Communications in Mathematical Sciences

Volume 21 (2023)

Number 2

KAM persistence for multiscale generalized Hamiltonian systems

Pages: 559 – 579

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n2.a12

Authors

Weichao Qian (College of Mathematics, Jilin University, Changchun, China)

Xue Yang (College of Mathematics, Jilin University, Changchun, China; School of Mathematics and Statistics & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, China)

Yong Li (College of Mathematics, Jilin University, Changchun, China; School of Mathematics and Statistics & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, China)

Abstract

This paper concerns the persistence of invariant tori for multiscale generalized Hamiltonian systems. A multiscale nondegenerate condition on Poisson manifold comparing Kolmogorov nondegenerate one on symplectic manifold and multiscale iso-energetically nondegenerate condition on Poisson manifold comparing iso-energetically nondegenerate one due to Arnold are introduced, hence some multiscale KAM theorems and multiscale iso-energetic KAM theorems on Poisson manifold are established. And we give three applications by a direct example, first order PDEs and steady Euler fluid path flow, respectively.

Keywords

multiscale generalized Hamiltonian systems, multiscale nondegenerate condition, multiscale iso-energetically nondegenerate condition, KAM persistence

2010 Mathematics Subject Classification

37J40, 70H08

Received 12 September 2021

Received revised 26 June 2022

Accepted 26 June 2022

Published 1 February 2023