An infinite-dimensional Kolmogorov theorem and the construction of almost periodic breathers

2025/08/19 by Zhicheng Tong, Yong Li, Tong, Zhicheng +1
Engineering · Mathematics · Physics and Astronomy · #37K55 #37K60 #37L60 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2508.13501

openalex publication_date 2025/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we present two infinite-dimensional Kolmogorov theorems based on non-resonant frequencies of Bourgain's Diophantine type or even weaker conditions. To be more precise, under a Legendre-type nondegeneracy condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with a universally prescribed frequency independent of any spectral asymptotics. As an application, we prove that for a class of perturbed networks with weakly coupled oscillators described by \frac\rm d²x_n\rm dt² + V'( x_n ) = ε_n W'( x_{n + 1} - x_n ) - ε_n-1W'( x_n - x_{n - 1} ) , n ∈ ℤ, or even for more general perturbed networks, frequency-preserving almost periodic breathers do persist, provided that the local potential V and the coupling potential W satisfy certain assumptions. In particular, this yields the first frequency-preserving result for the Aubry--MacKay conjecture [MA94,Aub95].

Citations

• Maximal tori in infinite-dimensional Hamiltonian systems: a Renormalization Group approach
• Quantitative Destruction and Persistence of Lagrangian Torus in Hamiltonian Systems
• An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS
• Linear Schrödinger equation with an almost periodic potential
• Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices
• Destruction of Lagrangian torus for positive definite Hamiltonian systems
• On invariant tori of full dimension for 1D periodic NLS
• Invariant Cantor Manifolds of Quasi-Periodic Oscillations for a Nonlinear Schrodinger Equation

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• KAM for Hamiltonian partial differential equations with weaker Spectral Asymptotics