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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 energies. They are obtained as minimizers of a modified energy with fixed momentum. The proof builds upon various ideas developed for the Gross–Pitaevskii equation (and more generally nonlinear Schrödinger equations with non zero limit at infinity). Even in the Schrödinger 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 Benzoni-Gavage (2013 Differ. Integral Equ. 26 439–85) actually implies nonlinear instability.