyticity, symmetry, and asymptotic behavior of the Jost function and scattering matrix, the RH problem (RHP) with ZBCs and NZBCs is constructed. Furthermore, the obtained RHP with ZBCs and NZBCs can be solved in the case that reflection coefficients have double or triple poles. Finally, we derive the general precise formulas of N-double and N-triple pole solutions corresponding to ZBCs and NZBCs, respectively. In addition, the asymptotic states of the one-double pole soliton solution and the one-triple pole soliton solution are analyzed when t tends to infinity. The dynamical behaviors for these solutions are further discussed by image simulation.