[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()

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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

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 u_tt-u_xx+(f(u)+u_xx)_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.

References:

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[2] Bona J, Sachs R. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation[J]. Comm Math Phys, 1988, 118(1): 15-29.
[3] Liu Y. Instability of solitary waves for generalized Boussinesq equations[J]. J Dynam Differential Equations, 1993, 5(3): 537-558.
[4] Liu Y. Instability and blow-up of solutions to a generalized Boussinesq equation[J]. SIAM J Math Anal, 1995, 26(6): 1527-1546.
[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]

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