  We investigate the existence and properties of traveling waves for the
Euler-Korteweg system with general capillarity and pressure. Our main result is
the existence in dimension two of waves with arbitrarily small energy. They are
obtained as minimizers of a modified energy with fixed momentum. The proof
follows various ideas developed for the Gross-Pitaevskii equation (and more
generally nonlinear Schr\"odinger equations with non zero limit at infinity).
Even in the Schr\"odinger case, the fact that we work with the hydrodynamical
variables and a general pressure law both brings new difficulties and some
simplifications. Independently, in dimension one we prove that the criterion
for the linear instability of traveling waves from [6] actually implies
nonlinear instability.
