QUASI-PERIODIC SOLUTIONS FOR 1D NONLINEAR WAVE EQUATION

Meina Gao

doi:10.11948/20220334

2023 Volume 13 Issue 3

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Meina Gao. QUASI-PERIODIC SOLUTIONS FOR 1D NONLINEAR WAVE EQUATION[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1505-1534. doi: 10.11948/20220334

Citation:

Meina Gao. QUASI-PERIODIC SOLUTIONS FOR 1D NONLINEAR WAVE EQUATION[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1505-1534. doi: 10.11948/20220334

QUASI-PERIODIC SOLUTIONS FOR 1D NONLINEAR WAVE EQUATION

Meina Gao^1, ,

1.
School of Mathematics Physics and Statistics, Shanghai Polytechnic University, Shanghai, 201209, China

Corresponding author: Email: mngao@sspu.edu.cn (M. Gao)
Fund Project: The author was supported by National Natural Science Foundation of China (11971299)

Abstract

Abstract

In this paper, one-dimensional (1D) nonlinear wave equation
$ u_{tt}-u_{xx}+mu+u^{7}=0 $
on the finite $ x $-interval $ [0, \pi] $ with Dirichlet boundary conditions is considered. It is proved that there are many $ 3 $-dimensional elliptic invariant tori, and thus quasi-periodic solutions for the above equation. This is an extension of the previous work [11] by the same author, where many $ 2 $-dimensional elliptic invariant tori for the above equation are obtained. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.
- Invariant tori /
- quasi-periodic solutions /
- KAM theory /
- nonlinear wave equation /
- Birkhoff normal form
MSC: 37K55, 35B10, 35B20

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