Some representative semi-rational bright–bright soliton, dark–bright soliton, and breather–breather solutions are examined in detail. The reflectionless potentials with double poles for the ZBCs and NZBCs are both carried out explicitly by means of determinants. In particular, the IST with NZBCs at infinity is proposed by a suitable uniformization variable, which allows the scattering problem to be solved on a standard complex plane instead of a two-sheeted Riemann surface. The inverse scattering problems are formulated and solved with the aid of the matrix Riemann–Hilbert problems, and the reconstruction formulae, trace formulae and theta conditions are also posed.
The direct scattering problem establishes the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix, and properties of discrete spectra. The scattering theories for both ZBCs and NZBCs are addressed. In this paper, we report a rigorous theory of the inverse scattering transforms (ISTs) for the derivative nonlinear Schrödinger (DNLS) equation with both zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs) at infinity and double zeros of analytical scattering coefficients.
