Infinite Dimensional KAM Theorem Research Articles

A KAM Theorem for Higher Dimensional Wave Equations Under Nonlocal Perturbation

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional wave equation under nonlocal perturbation $$\begin{aligned} u_{tt}-\triangle u +M_\xi u +\left( \int _{\mathbb {T}^d} u^2 dx\right) u=0,\quad t\in \mathbb {R},\ x\in \mathbb {T}^d\ \end{aligned}$$where $$M_\xi $$ is a real Fourier multiplier.

Kam Tori for Defocusing Kdv-Mkdv Equation

In this paper, we consider small perturbations of the KdV-mKdV equation $${u_t} = - {u_{xxx}} + 6u{u_x} + 6{u^2}{u_x}$$ on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family of small amplitude quasi-periodic solutions under such perturbations. The proof is based on an abstract infinite dimensional KAM theorem.

Quasi-Periodic Solutions for Non-Autonomous mKdV Equation

In this paper, we consider the non-autonomous modified Korteweg-de Vries (mKdV) equation $${u_t} = {u_{xxx}} - 6f\left( {\omega t} \right){u^2}{u_x},x \in \mathbb{R}/2\pi \mathbb{Z}$$ , where f(ωt) is real analytic and quasi-periodic in t with frequency vector ω = (ω1,ω2, · · ·; ω m ). Basing on an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, we obtain the existence of Cantor families of smooth quasi-periodic solutions.

Quasi-periodic solutions for a Schr\xf6dinger equation with a quintic nonlinear term depending on the time and space variables

This article is devoted to the study of a nonlinear Schrödinger equation with an x-periodic and t-quasi-periodic quintic nonlinear term. It is proved that the equation admits small-amplitude, linearly stable, real analytic, and quasi-periodic solutions for most values of 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 sixth order Birkhoff normal form. We show an infinite-dimensional KAM theorem for non-autonomous Schrödinger equations and apply the theorem to prove the existence of quasi-periodic solutions.

Invariant tori for the Schrödinger equation in the Heisenberg Ferromagnetic chain

In this paper, we consider the nonlinear Heisenberg Ferromagnetic chain equation under Dirichlet boundary conditions. By Taylor formula, the nonlinear Heisenberg Ferromagnetic chain equation can be described by the nonlinear Schrödinger type equation. Using an infinite dimensional KAM theorem for reversible system, we prove the existence of many n-dimensional invariant tori under sufficiently small perturbation and thus many time quasi-periodic solutions for the above equation.

Quasi-Periodic Solutions for Fractional Nonlinear Schrödinger Equation

We establish an infinite dimensional KAM theorem with dense normal frequency. As an application, we use this theorem to study the Fractional NLS $$\begin{aligned} iu_t-|\partial _x|^{1\over 2}u+M_\xi u=\epsilon f(|u|^2)u,\quad x\in {\mathbb T},t\in {\mathbb R}, \end{aligned}$$ where f is real analytic in a neighborhood of $$0\in {\mathbb C}$$ and $$\epsilon >0$$ is small enough. We obtain a family of small-amplitude quasi-periodic solutions with linear stability.

A KAM Theorem for Higher Dimensional Forced Nonlinear Schrödinger Equations

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional nonlinear Schrodinger equations with outer force $$\begin{aligned} iu_t-\triangle u +M_\xi u+f(\bar{\omega }t)|u|^2u=0, \quad t\in {{\mathbb R}}, x\in {{\mathbb T}}^d \end{aligned}$$ where \(M_\xi \) is a real Fourier multiplier,\(f({\bar{\theta }})({\bar{\theta }}={\bar{\omega }} t)\) is real analytic and the forced frequencies \(\bar{\omega }\) are fixed Diophantine vectors.

Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay

In this paper we prove an infinite dimensional KAM theorem, in which the assumptions on the derivatives of the perturbation in [] are weakened from polynomial decay to logarithmic decay. As a consequence, we can apply it to 1D quantum harmonic oscillators and prove the reducibility of the linear harmonic oscillator, , on perturbed by the quasi-periodic in the time potential with logarithmic decay. This proves the pure-point nature of the spectrum of the Floquet operator K, whereK:=−i∑k=1nωk∂∂θk−d2dx2+x2+εV(x,θ;ω)is defined on , and the potential has logarithmic decay as well as its gradient in ω.

Quasi-periodic solutions of beam equations with the nonlinear terms depending on the time and space variables

This article is devoted to the study of a beam equation with a constant potential and an $x$-periodic and $t$-quasiperiodic nonlinear term. It is proved that the equation admits small amplitude, linearly stable and $t$-quasiperiodic solutions under periodic boundary conditions for most values of the frequency vector and for most potentials. By utilizing the measure estimation of essentially finitely many small divisors, we construct a real analytic, symplectic change of coordinates which can transform the Hamiltonian into some Birkhoff normal form. Based on an infinite dimensional KAM theorem, we prove the existence of quasi-periodic solutions.

Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

In this paper, we consider the perturbed KdV equation with Fourier multiplier $$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$ with analytic data of size \(\varepsilon \). We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with \(\tilde{J}\) Diophantine frequencies, where the order of \(\tilde{J}\) is \(O(\frac{1}{\varepsilon })\). The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Toplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Toplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.

Invariant Tori of Full Dimension for Second KdV Equations with the External Parameters

In this paper, we consider the second KdV equation with the external parameters $$\begin{aligned} u_{t} =\partial _x^5 u +(M_{\sigma }u+u^3)_{x}, \end{aligned}$$ under zero mean-value periodic boundary conditions $$\begin{aligned} u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$ where \(M_\sigma \) is a real Fourier multiplier. It is proved that the equations admit a Whitney smooth family of small amplitude, real analytic almost periodic solutions with all frequencies. The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Toplitz–Lipschitz property of the perturbation and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Toplitz–Lipschitz property of the perturbation, our normal form part is independent of angle variables in spite of the unbounded perturbation. This is the first attempt to prove the almost periodic solutions for the unbounded perturbation case.

Quasi-Periodic Solutions of Nonlinear Wave Equation with x-Dependent Coefficients

This paper is devoted to the study of quasi-periodic solutions of a nonlinear wave equation with x-dependent coefficients. Such a model arises from the forced vibration of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. Based on the partial Birkhoff normal form and an infinite-dimensional KAM theorem, we can obtain the existence of quasi-periodic solutions for this model under the general boundary conditions.

KAM tori for defocusing modified KDV equation

In this paper, one-dimensional defocusing modified KDV equation: u t + u x x x − 6 u 2 u x = 0 with periodic boundary condition is considered. It is proved that the above equation admits a Cantor family of small amplitude, quasi-periodic solutions. The proof is based on an infinite dimensional KAM theorem and partial Birkhoff normal form.

On the quasi-periodic solutions for generalized Boussinesq equation with higher order nonlinearity

In this paper, one-dimensional generalized Boussinesq equation with hinged boundary conditions is considered, where . It is proved that for each prescribed integer , the above equation admits a Whitney-smooth family of small amplitude, quasi-periodic solutions with -dimensional Diophantine frequencies. The proof is based on an infinite-dimensional KAM theorem, partial Birkhoff normal form and scaling skills.

THE CANTOR MANIFOLD THEOREM WITH SYMMETRY AND APPLICATIONS TO PDEs

In this paper we introduce a new Cantor manifold theorem and then apply it to one new type of one-dimensional (1 d ) beam equations $$u_{tt}+u_{xxxx}+mu-2u^2u_{xx}-2uu_x^2=0, m>0,$$ with periodic boundary conditions. We show that the above equation admits small-amplitude linearly stable quasi-periodic solutions corresponding to finite dimensional invaraint tori of an associated infinite dimensional dynamical system. The proof is based on a partial Birkhoff normal form and an infinite dimensional KAM theorem for Hamiltonians with symmetry (cf. [19]).

On the quasi-periodic solutions of generalized Kaup systems

In this paper we analyze the behavior of small amplitude solutions of the variant of the classical Kap system given by \[ \partial_t u = \partial_x v - 2 \partial_x(v^3), \quad \partial_t v = \partial_x u - \frac 1 3 \partial_{xxx} u. \] It is proved that the above equation admits small-amplitude solutions that are quasiperiodic in time and that correspond to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof relies on the Hamiltonian formulation of the problem, the study of its Birkhoff normal form and an infinite dimensional KAM theorem. This is the abstract of your paper and it should not exceed.

Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations

In this paper, we establish an estimate for the solutions of a small-divisor equation with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem whichallows for multiple normal frequencies and unbounded perturbations, we prove that there are manyperiodic solutions for the derivative nonlinear Schrödinger equations subject to small Hamiltonian perturbations.

On quasi-periodic solutions for a generalized Boussinesq equation

In this paper a one-dimensional generalized Boussinesq equation utt−uxx+(u3+uxx)xx=0 with hinged boundary conditions is considered. It is proved that the above equation admits 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 the Birkhoff normal form.

KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions

This paper is concerned with the derivative nonlinear Schrödinger equation with periodic boundary conditionsiut+uxx+i(f(|u|2)u)x=0,x∈T:=R/2πZ, where f is real analytic in some neighborhood of the origin in C, f(0)=0, and f′(0)≠0. 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.

Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation

In this paper, we consider the non-autonomous Benjamin-Ono equation $$u_t+\mathscr{H}u_{xx}- uu_x- (F(\omega t,x,u))_x=0$$ under periodic boundary conditions. Using an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field and partial Birkhoff normal form, we will prove that there exists a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.