LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY∗

doi:10.1007/s10473-023-0412-0

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1633-1644.doi: 10.1007/s10473-023-0412-0

LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY^∗

Wei LUO^†, Zhaoyang YIN

Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

收稿日期:2021-12-10 修回日期:2022-06-06 发布日期:2023-08-08
通讯作者: †Wei LUO, E-mail: luowei23@mail2.sysu.edu.cn
作者简介:Zhaoyang YIN, mcsyzy@mail.sysu.edu.cn
基金资助:
*National Key R&D Program of China (2021YFA1002100), the NSFC (12171493, 11701586), the FDCT (0091/2018/A3), the Guangdong Special Support Program (8-2015) and the Key Project of NSF of Guangdong Province (2021A1515010296).

LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY^∗

Wei LUO^†, Zhaoyang YIN

Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

Received:2021-12-10 Revised:2022-06-06 Published:2023-08-08
Contact: †Wei LUO, E-mail: luowei23@mail2.sysu.edu.cn
About author:Zhaoyang YIN, mcsyzy@mail.sysu.edu.cn
Supported by:
*National Key R&D Program of China (2021YFA1002100), the NSFC (12171493, 11701586), the FDCT (0091/2018/A3), the Guangdong Special Support Program (8-2015) and the Key Project of NSF of Guangdong Province (2021A1515010296).

摘要/Abstract

摘要： In this paper we investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed. We assume that the free surface is almost periodic in the horizontal direction. Using conformal mappings, one can change the free boundary problem into a fixed boundary problem for some unknown functions with the boundary condition. By virtue of the Hilbert transform, the problem is equivalent to a quasilinear pseudodifferential equation for an almost periodic function of one variable. The bifurcation theory ensures that we can obtain an existence result. Our existence result generalizes and covers the recent result in [15]. Moreover, our result implies a non-uniqueness result at the same bifurcation point.

关键词: water waves, almost periodic functions, bifurcation theory, constant vorticity

Abstract: In this paper we investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed. We assume that the free surface is almost periodic in the horizontal direction. Using conformal mappings, one can change the free boundary problem into a fixed boundary problem for some unknown functions with the boundary condition. By virtue of the Hilbert transform, the problem is equivalent to a quasilinear pseudodifferential equation for an almost periodic function of one variable. The bifurcation theory ensures that we can obtain an existence result. Our existence result generalizes and covers the recent result in [15]. Moreover, our result implies a non-uniqueness result at the same bifurcation point.

Key words: water waves, almost periodic functions, bifurcation theory, constant vorticity

Wei LUO, Zhaoyang YIN. LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1633-1644.

Wei LUO, Zhaoyang YIN. LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1633-1644.

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LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY^∗

LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY^∗

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