Double and triple poles solutions for the Gerdjikov-Ivanov type of derivative nonlinear Schr\"{o}dinger equation with zero/nonzero boundary conditions

Abstract

In this work, the double and triple poles soliton solutions for the Gerdjikov-Ivanov(GI) type of derivative nonlinear Schr\"{o}dinger equation with zero boundary conditions(ZBCs) and nonzero boundary conditions(NZBCs) are studied via Riemann-Hilbert (RH) method. Though spectral problem analysis, we first give out the Jost function and scattering matrix under ZBCs and NZBCs. Then according to the analyticity, symmetry and asymptotic behavior of Jost function and scattering matrix, the Riemann-Hilbert problem(RHP) with ZBCs and NZBCs are constructed. Further, 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 formulae of N-double and N-triple poles solutions corresponding to ZBCs and NZBCs, respectively. The dynamical behaviors for these solutions are further discussed by image simulation.