Nonlinear kink modes for nonisentropic supersonic vortex sheets
Abstract
In this paper, planar nonisentropic vortex sheets of the two-dimensional Euler equations under the stability condition are studied via nonlinear geometric optics, focusing on the interaction induced by a small-amplitude, high-frequency incident wave impinging on a planar vortex sheet. It is found that there exist three asymmetric critical incident angles at which resonance occurs, leading to a significant amplification of the reflected and refracted oscillatory waves. Moreover, by refining the asymptotic expansion, we deduce that the leading profiles of the highly oscillatory waves are described by initial-boundary value problems for Burgers transport equations, while the leading profile of the contact discontinuity front satisfies an initial value problem for a Hamilton--Jacobi-type equation, which reduces to a forced linear transport equation on one resonant branch. These results extend the resonant amplification mechanism from the isentropic setting to the nonisentropic Euler flow and reveal the asymmetric influence of entropy variations on the nonlinear evolution of compressible vortex sheets.
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BibTeX
@misc{li2026nonlinear,
title = {{Nonlinear kink modes for nonisentropic supersonic vortex sheets}},
author = {Zihao Li and Yang Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01036},
eprint = {2611.01036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01036},
}
RIS
TY - UNPB
TI - Nonlinear kink modes for nonisentropic supersonic vortex sheets
AU - Zihao Li
AU - Yang Xu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01036
UR - https://arxiv.org/abs/2611.01036
AB - In this paper, planar nonisentropic vortex sheets of the two-dimensional Euler equations under the stability condition are studied via nonlinear geometric optics, focusing on the interaction induced by a small-amplitude, high-frequency incident wave impinging on a planar vortex sheet. It is found that there exist three asymmetric critical incident angles at which resonance occurs, leading to a significant amplification of the reflected and refracted oscillatory waves. Moreover, by refining the asymptotic expansion, we deduce that the leading profiles of the highly oscillatory waves are described by initial-boundary value problems for Burgers transport equations, while the leading profile of the contact discontinuity front satisfies an initial value problem for a Hamilton--Jacobi-type equation, which reduces to a forced linear transport equation on one resonant branch. These results extend the resonant amplification mechanism from the isentropic setting to the nonisentropic Euler flow and reveal the asymmetric influence of entropy variations on the nonlinear evolution of compressible vortex sheets.
ER -