A degenerate KAM theorem for partial differential equations with periodic boundary conditions

Meina Gao; Jianjun Liu

doi:10.3934/dcds.2020252

Article Contents

2020, Volume 40, Issue 10: 5911-5928. Doi: 10.3934/dcds.2020252

This issue Previous Article Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz–Krieger algebras Next Article The two-component

$ \mu $

-Camassa–Holm system with peaked solutions

A degenerate KAM theorem for partial differential equations with periodic boundary conditions

Meina Gao^1, and
Jianjun Liu^2, ,

1.
College of Arts and Sciences, Shanghai Polytechnic University, Shanghai 201209, China
2.
School of Mathematics, Sichuan University, Chengdu 610065, China

^* Corresponding author: Jianjun Liu
^* Corresponding author: Jianjun Liu

Received: December 2019

Revised: April 2020

Published: June 2020

Meina Gao is supported by NNSFC 11971299. Jianjun Liu is supported by NNSFC 11671280, NNSFC 11822108, Fok Ying Tong Education Foundation 161002

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Abstract

In this paper, an infinite dimensional KAM theorem with double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with simple normal frequencies in [3] by Bambusi, Berti and Magistrelli. As applications, for nonlinear wave equation and nonlinear Schr$ \ddot{\mbox{o}} $dinger equation with periodic boundary conditions, quasi-periodic solutions of small amplitude and quasi-periodic solutions around plane wave are obtained respectively.

Keywords:

Degenerate KAM theory,
multiple normal frequencise.

Mathematics Subject Classification: Primary: 37K55; Secondary: 35Q55.

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References

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