arXiv subjects
Publications and source records attributed to Xiaoping Yuan.
In this paper, we establish Anderson localization for the quantum kicked rotor model. More precisely, we proved that \begin{equation*} H=\tanπ\left(x_0+my_0+\frac{m(m-1)}{2}ω\right) δ_{mn}+εS_ϕ\end{equation*} has pure point spectrum with exponentially decaying eigenfunctions for almost all $ω\in DC$ (diophantine condition).
In this paper, we establish Anderson localization for the Maryland model with long range interactions.
In this paper, we investigate the existence of KAM tori for an infinite dimensional Hamiltonian system with finite number of zero normal frequencies. By constructing a constant quantity we show that, for "most" frequencies in the sense of Lebesgue measure, either if the quantity is zero, there is a KAM tori or if the quantity is not zero, there is no KAM tori in some domain. As application, we show that the nonlinear Schrödinger equation with a zero frequency possesses many quasi-periodic solutions.
In this paper, we establish Anderson localization for a class of Jacobi matrices associated with skew shifts on $\mathbb{T}^{d}$, $d\geq3$.
In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed by Bourgain \cite{BJFA2005}.
We employ the Craig-Wayne-Bourgain method to construct quasi-periodic solutions for delayed perturbation differential equations. Our results not only implement the existing literatures on constructing quasi-periodic solutions for DDE by the KAM method and space splitting technique, but also provide an example of application of multi-scale analysis method to non-selfadjoint problem.
In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighborhood of a given $ I^{0}$ for nearly integrable and analytic Hamiltonian system $ H(I,θ) = H_{0}(I)+ εH_{1}(I,θ)$ of infinite degree of freedom and of short range. That is to say, we will give an extension of the original Kolmogorov theorem to the infinite-dimensional case of short range. The proof is based on the approximation of finite-dimensional Kolmogorov theorem and an improved KAM machinery which works for the normal form depending on initial $ I^{0}$.
By constructing an infinite dimensional KAM theorem of the normal frequencies being dense at finite-point, we show that some shallow water equations such as Benjamin-Bona-Mahony equation and the generalized $d$-Dim. Pochhammer-Chree equation subject to some boundary conditions possess many (a family of initial values of positive Lebesgue measure of finite dimension) smooth solutions which are quasi-periodic in time.
In this paper, we study the quasi-periodic operators $H_{ε,ω}(x)$: $$(H_{ε,ω}(x)\vecψ)_n=ε\sum_{k\in\mathbb{Z}}W_k\vecψ_{n-k}+V(x+nω)\vecψ_n,$$ where $$\vecψ=\{\vecψ_n\}\in\ell^2(\mathbb{Z},\mathbb{C}^l),\ V(x)=\text{diag}\left(v_1(x),\cdots,v_l(x)\right)$$ with $v_i$ ($1\leq i \leq l$) being real analytic functions on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and $W_k$ ($k\in\mathbb{Z}$) being $l\times l$ matrices satisfying $\|W_k\|\leq C_0e^{-ρ|k|}$. Using techniques developed by Bourgain and Goldstein [\textit{Ann. of Math. 152(3):835--879, 2000}], we show that for $|ε|\leq ε_{0}(V,ρ,l,C_0)$ ( depending only on $V,ρ, l, C_0$) and $x\in \mathbb{R}/\mathbb{Z}$, there is some full Lebesgue measure subset $\mathcal{F}$ of the Diophantine frequencies such that $H_{ε,ω}(x)$ exhibits Anderson localization if $ω\in \mathcal{F}$.
This work discusses the boundedness of solutions for impulsive Duffing equation with time-dependent polynomial potentials. By KAM theorem, we prove that all solutions of the Duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many (positive Lebesgue measure) quasi-periodic solutions clustering at infinity. This result extends some well-known results on Duffing equations to impulsive Duffing equations.
In this paper, we study the non-self dual extended Harper's model with Liouvillean frequency. By establishing quantitative reducibility results together with the averaging method, we prove that the lengths of spectral gaps decay exponentially.
In this paper, we prove the existence of quasi-periodic small-amplitude solutions for quasi-linear Hamiltonian perturbation of the fifth-order KdV equation on the torus in presence of a quasi-periodic forcing.
It is shown that all solutions are bounded for Duffing equation $\ddot{x}+ x^{2n+1}+\sum_{j=0}^{2n}P_{j}(t)x^{j}=0,$ provided that for each $n+1\le j\le 2n$, $P_j(t)\in C^γ(\mathbb T)$ with $γ>1-\frac1n$ and for each $0\le j\le n$, $P_j\in L(\mathbb T^1)$.
In this paper, it is proved that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., \textbf{229} (2005), no. 1, 62-94.] is stable in a very long time for 1D nonlinear Schrödinger equation with periodic boundary conditions.
{\bf Abstract}: The existence of 2-dimensional KAM tori is proved for the perturbed generalized nonlinear vibrating string equation with singularities $u_{tt}=((1-x^2)u_x)_x-mu-u^3$ subject to certain boundary conditions by means of infinite-dimensional KAM theory with the help of partial Birkhoff normal form, the characterization of the singular function space and the estimate of the integrals related to Legendre basis.
To study the variation problem related to the incompressible fluid mechanics, Brenier brings the concept of generalized flow and shows that the generalized incompressible flow (GIF) is deeply related to the classical solution of the incompressible Euler equations. In this paper, we will study the ergodic theory of the GIF which may help us understand the dynamic property of the classical solution of the incompressible Euler equations. First, we show that the GIF has the weak recurrent property rather than the classical one. Then, we define the ergodicity of the GIF and discuss its relation with the classical ergodic flow. Next, we prove some ergodic theorems of the GIF. Finally, we give a theorem about the structure of the set of all GIFs.
In this paper, we explore a topological system $f:M\rightarrow M$ with average shadowing property. We extend Sigmund's results and show that every non-empty, compact and connected subset $V\subseteq\mathcal {M}_{inv}(f)$ coincides with $V_f(y)$, where $\mathcal {M}_{inv}(f)$ denotes the space of invariant Borel probability measures on M, and $V_f(y)$ denotes the accumulation set of time average of Dirac measures supported at the orbit of $y$. We also show that the set $M_{V}=\{y\in M\,\,|\,\,V_{f}(y)=V\}$ is dense in $Δ_{V}=\bigcup_{ν\in V}supp(ν)$. In particular, if $Δ_{max}=\bigcup_{ν\in\mathcal {M}_{inv}(f)}supp(ν)$ is isolated or coincides with $M$, then $M_{max}=\{y: V_{f}(y)=\mathcal {M}_{inv}(f)\}$ is residual in $Δ_{max}$.
Avila and Jitomirskaya prove that the quasi-periodic Schrödinger operator $H_{λv,α,θ}$ has purely absolutely continuous spectrum for $α$ in sub-exponential regime (i.e., $β(α)=0$) with small $λ$, if $v$ is real analytic in a strip of real axis. In the present paper, we show that for all $α$ with $0<β(α)<\infty$, $H_{λv,α,θ}$ has purely absolutely continuous spectrum with small $λ$, if $v$ is real analytic in strip $ |\Im x|< Cβ$, where $C$ is a large absolute constant.