Compressible steady water waves
Abstract
In this paper, we consider the existence of small- and large-amplitude compressible water waves. These are solutions of the two-dimensional free boundary isentropic Euler equations subjected to a constant gravitational force and neglecting the effects of surface tension. We study periodic and solitary waves beneath vacuum as well as front-type solutions of the two-phase problem in a channel. Using local and global bifurcation techniques, we construct large families of these waves beginning at an incompressible solution and treating the Mach number (defined in a suitable sense) as the bifurcation parameter. This gives a rigorous version of the classical Janzen--Rayleigh expansion, but also furnishes solution curves that may limit to transsonic flow.
Related
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- SCATTERING OF WATER WAVES IN PRESENCE OF A PARTIALLY IMMERSED OBSTACLE
- Stochastic interpretation of the Water Waves equations
BibTeX
@misc{ramadan2026compressible,
title = {{Compressible steady water waves}},
author = {Abba Ramadan and Samuel Walsh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01883},
eprint = {2610.01883},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01883},
note = {CC0},
}
RIS
TY - UNPB
TI - Compressible steady water waves
AU - Abba Ramadan
AU - Samuel Walsh
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01883
UR - https://arxiv.org/abs/2610.01883
AB - In this paper, we consider the existence of small- and large-amplitude compressible water waves. These are solutions of the two-dimensional free boundary isentropic Euler equations subjected to a constant gravitational force and neglecting the effects of surface tension. We study periodic and solitary waves beneath vacuum as well as front-type solutions of the two-phase problem in a channel. Using local and global bifurcation techniques, we construct large families of these waves beginning at an incompressible solution and treating the Mach number (defined in a suitable sense) as the bifurcation parameter. This gives a rigorous version of the classical Janzen--Rayleigh expansion, but also furnishes solution curves that may limit to transsonic flow.
ER -