[1] Shi Yanling, Lu Xuezhu,. KAM tori for generalized Boussinesq equation [J]. Journal of Southeast University (English Edition), 2015, 31 (1): 157-162. [doi:10.3969/j.issn.1003-7985.2015.01.026]
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KAM tori for generalized Boussinesq equation(
)
广义Boussinesq方程的KAM环面
)Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]
- Volumn:
- 31
- Issue:
- 2015 1
- Page:
- 157-162
- Research Field:
- Mathematics, Physics, Mechanics
- Publishing date:
- 2015-03-30
Info
- Title:
- KAM tori for generalized Boussinesq equation
- 广义Boussinesq方程的KAM环面
- Author(s):
- Shi Yanling1, 2, Lu Xuezhu1
-
1Department of Mathematics, Southeast University, Nanjing 210096, China
2Department of Basic Science, Yancheng Institute of Technology, Yancheng 224003, China
- Keywords:
- generalized Boussinesq equation; quasi-periodic solution; Hamiltonian system; invariant tori
- PACS:
- O175.2
- DOI:
- 10.3969/j.issn.1003-7985.2015.01.026
- Abstract:
- One-dimensional generalized Boussinesq equation utt-uxx+(f(u)+uxx)xx=0 with periodic boundary condition is considered, where f(u)=u3. First, the above equation is written as a Hamiltonian system, and then by choosing the eigenfunctions of the linear operator as bases, the Hamiltonian system in the coordinates is expressed. Because of the intricate resonance between the tangential frequencies and normal frequencies, some quasi-periodic solutions with special structures are considered. Secondly, the regularity of the Hamiltonian vector field is verified and then the fourth-order terms are normalized. By the Birkhoff normal form, the non-degeneracy and non-resonance conditions are obtained. Applying the infinite dimensional Kolmogorov-Arnold-Moser(KAM)theorem, the existence of finite dimensional invariant tori for the equivalent Hamiltonian system is proved. Hence many small-amplitude quasi-periodic solutions for the above equation are obtained.
- 考虑周期边界下具有非线性项f(u)=u3的一维广义Boussinesq方程utt-uxx+(f(u)+uxx)xx=0.首先, 将上述方程转化为一个哈密顿系统, 并将该系统在线性算子的特征基上展开得到坐标形式下的哈密顿系统. 鉴于切频与法频之间复杂的共振关系, 考虑一类具有特殊结构的拟周期解.其次, 验证了哈密顿向量场的正则性, 并对四次项进行规范化, 从规范形中可以得到无穷维KAM定理所要求的非退化和非共振条件.利用一个KAM定理证明与方程等价的无穷维哈密顿系统存在许多有限维不变环面, 故原方程有许多小振幅的拟周期解.
References:
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[5] Yusufoglu E. Blow-up solutions of the generalized Boussinesq equation obtained by variational iteration method[J]. Nonlinear Dynam, 2008, 52(4): 395-402.
[6] Chierchia L, You J. KAM tori for 1D nonlinear wave equations with periodic boundary conditions [J]. Comm Math Phys, 2000, 211(2): 497-525.
[7] Geng J, You J. A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions[J]. J Differential Equations, 2005, 209(1): 1-56.
[8] Eliasson L, Kuksin S. KAM for the nonlinear Schrödinger equation[J]. Annals of Mathematics, 2010, 172(1): 371-435.
[9] Kuksin S, Poschel J. Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation[J]. Annals of Mathematics, 1996, 143(1): 149-179.
Memo
- Memo:
-
Biography: Shi Yanling(1982—), female, graduate, shiyanling96998@163.com.
Foundation items: The National Natural Science Foundation of China(No.11301072), the Natural Science Foundation of Jiangsu Province(No.BK20131285), the Research and Innovation Project for College Graduates of Jiangsu Province(No.CXZZ12-0083, CXLX13-074).
Citation: Shi Yanling, Lu Xuezhu. KAM tori for generalized Boussinesq equation[J].Journal of Southeast University(English Edition), 2015, 31(1):157-162.[doi:10.3969/j.issn.1003-7985.2015.01.026]