Existence of multi-traveling waves in capillary fluids

Abstract

We prove the existence of multi-soliton and kink-multi-soliton solutions of the Euler-Korteweg system in dimension one. Such solutions behaves asymptotically in time like several traveling waves far away from each other. A kink is a traveling wave with different limits at $\pm$$\infty$. The main assumption is the linear stability of the solitons, and we prove that this assumption is satisfied at least in the transonic limit. The proof relies on a classical approach based on energy estimates and a compactness argument.