Components of Invariant Variety of Periodic Points and Fundamental Domains of Recurrence Equation

Localization and Recurrence of Quantum Walk in Periodic Potential on a Line

We present numerical study of a model of quantum walk in periodic potential on the line. We take the simple view that different potentials affect differently the way the coin state of the walker is changed. For simplicity and definiteness, we assume the walker's coin state is unaffected at sites without potential, and is rotated in an unbiased way according to Hadamard matrix at sites with potential. This is the simplest and most natural model of a quantum walk in a periodic potential with two coins. Six generic cases of such quantum walks were studied numerically. It is found that of the six …

The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, the main physical mechanism for the generation of rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the $x$-periodic Cauchy problem for NLS for a generic periodic initial perturbation of the unstable constant background solution, in the case of $N=1,2$ unstable modes. We use matched asymptotic expansion techniques to show that the solution of this problem describes an exact deterministic alternat…

Recurrence in 2D Inviscid Channel Flow

I will prove a recurrence theorem which says that any $H^s$ ($s>2$) solution to the 2D inviscid channel flow returns repeatedly to an arbitrarily small $H^0$ neighborhood. Periodic boundary condition is imposed along the stream-wise direction. The result is an extension of an early result of the author [Li, 09] on 2D Euler equation under periodic boundary conditions along both directions.

Ablowitz-Ladik 晶格中的调制不稳定性、周期异常波重现与爆破

Modulation instability, periodic anomalous wave recurrence, and blow up in the Ablowitz-Ladik lattices

研究Ablowitz-Ladik晶格中背景解的调制不稳定性，发现AL_-在振幅大于1时对任意波数扰动不稳定，并构造精确周期解，证明AL_-解奇异，用匹配渐近展开描述周期扰动演化，解析与数值完全一致。

Periodic Cluster Mutations and Related Integrable Maps

One of the remarkable properties of cluster algebras is that any cluster, obtained from a sequence of mutations from an initial cluster, can be written as a Laurent polynomial in the initial cluster (known as the "Laurent phenomenon"). There are many nonlinear recurrences which exhibit the Laurent phenomenon and thus unexpectedly generate integer sequences. The mutation of a typical quiver will not generate a recurrence, but rather an erratic sequence of exchange relations. How do we "design" a quiver which gives rise to a given recurrence? A key role is played by the concept of "periodic clus…

多维非线性薛定谔方程中的x周期准一维反常（流氓）波：裂变、聚变与重现

x-periodic Quasi One Dimensional Anomalous (Rogue) Waves in Multidimensional Nonlinear Schrödinger Equations: Fission, Fusion, and Recurrence

研究多维非线性薛定谔方程在准一维区域中x周期反常波的非线性调制不稳定性，揭示首次非线性阶段的普适性及后续重现动力学中裂变与聚变过程的模型依赖性，并利用有限间隙微扰理论提供解析描述。

Davey-Stewartson方程的周期N呼吸子反常波解：首次出现、重现与爆破性质

The periodic N breather anomalous wave solution of the Davey-Stewartson equations; first appearance, recurrence, and blow up properties

该研究构建了Davey-Stewartson方程的Akhmediev型N呼吸子反常波解，分析了其极限情形与爆破约束，并用匹配渐近展开法探讨了双周期柯西问题中的解相关性及双不稳定模式下的无普遍爆破特性。

The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1

The focusing NLS equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, considered the main physical mechanism for the appearance of rogue (anomalous) waves (RWs) in Nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for periodic initial perturbations of the unstable background solution of NLS exciting just one of the unstable modes. We distinguish two cases. In the case in which only the corresponding unstable gap is theoretically open, the solution describes an exact determin…

Double tangent method for two-periodic Aztec diamonds

We use the octahedron recurrence, which generalizes the quadratic recurrence found by Kuo for standard Aztec diamonds, in order to compute boundary one-refined and two-refined partition functions for two-periodic Aztec diamonds. In a first approach, the geometric tangent method allows to derive the parametric form of the arctic curve, separating the solid and liquid phases. This is done by using the recently reformulation of the tangent method and the one-refined partition functions without extension of the domain. In a second part, we use the two-refined tangent method to rederive the arctic …

Traveling fronts for Fisher-KPP lattice equations in almost periodic media

This paper investigates the existence of almost periodic traveling fronts for Fisher-KPP lattice equations in one-dimensional almost periodic media. By the Lyapunov exponent of the linearized operator near the unstable steady state, we give sufficient condition of the existence of minimal speed of traveling fronts. Furthermore, it is showed that almost periodic traveling fronts share the same recurrence property as the structure of the media. As applications, we give some typical examples which have minimal speed, and the proof of this depends on dynamical system approach to almost periodic Sc…

Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues

Let $H_0$ be a discrete periodic Schrödinger operator on $\ell^2(\mathbb{Z}^d)$: $$H_0=-Δ+V,$$ where $Δ$ is the discrete Laplacian and $V: \mathbb{Z}^d\to \mathbb{C}$ is periodic. We prove that for any $d\geq3$, the Fermi variety at every energy level is irreducible (modulo periodicity). For $d=2$, we prove that the Fermi variety at every energy level except for the average of the potential is irreducible (modulo periodicity) and the Fermi variety at the average of the potential has at most two irreducible components (modulo periodicity). This is sharp since for $d=2$ and a constant potential …