On the rotation-two-component b-family shallow-water system: derivation and wave-breaking analysis
Abstract
In this paper, we derive the rotation-two-component b-family shallow-water wave system from the Green--Naghdi type equations incorporating the effects of the Coriolis force and constant vorticity. The system admits, among its special cases, the two-component Degasperis--Procesi (2DP) system as a reduction, thereby confirming its genuine role as a rotational extension of the b-family. We then focus on a specific reduction, the rotation-two-component Degasperis-Procesi system, whose particular interest lies in the fact that it combines two independent sources of difficulty: unlike the two-component Camassa-Holm type system, it possesses no conserved H1× L2-type quantity, and in contrast to the standard 2DP system, it contains a higher-order nonlinear coupling term induced by the Coriolis force. To overcome this double difficulty, we introduce an auxiliary variable and establish a local L∞-estimate for u without using any conservation law, based on which we derive two distinct blow-up criteria. When the Coriolis force vanishes, our results recover known criteria for the 2DP system and also provide new criteria for both the 2DP system and Degasperis-Procesi equation. The results extend naturally to the periodic setting and to relevant special cases of the rotational b-family system.
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BibTeX
@misc{liu2026rotationtwocomponent,
title = {{On the rotation-two-component $b$-family shallow-water system: derivation and wave-breaking analysis}},
author = {Xingxing Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09900},
eprint = {2609.09900},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09900},
}
RIS
TY - UNPB
TI - On the rotation-two-component b-family shallow-water system: derivation and wave-breaking analysis
AU - Xingxing Liu
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09900
UR - https://arxiv.org/abs/2609.09900
AB - In this paper, we derive the rotation-two-component b-family shallow-water wave system from the Green--Naghdi type equations incorporating the effects of the Coriolis force and constant vorticity. The system admits, among its special cases, the two-component Degasperis--Procesi (2DP) system as a reduction, thereby confirming its genuine role as a rotational extension of the b-family. We then focus on a specific reduction, the rotation-two-component Degasperis-Procesi system, whose particular interest lies in the fact that it combines two independent sources of difficulty: unlike the two-component Camassa-Holm type system, it possesses no conserved H^1× L^2-type quantity, and in contrast to the standard 2DP system, it contains a higher-order nonlinear coupling term induced by the Coriolis force. To overcome this double difficulty, we introduce an auxiliary variable and establish a local L^∞-estimate for u without using any conservation law, based on which we derive two distinct blow-up criteria. When the Coriolis force vanishes, our results recover known criteria for the 2DP system and also provide new criteria for both the 2DP system and Degasperis-Procesi equation. The results extend naturally to the periodic setting and to relevant special cases of the rotational b-family system.
ER -