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Sharp phase transitions for the almost Mathieu operator
It is known that the spectral type of the almost Mathieu operator depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We studyExpand
A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces
In this paper, we give a KAM theorem for a class of infinite dimensional nearly integrable Hamiltonian systems. The theorem can be applied to some Hamiltonian partial differential equations in higherExpand
Perturbations of Lower Dimensional Tori for Hamiltonian Systems
  • J. You
  • Mathematics
  • 10 February 1999
Abstract This paper provides a generalized Kolmogorov–Arnold–Moser theorem for lower dimensional tori in Hamiltonian systems, which applies to multiple normal frequency case. The proof is based onExpand
KAM Tori for 1D Nonlinear Wave Equations¶with Periodic Boundary Conditions
Abstract:In this paper, one-dimensional (1D) nonlinear wave equations with periodic boundary conditions are considered; V is a periodic smooth or analytic function and the nonlinearity f is anExpand
Invariant tori and Lagrange stability of pendulum-type equations
Abstract In this paper we prove that the pendulum-type equation x″ + g(t, x) = 0 possesses infinitely many invariant tori whenever g(t, x) has zero mean value on the torus T2, where g(t, x) belongsExpand
Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems
In this paper, we prove that a quasi-periodic linear differential equation in sl(2,ℝ) with two frequencies (α,1) is almost reducible provided that the coefficients are analytic and close to aExpand
An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation
Abstract We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the two dimensional nonlinear Schrodinger equation i u t − Δ u + | u | 2 u = 0 , t ∈ R , x ∈ T 2Expand
Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications
In this paper, we prove that any analytic quasi-periodic cocycle close to constant is the Poincaré map of an analytic quasi-periodic linear system close to constant, which bridges both methods andExpand
KAM tori for higher dimensional beam equations with constant potentials
In this paper, we consider the higher dimensional nonlinear beam equations with periodic boundary conditions, where the nonlinearity f(u) is a real–analytic function near u = 0 with f(0) = f'(0) = 0Expand
Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles
We show that if the base frequency is Diophantine, then the Lyapunov exponent of a $$C^{k}$$Ck quasi-periodic $$SL(2,{\mathbb {R}})$$SL(2,R) cocycle is 1 / 2-Hölder continuous in the almost reducibleExpand
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