Infinite Dimensional KAM Theorem Research Articles

An infinite dimensional KAM theorem with normal degeneracy

In this paper, we consider a classical Hamiltonian normal form with degeneracy in the normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy in the normal form. Instead of that, we introduce a topological degree condition and a weak convexity condition, which are easy to verify, and we prove the persistence of lower dimensional tori without any restriction on perturbation but only smallness and analyticity.

Linearly stable KAM tori for one dimensional forced Kirchhoff equations with refined Töplitz-Lipschitz property

We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for one dimensional forced Kirchhoff equations with Dirichlet boundary conditionsutt−(1+∫0π|ux|2dx)uxx+Mξu+ϵg(ω¯t,x)=0,u(t,0)=u(t,π)=0, where Mξ is a real Fourier multiplier, g(ω¯t,x)=−g(ω¯t,−x) is real analytic with forced Diophantine frequencies ω¯, ϵ is a small parameter. The proof is based on an improved Kuksin lemma and the off-diagonal decay together with the refined Töplitz-Lipschitz property of the forcing term.

Quasi-periodic solutions for a class of wave equation system

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the 1D wave equation system { u 1 tt − u 1 xx + σ u 1 + u 1 u 2 2 = 0 u 2 tt − u 2 xx + μ u 2 + u 1 2 u 2 = 0 under Dirichlet boundary conditions, where 0 < σ ∈ [ σ 1 , σ 2 ] ⊂ [ 0 , 1 ] , 0 < μ ∈ [ μ 1 , μ 2 ] ⊂ [ 0 , 1 ] are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.

Quasi-periodic solutions for a Schrödinger equation under periodic boundary conditions with given potential

This paper is concerned with the cubic nonlinear Schrödinger equationiut−uxx+V(x)u+|u|2u=0,x∈T=R/2πZ subject to periodic boundary conditions, where the potential V(x) is real analytic in |Imx|<r and i=−1. It is proved that there exist many quasi-periodic solutions of the nonlinear Schrödinger equation with given potential V(x) by a infinite dimensional KAM theorem dealing with multiple eigenvalues.

Reducible KAM tori for two-dimensional nonlinear Schrödinger equations with explicit dependence on the spatial variable

We study the two-dimensional nonlinear Schrödinger equationiut−△u+|u|2u+∂f(x,u,u¯)∂u¯=0,t∈R,x∈T2 with periodic boundary conditions. The nonlinearity f(x,u,u¯)=∑j,l,j+l≥6ajl(x)uju¯l, ajl=alj is a real analytic function in a neighborhood of the origin. We obtain, through an infinite dimensional KAM theorem, a Whitney smooth family of small–amplitude quasi–periodic solutions.

Linearly stable KAM tori for higher dimensional Kirchhoff equations

We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for higher dimensional Kirchhoff equations with periodic boundary conditions. The proof is based on an improved Kuksin lemma and the special form of the nonlinear term.

Reducibility and quasi-periodic solutions for a two dimensional beam equation with quasi-periodic in time potential

This article is devoted to the study of a two-dimensional $(2D)$ quasi-periodically forced beam equation $ u_{tt}+\Delta^2 u+ \varepsilon\phi(t)(u+{u}^3) = 0, \quad x\in\mathbb{T}^2, \quad t\in\mathbb{R} $ under periodic boundary conditions, where $\varepsilon$ is a small positive parameter, $\phi(t)$ is a real analytic quasi-periodic function in $t$ with frequency vector $\omega = (\omega_1, \omega_2 \ldots, \omega_m)$. We prove that the equation possesses a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form. By solving the measure estimation of infinitely many small divisors, we construct a symplectic coordinate transformation which can reduce the linear part of Hamiltonian system to constant coefficients. And we construct some conversion of coordinates which can change the Hamiltonian of the equation into some Birkhoff normal form depending sparse angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. Lastly, we prove that there are many quasi-periodic solutions for the above equation via an abstract KAM theorem.

Quasi-Periodic Solution of Nonlinear Beam Equation on $${\mathbb T}^d$$ with Forced Frequencies

In this paper, we study the higher dimensional nonlinear beam equation under periodic boundary condition: utt+Δx2u+Mξu+f(ω¯t,u)=0,u=u(t,x),t∈R,x∈Td,d≥2where Mξ is a real Fourier multiplier, f=f(θ¯,u) is a real analytic function with respect to (θ¯,u), and f(θ¯,u)=O(|u|2). This equation can be viewed as an infinite dimensional nearly integrable Hamiltonian system. We establish an infinite dimensional KAM theorem, and apply it to this equation to prove that there exist a class of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori.

Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency

In this paper, one dimensional nonlinear wave equation \begin{document}$ u_{tt}-u_{xx} +mu +\varepsilon f(\omega t,x,u;\xi) = 0 $\end{document} with Dirichlet boundary condition is considered, where \begin{document}$ \varepsilon $\end{document} is small positive parameter, \begin{document}$ \omega = \xi \bar{\omega}, $\end{document} \begin{document}$ \bar{\omega} $\end{document} is weak Liouvillean frequency. It is proved that there are many quasi-periodic solutions with Liouvillean frequency for the above equation. The proof is based on an infinite dimensional KAM Theorem.

Bounded Non-response Solutions with Liouvillean Forced Frequencies for Nonlinear Wave Equations

In this paper, one dimensional nonlinear wave equations $$\begin{aligned} u_{tt}-u_{xx}+M_{\xi _{2}}u +\varepsilon f(\omega _{1} t, x, u)=0 \end{aligned}$$ with Dirichlet boundery conditions are considered, where $$M_{\xi _{2}}$$ is a real Fourier multiplier, f is a real analytic function with $$f(\omega _{1} t, -x, -u)=-f(\omega _{1} t, x, u)$$ and the forced frequencies $$\omega _1=(1,\alpha )$$ are Liouvillean. We obtain a family of $$C^{\infty }$$ smooth, bounded non-response solutions with Liouvillean forced frequencies. This is based on an infinite dimensional KAM theorem with angle-dependent normal form.

Quasi-periodic solutions of fractional nonlinear Schrödinger equation systems

In this paper, the fractional nonlinear Schrödinger equation system with periodic boundary conditions is considered. I establish an abstract infinite-dimensional KAM theorem and apply it to fractional nonlinear Schrödinger equation system with real Fourier multiplier. By establishing a block diagonal normal form, we prove the existence of a class of Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite-dimensional invariant tori of an associated infinite-dimensional dynamic system.

The relation between the size of perturbations and the dimension of tori in an infinite-dimensional KAM theorem of Pöschel

In this paper, we discuss the relation between the size of perturbations and the dimension of tori based on the KAM theorem of Pöschel (1996) which can be applied to infinite dimensional PDEs, in other words, we give a quantitative version of the KAM theorem of Pöschel (1996). Moreover, motivated by Chung and Yuan (2007, 2008) and Li and Yuan (2012), after infinite KAM iterations, we make action and normal variables of KAM tori of the reduced Hamiltonian system in a domain instead of shrinking them to the origin.

Quasi-periodic Solutions for Two Dimensional Modified Boussinesq Equation

In this paper, two dimensional modified Boussinesq equation $$\begin{aligned} u_{tt} +\Delta ^2u+ \Delta ( u^3 ) =0, \quad x\in {\mathbb {T}}^2,~t\in {\mathbb {R}} \end{aligned}$$ under periodic boundary conditions is considered. It is proved that the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form.

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

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 \begin{document}$ \ddot{\mbox{o}} $\end{document} dinger equation with periodic boundary conditions, quasi-periodic solutions of small amplitude and quasi-periodic solutions around plane wave are obtained respectively.

PERIODIC AND QUASI-PERIODIC SOLUTIONS FOR THE COMPLEX SWIFT-HOHENBERG EQUATION

In this paper, we consider the complex Swift-Hohenberg(CSH) equation $ \frac{\partial u}{\partial t} = \lambda u-(\alpha+\mathrm{i}\beta)\left(1+\frac{\partial^2}{\partial x^2}\right)^2u-(\sigma+\mathrm{i}\rho)|u|^2u $ subject to periodic boundary conditions. Using an infinite dimensional KAM theorem, we prove that there exist a continuous branch of periodic solutions and a Cantorian branch of quasi-periodic solutions for the above equation.

KAM Theorem for a Hamiltonian System with Sublinear Growth Frequencies at Infinity

We prove an infinite-dimensional KAM theorem for a Hamiltonian system with sublinear growth frequencies at infinity. As an application, we prove the reducibility of the linear fractional Schr\"odinger equation with quasi-periodic time-dependent forcing.

Quasi-periodic solutions for a completely resonant beam equation with a nonlinear term depending on the time and space variables

This article is devoted to the study of a completely resonant beam equation with an x-periodic and t-quasi-periodic nonlinear term. It is proved that the equation admits small amplitude, linearly stable and quasi-periodic solutions for most values of the frequency vector. By utilizing the measure estimation of infinitely many small divisors, we construct a real analytic, symplectic change of coordinates which can transform the Hamiltonian into some Birkhoff normal form. We show an infinite dimensional KAM theorem for non-autonomous beam equations, which is applied to prove the existence of quasi-periodic solutions.

Quasi-periodic solutions for a class of beam equation system

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional beam equation system \begin{document}$ \left\{ \begin{array}{lll} u_{1tt}+ \Delta^2 u_1 +\sigma u_1 +u_1u_2^2 &amp; = &amp; 0 \\ &amp;&amp;\\ u_{2tt}+ \Delta^2 u_2 +\mu u_2 +u_1^2 u_2 &amp; = &amp; 0 \end{array} \right. $\end{document} under periodic boundary conditions, where $ 0&lt;\sigma \in [ \sigma_1,\sigma_2 ], $ $ 0&lt;\mu\in [ \mu_1,\mu_2 ] $ are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.

Invariant tori for a nonlinearly modified Kawahara equation with periodic boundary conditions

This paper is concerned with a nonlinearly modified Kawahara equation with periodic boundary conditions \t\t\tut+uxxx−u5x+αu2ux=0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{t}+u_{xxx}-u_{5x}+\\alpha u^{2}u_{x}=0. $$\\end{document} Based on an abstract infinite-dimensional KAM theorem for unbounded perturbation vector fields and partial Birkhoff normal form, we prove the existence of n-dimensional invariant tori for the equation.

Invariant Cantor manifolds of quasi-periodic solutions for the derivative nonlinear Schrödinger equation

This paper is concerned with the derivative nonlinear Schrödinger equation with periodic boundary conditionsiut+uxx+i(f(x,u,u¯))x=0,x∈T:=R/2πZ, where f is an analytic function of the formf(x,u,u¯)=μ|u|2u+f≥4(x,u,u¯),0≠μ∈R, and f≥4(x,u,u¯) denotes terms of order at least four in u,u¯. We show the above equation possesses Cantor families of smooth quasi-periodic solutions of small amplitude. The proof is based on an infinite dimensional KAM theorem for unbounded perturbation vector fields.