Quasi-periodic solutions for 1D resonant beam equation

Zhenguo Liang; Jiansheng Geng

doi:10.3934/cpaa.2006.5.839

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2006, Volume 5, Issue 4: 839-853. Doi: 10.3934/cpaa.2006.5.839

This issue Previous Article Convergence to stationary solutions for a parabolic-hyperbolic phase-field system Next Article A Liouville type Theorem for an integral system

Quasi-periodic solutions for 1D resonant beam equation

Zhenguo Liang^1, and
Jiansheng Geng^2,

1.
School of Mathematical Sciences, Institute of Mathematics, Fudan University, Shanghai 200433
2.
Department of Mathematics, Nanjing University, Nanjing 210093

Received: January 2006

Revised: June 2006

Published: December 2006

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, one--dimensional ($1D$) resonant beam equation

$u_{t t} +u_{x x x x} +u^3=0,$

with hinged boundary conditions is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on infinite dimensional KAM theorem, partial normal form and scaling skills.

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Mathematics Subject Classification: 37K55.

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