具有一般非线性的广义Boussinesq方程的概周期解
Almost Periodic Solutions of the Generalized Boussinesq Equations with General Nonlinearities
DOI: 10.12677/aam.2024.1311490, PDF,    科研立项经费支持
作者: 汪 崧：南京航空航天大学数学学院，江苏 南京
关键词: 概周期解；广义Boussinesq方程；KAM；Almost Periodic Solution； Generalized Boussinesq Equation； KAM
摘要: 本文考虑了一类在铰接边界条件下具有一般非线性的广义Boussinesq方程。利用无限维KAM (Kolmogorov-Arnold-Moser)理论，我们证明了在足够小的扰动下，这类方程存在无限多个小振幅、实解析且线性稳定的概周期解。
Abstract: This paper considers a class of generalized Boussinesq equations with general nonlinearities under hinged boundary conditions. Using infinite-dimensional KAM (Kolmogorov-Arnold-Moser) theory, we prove that for sufficiently small perturbations, the equations admit infinitely many of small amplitude, real analytic and linearly stable almost periodic solutions.
文章引用：汪崧. 具有一般非线性的广义Boussinesq方程的概周期解[J]. 应用数学进展, 2024, 13(11): 5073-5088. https://doi.org/10.12677/aam.2024.1311490

参考文献

[1] Shi, Y. and Xu, J. (2023) Quasi-Periodic Solutions for a Generalized Higher-Order Boussinesq Equation. Qualitative Theory of Dynamical Systems, 22, Article No. 139. [Google Scholar] [CrossRef] 
[2] Shi, Y., Xu, J. and Xu, X. (2014) On Quasi-Periodic Solutions for a Generalized Boussinesq Equation. Nonlinear Analysis: Theory, Methods & Applications, 105, 50-61. [Google Scholar] [CrossRef] 
[3] Shi, Y., Xu, J. and Xu, X. (2015) On Quasi-Periodic Solutions for Generalized Boussinesq Equation with Quadratic Nonlinearity. Journal of Mathematical Physics, 56, Article ID: 022703. [Google Scholar] [CrossRef] 
[4] Shi, Y., Xu, J. and Xu, X. (2017) Quasi-periodic Solutions of Generalized Boussinesq Equation with Quasi-Periodic Forcing. Discrete & Continuous Dynamical Systems—B, 22, 2501-2519. [Google Scholar] [CrossRef] 
[5] Shi, Y., Xu, J., Xu, X. and Jiang, S. (2014) On the Quasi-Periodic Solutions for Generalized Boussinesq Equation with Higher Order Nonlinearity. Applicable Analysis, 94, 1977-1996. [Google Scholar] [CrossRef] 
[6] Wang, S., Lou, Z. and Si, J. (2020) Long Time Stability of KAM Tori for the Generalized Boussinesq Equation. Nonlinear Analysis, 200, Article ID:112084. [Google Scholar] [CrossRef] 
[7] Pöschel, J. (2002) On the Construction of Almost Periodic Solutions for a Nonlinear Schrödinger Equation. Ergodic Theory and Dynamical Systems, 22, 1537-1549. [Google Scholar] [CrossRef] 
[8] Bourgain, J. (2005) On Invariant Tori of Full Dimension for 1D Periodic NLS. Journal of Functional Analysis, 229, 62-94. [Google Scholar] [CrossRef] 
[9] Niu, H. and Geng, J. (2007) Almost Periodic Solutions for a Class of Higher-Dimensional Beam Equations. Nonlinearity, 20, 2499-2517. [Google Scholar] [CrossRef] 
[10] Geng, J. (2009) Almost Periodic Solutions for a Class of Higher Dimensional Schrödinger Equations. Frontiers of Mathematics in China, 4, 463-482. [Google Scholar] [CrossRef] 
[11] Geng, J. (2012) Invariant Tori of Full Dimension for a Nonlinear Schrödinger Equation. Journal of Differential Equations, 252, 1-34. [Google Scholar] [CrossRef] 
[12] Geng, J. and Xu, X. (2013) Almost Periodic Solutions of One Dimensional Schrödinger Equation with the External Parameters. Journal of Dynamics and Differential Equations, 25, 435-450. [Google Scholar] [CrossRef] 
[13] Cong, H., Mi, L., Shi, Y. and Wu, Y. (2019) On the Existence of Full Dimensional KAM Torus for Nonlinear Schrödinger Equation. Discrete & Continuous Dynamical Systems—A, 39, 6599-6630. [Google Scholar] [CrossRef] 
[14] Cong, H. and Yuan, X. (2021) The Existence of Full Dimensional Invariant Tori for 1-Dimensional Nonlinear Wave Equation. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 38, 759-786. [Google Scholar] [CrossRef] 
[15] Massetti, J.E., Procesi, M. and Biasco, L. (2021) Almost Periodic Invariant Tori for the NLS on the Circle. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 38, 711-758. [Google Scholar] [CrossRef] 
[16] Biasco, L., Massetti, J.E. and Procesi, M. (2023) Small Amplitude Weak Almost Periodic Solutions for the 1D NLS. Duke Mathematical Journal, 172, 2643-2714. [Google Scholar] [CrossRef] 
[17] Pöschel, J. (1996) A KAM-Theorem for Some Nonlinear Partial Differential Equations. Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze, 23, 119-148.
http://www.numdam.org/item?id=ASNSP_1996_4_23_1_119_0

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