  We systematically report a rigorous theory of the inverse scattering
transforms (ISTs) for the derivative nonlinear Schrodinger (DNLS) equation with
both zero boundary condition (ZBC)/non-zero boundary conditions (NZBCs) at
infinity and double poles of analytical scattering coefficients. The scattering
theories for both ZBC and NZBCs are addressed. The direct problem establishes
the analyticity, symmetries and asymptotic behavior of the Jost solutions and
scattering matrix, and properties of discrete spectra. The inverse 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. 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
reflectionless potentials with double poles for the ZBC and NZBCs are both
carried out explicitly by means of determinants. Some representative
semi-rational bright-bright soliton, dark-bright soliton, and breather-breather
solutions are examined in detail. These results will be useful to further
explore and apply the related nonlinear wave phenomena.
