具有恒定涡度的小振幅水弹性孤波与广义孤波——献给陈化教授 70 寿辰

摘要/Abstract

摘要：

该文证明了具有恒定涡度的二维小振幅水弹性孤波及广义孤波的存在性. 该波面下方水流的深度有限, 且水流中不出现临界层. 利用空间动力学方法, 原问题被转化为一个等价的动力系统, 其中水平空间方向作为类时变量. 随后应用中心流形约化与正规形理论, 得到了约化系统的同宿解, 这些解对应原水弹性波问题的孤波或广义孤波.

关键词: 水弹性孤立波, 恒定涡度, 空间动力系统

Abstract:

In this paper, we prove the existence of two-dimensional hydroelastic solitary waves with constant vorticity by using spatial dynamics method. The flow beneath the wave is of finite depth, and critical layers are absent throughout the flow. The problem is converted to an equivalent dynamical system in which the horizontal spatial direction is the time like variable. Then application of a center-manifold reduction technique and normal-form theory yields the existence of homoclinic solutions to the reduced system, which correspond to solitary or generalized solitary waves of the hydroelastic problem.

Key words: hydroelastic solitary waves, constant vorticity, spatial dynamics

中图分类号:

O175.29

王灵君. 具有恒定涡度的小振幅水弹性孤波与广义孤波——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 628-645.

Lingjun Wang. Hydroelastic Small-Amplitude Solitary and Generalized Solitary Waves with Constant Vorticity[J]. Acta mathematica scientia,Series A, 2026, 46(2): 628-645.

图/表 1

参考文献 37

[1]	Ahmad R, Groves M D. Spatial dynamics and solitary hydroelastic surface waves. Water Waves, 2024, 6: 5-47
[2]	Ahmad R, Groves M D, Nilsson D. A resonant lyapunov centre theorem with an application to doubly periodic travelling hydroelastic waves. J Nonlinear Sci, 2024, 34: Art 104
[3]	Buffoni B, Groves M D. A multiplicity result for solitary gravity-capillary waves in deep water via critical-point theory. Arch Ration Mech Anal, 1999, 146: 183-220
[4]	Chen R M, Walsh S, Wheeler M H. Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics. Nonlinearity, 2022, 35(4): 1927-1985
[5]	Constantin A, Strauss W. Exact steady periodic water waves with vorticity. Comm Pure Appl Math, 2004, 57: 481-527
[6]	Constantin A, Strauss W, Varvaruca E. Global bifurcation of steady gravity water waves with critical layers. Acta Math, 2016, 217: 195-262
[7]	Faye G, Scheel A. Center manifolds without a phase space. Trans Am Math Soc, 2018, 370: 5843-5885
[8]	Faye G, Scheel A. Corrigendum to center manifolds without a phase space. arXiv:2007.14260
[9]	Gao T, Wang Z, Milewski P A. Nonlinear hydroelastic waves on a linear shear current at finite depth. J Fluid Mech, 2019, 876: 55-86
[10]	Gao T, Wang Z, Vanden-Broeck J M. New hydroelastic solitary waves in deep water and their dynamics. J Fluid Mech, 2016, 788: 469-491
[11]	Groves M D, Hewer B, Wahlén E. Variational existence theory for hydroelastic solitary waves. C R Acad Sci Paris, 2016, 354: 1078-1086
[12]	Groves M D, Lloyd D J B, Stylianou A. Pattern formation on the free surface of a ferrofluid: Spatial dynamics and homoclinic bifurcation. Physica D, 2017, 350: 1-12
[13]	Groves M D, Mielke A. A spatial dynamics approach to three-dimensional gravity-capillary steady water waves. Proc Roy Soc Edinburgh, 2001, 131(1): 83-136
[14]	Groves M D, Nilsson D V. Spatial dynamics methods for solitary waves on a ferrofluid jet. J Math Fluid Mech, 2018, 20: 1427-1458
[15]	Groves M D, Wahlén E. Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity. SIAM J Math Anal, 2007, 39: 932-964
[16]	Guyenne P, Pǎrǎu E I. Computations of fully nonlinear hydroelastic solitary waves on deep water. J Fluid Mech, 2012, 713: 307-329
[17]	Guyenne P, Pǎrǎu E I. Finite-depth effects on solitary waves in a floating ice sheet. J Fluid Struct, 2014, 49: 242-262
[18]	Haragus M, Iooss G. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. London: Springer-Verlag, 2011
[19]	Iooss G, Pérouème M C. Perturbed homoclinic solutions in reversible 1 : 1 resonance vector fields. J Differential Equations, 1993, 102: 62-88
[20]	Johnson M A, Truong T, Wheeler M H. Solitary waves in a Whitham equation with small surface tension. Stud Appl Math, 2022, 148: 773-812
[21]	Kirchgässner K. Wave-solutions of reversible systems and applications. J Differential Equations, 1982, 45: 113-127
[22]	Kozlov V, Kuznetsov N, Lokharu E. Solitary waves on rotational flows with an interior stagnation point. arXiv:1904.00401
[23]	Kozlov V, Lokharu E. Small-amplitude steady water waves with critical layers: Non-symmetric waves. J Differential Equations, 2019, 267: 4170-4191
[24]	Lombardi E. Oscillatory Integrals and Phenomena Beyond all Algebraic Orders:With Applications to Homoclinic Orbits in Reversible Systems. New York: Springer-Verlag, 2000 %MR1770093 (2002f: 34081).
[25]	Martin C I, Matioc B V. Steady periodic water waves with unbounded vorticity: equivalent formulations and existence results. J Nonlinear Sci, 2014, 24: 633-659
[26]	Mielke A. Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math Methods Appl Sci, 1988, 10: 51-66
[27]	Milewski P A, Vanden-Broeck J M, Wang Z. Hydroelastic solitary waves in deep water. J Fluid Mech. 2011, 679: 628-640
[28]	Milewski P A, Vanden-Broeck J M, Wang Z. Steady dark solitary flexural gravity waves. Procedings of the Royal Society A, 2013, 469: Art 20120485
[29]	Nilsson D V. Internal gravity-capillary solitary waves in finite depth. Math Methods Appl Sci, 2017, 40: 1053-1080
[30]	Plotnikov P, Toland J F. Modelling nonlinear hydroelastic waves. Philos Trans R Soc Lond Ser A, 2011, 369: 2942-2956
[31]	Toland J F. Steady periodic hydroelastic waves. Arch Rational Mech Anal, 2008, 189: 325-362
[32]	Truong T, Wahlén E, Wheeler M H. Global bifurcation of solitary waves for the Whitham equation. Mathematische Annalen, 2022, 383: 1521-1565
[33]	Wahlén E, Weber J. Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticity. International Mathematics Research Notices, 2023, 2023(20): 17377-17410
[34]	Wahlén E, Weber J. Large-amplitude steady gravity water waves with general vorticity and critical layers. Duke Mathematical Journal, 2024, 173(11): 2197-2258
[35]	Small-amplitude solitary and generalized solitary traveling waves in a gravity two-layer fluid with vorticity. Nonlinear Anal, 2017, 150: 159-193
[36]	Wang Z, Guan X, Vanden-Broeck J M. Progressive flexural-gravity waves with constant vorticity. J Fluid Mech, 2020, 905: Art 12
[37]	Wheeler M. On stratified water waves with critical layers and Coriolis forces. Discrete Cont Dyn Syst, 2019, 39(8): 4747-4770