Existence of multi-traveling waves in capillary fluids

TL;DR

This paper proves the existence of multi-traveling wave solutions, including solitons and kinks, in the one-dimensional Euler-Korteweg system, demonstrating their asymptotic behavior and stability under certain conditions.

Contribution

It establishes the existence of multi-traveling wave solutions in the Euler-Korteweg system, including kinks, under the assumption of soliton stability, especially in the transonic limit.

Findings

Existence of multi-soliton solutions proven.

Existence of kink-multi-soliton solutions established.

Stability of solitons confirmed in the transonic limit.

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.