  We study a defocusing quasilinear Schr\"odinger equation with nonzero
conditions at infinity in dimension one. This quasilinear model corresponds to
a weakly nonlocal approximation of the nonlocal Gross--Pitaevskii equation, and
can also be derived by considering the effects of surface tension in
superfluids. When the quasilinear term is neglected, the resulting equation is
the classical Gross-Pitaevskii equation, which possesses a well-known stable
branch of subsonic traveling waves solution, given by dark solitons.
  Our goal is to investigate how the quasilinear term affects the
traveling-waves solutions. We provide a complete classification of finite
energy traveling waves of the equation, in terms of the two parameters: the
speed and the strength of the quasilinear term. This classification leads to
the existence of dark and antidark solitons, as well as more exotic localized
solutions like dark cuspons, compactons, and composite waves, even for
supersonic speeds. Depending on the parameters, these types of solutions can
coexist, showing that finite energy solutions are not unique. Furthermore, we
prove that some of these dark solitons can be obtained as minimizers of the
energy, at fixed momentum, and that they are orbitally stable.
