Small energy traveling waves for the Euler-Korteweg system

Abstract

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 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 [6] actually implies nonlinear instability.

43 pages

References in corpus (3)

• On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics
• Travelling waves for the Gross-Pitaevskii equation II
• Global well-posedness of the Euler-Korteweg system for small irrotational data

Cited by in corpus (1)

• Exotic traveling waves for a quasilinear Schrödinger equation with nonzero background