Small energy traveling waves for the Euler-Korteweg system

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.