  In this work, the higher-order dispersive nonlinear Schr\"{o}dinger equation
with non-zero boundary conditions at infinity is investigated including the
simple and double zeros of the scattering coefficients. We introduce a
appropriate Riemann surface and uniformization variable in order to deal with
the double-valued functions occurring in the process of direct scattering.
Then, the direct scattering problem is analyzed involving the analyticity,
symmetries and asymptotic behaviors. Moreover, for the cases of simple and
double poles, we study the discrete spectrum and residual conditions, trace
foumulae and theta conditions and the inverse scattering problem which is
solved via the Riemann-Hilbert method. Finally, for the both cases, we
construct the soliton and breather solutions under the condition of
reflection-less potentials. Some interesting phenomena of the soliton and
breather solutions are analyzed graphically by considering the influences of
each parameters.
