广义正则化长波方程行波解的轨道不稳定性
Orbital Instability of Traveling Waves for the Generalized Regularized Long Wave Equation
DOI: 10.12677/AAM.2025.147350, PDF,   
作者: 罗霈伶：长沙理工大学数学与统计学院，湖南 长沙
关键词: 行波解；轨道不稳定性；Virial 等式；Traveling Wave； Orbital Instability； Virial Identity
摘要: 本文主要研究一类一维广义正则化长波方程的行波解轨道稳定性问题。 研究中涉及的一些参数为 正整数, 函数为实值函数。 在特定参数条件下，本文考察了该类方程的行波解的轨道稳定性。 文中 基于反证法, 结合强制性条件与调制分析，构造了具有良好单调性的 virial 等式，最终证明了该类 方程的行波解在轨道意义下是不稳定的。
Abstract: In this paper, we investigate the orbital stability of traveling wave solutions for a class of one-dimensional generalized regularized long wave equations. The study involves certain parameters that are positive integers, and the functions considered are real- valued. Under specific parameter conditions, the orbital stability of traveling wave solutions is analyzed.By employing a contradiction argument, together with coerciv- ity conditions and modulation analysis, the paper constructs a bounded virial-type identity with good monotonicity properties. It is ultimately shown that the traveling wave solutions of this equation are orbitally unstable.
文章引用：罗霈伶. 广义正则化长波方程行波解的轨道不稳定性[J]. 应用数学进展, 2025, 14(7): 96-115. https://doi.org/10.12677/AAM.2025.147350

参考文献

[1] Peregrine, D.H. (1966) Calculations of the Development of an Undular Bore. Journal of Fluid Mechanics, 25, 321-330.
https://doi.org/10.1017/s0022112066001678
[2] Peregrine, D.H. (1967) Long Waves on a Beach. Journal of Fluid Mechanics, 27, 815-827.
https://doi.org/10.1017/s0022112067002605
[3] Benjamin, T.B., Bona, J.L. and Mahony, J.J. (1972) Model Equations for Long Waves in Nonlinear Dispersive Systems. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 272, 47-78.
[4] Souganidis, P.E. and Strauss, W.A. (1990) Instability of a Class of Dispersive Solitary Waves. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 114, 195-212.
https://doi.org/10.1017/s0308210500024380
[5] Bona, J. and Tzvetkov, N. (2009) Sharp Well-Posedness Results for the BBM Equation. Dis- crete & Continuous Dynamical Systems-A, 23, 1241-1252.
https://doi.org/10.3934/dcds.2009.23.1241
[6] Grillakis, M., Shatah, J. and Strauss, W. (1987) Stability Theory of Solitary Waves in the Presence of Symmetry, I. Journal of Functional Analysis, 74, 160-197.
https://doi.org/10.1016/0022-1236(87)90044-9
[7] Grillakis, M., Shatah, J. and Strauss, W. (1990) Stability Theory of Solitary Waves in the Presence of Symmetry, II. Journal of Functional Analysis, 94, 308-348.
https://doi.org/10.1016/0022-1236(90)90016-e
[8] Bona, J.L., Souganidis, P.E. and Strauss, W.A. (1987) Stability and Instability of Solitary Waves of Korteweg-de Vries Type. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 411, 395-412.
[9] Comech, A. and Pelinovsky, D. (2003) Purely Nonlinear Instability of Standing Waves with Minimal Energy. Communications on Pure and Applied Mathematics, 56, 1565-1607.
https://doi.org/10.1002/cpa.10104
[10] Zhang, W., Shi, G., Qin, Y., Wei, G. and Guo, B. (2011) Orbital Stability of Solitary Waves for the Compound KdV Equation. Nonlinear Analysis: Real World Applications, 12, 1627-1639.
https://doi.org/10.1016/j.nonrwa.2010.10.017
[11] Lin, C. (1998) Stability and Instability of Solitary Waves for Generalized Symmetric Regularized-Long-Wave Equations. Physica D: Nonlinear Phenomena, 118, 53-68.
https://doi.org/10.1016/s0167-2789(97)00325-4
[12] Zheng, X., Shang, Y. and Peng, X. (2016) Orbital Stability of Solitary Waves of the Coupled Klein-Gordon-Zakharov Equations. Mathematical Methods in the Applied Sciences, 40, 2623- 2633.
https://doi.org/10.1002/mma.4187
[13] Wu, Y. (2023) Instability of the Standing Waves for the Nonlinear Klein-Gordon Equations in One Dimension. Transactions of the American Mathematical Society, 376, 4085-4103.
https://doi.org/10.1090/tran/8852
[14] Guo, Z., Ning, C. and Wu, Y. (2020) Instability of the Solitary Wave Solutions for the General- ized Derivative Nonlinear Schro¨dinger Equation in the Critical Frequency Case. Mathematical Research Letters, 27, 339-375.
https://doi.org/10.4310/mrl.2020.v27.n2.a2
[15] Li, B., Ohta, M., Wu, Y. and Xue, J. (2020) Instability of the Solitary Waves for the Gener- alized Boussinesq Equations. SIAM Journal on Mathematical Analysis, 52, 3192-3221.
https://doi.org/10.1137/18m1199198
[16] Yin, S. (2018) Stability and Instability of the Standing Waves for the Klein-Gordon-Zakharov System in One Space Dimension. Mathematical Methods in the Applied Sciences, 41, 4428- 4447.
https://doi.org/10.1002/mma.4905
[17] Li, J., Liu, Y., Wu, Y. and Zheng, H. (2025) Stability and Instability of Solitary-Wave Solutions for the Nonlinear Klein-Gordon Equation. Journal of Functional Analysis, 289, Article 110981.
https://doi.org/10.1016/j.jfa.2025.110981
[18] Strauss, W.A. (1977) Existence of Solitary Waves in Higher Dimensions. Communications in Mathematical Physics, 55, 149-162.
https://doi.org/10.1007/bf01626517
[19] Jia, R. and Wu, Y. (2025) Instability of the Solitary Waves for the Generalized Benjamin-Bona- Mahony Equation. Calculus of Variations and Partial Differential Equations, 64, Article No. 124.
https://doi.org/10.1007/s00526-025-02981-z
[20] Weinstein, M.I. (1985) Modulational Stability of Ground States of Nonlinear Schro¨dinger Equa- tions. SIAM Journal on Mathematical Analysis, 16, 472-491.
https://doi.org/10.1137/0516034

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