M. Schwarz
In this work, we explain in what sense the generic level set of the constants of motion for the periodic nonlinear Schrodinger equation is an infinite dimensional torus on which each generalized nonlinear Schrodinger flow is reduced to straight line almost periodic motion, and describe how neighboring generic infinite dimensional tori are connected.
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In this paper, we prove new Strichartz estimates for linear Schrodinger equations posed on d-dimensional irrational tori. Then, we use these estimates to prove subcritical and critical local well-posedness results for nonlinear Schrodinger equations (NLS) on irrational tori.
We revisit the work of Bourgain on the invariance of the Gibbs measure for the cubic, defocusing nonlinear Schrödinger equation in 2D on a square torus, and we prove the equivalent result on any tori.
In this paper we consider the cubic Schrodinger equation in two space dimensions on irrational tori. Our main result is an improvement of the Strichartz estimates on irrational tori. Using this estimate we obtain a local well-posedness result in H^s for s>131/416. We also obtain improved growth bounds for higher order Sobolev norms.
We study the time of existence of the solutions of the following Schrödinger equation $$iψ_t = (-Δ)^s ψ+f(|ψ|^2)ψ, x \in \mathbb S^d, or x\in\T^d$$ where $(-Δ)^s$ stands for the spectrally defined fractional Laplacian with $s>1/2$ and $f$ a smooth function. We prove an almost global existence result for almost all $s>1/2$.
Energy transfer for solutions to the nonlinear Schrödinger equation on irrational tori
本文研究二维无理环面上散焦三次非线性薛定谔方程解的能量转移，结合解析分析与数值实验，并证明聚焦与散焦情形下准共振部分对有限质量初值全局适定。
Improved global well-posedness for mass-critical nonlinear Schrödinger equations on tori
本文证明了环面上质量临界非线性薛定谔方程的新全局适定性结果，涵盖一维和二维情形，并给出了聚焦与散焦情形下的质量条件。
A characteristic of the defocusing cubic nonlinear Schrödinger equation (NLSE), when defined so that the space variable is the multi-dimensional square (hence rational) torus, is that there exist solutions that start with arbitrarily small norms Sobolev norms and evolve to develop arbitrarily large modes at later times; this phenomenon is recognized as a weak energy transfer to high modes for the NLSE. In this paper, we show that when the system is considered on an irrational torus, energy transfer is more difficult to detect.
We consider a class of linear time dependent Schrödinger equations and quasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon and Schrödinger equations on arbitrary flat tori. For the linear Schrödinger equation, we prove a $t^ε$ $(\forall ε>0)$ upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasi-periodic solutions. Both results are based on "clusterization properties" of the eigenvalues of the Laplacian on a flat torus and on suitable "separation properties" of the singular sites of Sch…
We prove Strichartz estimates over large time scales for the Schrodinger equation set on irrational tori. They are optimal for Lebesgue exponents $p > 6$.
In this paper, we prove the long time stability of KAM tori for the nonlinear Schrödinger equation on the torus with arbitrary dimensions.
Eliasson and Kuksin developed a KAM approach to study the persistence of the invariant tori for nonlinear Schrödinger equation on $\mathbb{T}^{d}$. In this note, we improve Eliasson and Kuksin's KAM theorem by using Kolmogorov's iterative scheme and obtain a local normal form for the transformed Hamiltonian. As a consequence, we are able to derive the time $δ^{-1}$ stability of the obtained KAM tori.
Infinite dimensional invariant tori for nonlinear Schrödinger equations
本研究证明无外部参数的圆上非线性薛定谔方程存在大量概周期解，在Kuksin--Poschel 1996年构造的多数有限维KAM环面附近存在无穷维非共振克罗内克环面，回答了哈密顿PDE领域长期存在的问题。
We prove Strichartz estimates on general flat d-torus for arbitrary d. Using these estimates, we prove local wellposedness for the cubic nonlinear Schrödinger equations in appropriate Sobolev spaces. In dimensions 2 and 3, we prove polynomial bounds on the possible growth of Sobolev norms of smooth solutions.
We consider the stochastic nonlinear Schrödinger equations (SNLS) posed on $d$-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in $L^2(\mathbb{T})$. As for other power-type nonlinearities, namely (i) (super)quintic when $d = 1$ and (ii) (super)cubic when $d \geq 2$, we prove local well-posedness in all scaling-subcritical Sobolev spaces and global well-posedness in the energy space for the defocusing, energy-subcritical problems.
Long time stability for cubic nonlinear Schrödinger equations on non-rectangular flat tori
本文针对满足丢番图非退化条件的平坦环面上的三次非线性薛定谔方程，证明了高正则Sobolev空间中大多数小解的几乎全局存在性与稳定性，并发展了可调制频率的正规形方法。
We consider the energy critical nonlinear Schrodinger equation on generic irrational tori. Using the long-time Strichartz estimates proved in [8], we establish polynomial upper bounds for higher Sobolev norms for solutions with small energy.
We prove scale-invariant Strichartz inequalities for the Schrodinger equation on rectangular tori (rational or irrational) in all dimensions. We use these estimates to give a unified and simpler treatment of local well-posedness of the energy-critical nonlinear Schrodinger equation in dimensions three and four.
The energy-critical defocusing nonlinear Schrödinger equation on 3-dimensional rectangular tori is considered. We prove that the global well-posedness result for the standard torus of Ionescu and Pausader extends to this class of manifolds, namely, for any initial data in $H^1$ the solution exists globally in time.
On the existence of full dimensional KAM tori for 1D periodic nonlinear Schrödinger equation
本文证明了1维周期非线性薛定谔方程在非线性扰动显式依赖于空间变量的情况下，全维KAM环的存在性。
In this paper, we will prove the existence of full dimensional tori for 1-dimensional nonlinear Schrödinger equation with periodic boundary conditions \begin{equation*}\label{L1} \mathbf{i}u_t-u_{xx}+V*u+ε|u|^4u=0,\hspace{12pt}x\in\mathbb{T}, \end{equation*} where $V*$ is the convolution potential. Here the radius of the invariant torus satisfies a slower decay, i.e. \begin{equation*}\label{031601} I_n\sim e^{- \ln^σ|n|},\qquad \mbox{as}\ |n|\rightarrow\infty, \end{equation*} for any $σ>2$, which improves the result given by Bourgain (J. Funct. Anal. 229 (2005), no.1, 62-94).
On Strichartz estimates for many-body Schrödinger equation in the periodic setting
本文在周期环面上证明多体薛定谔方程的Strichartz估计，适用于有理和无理环面及小相互作用势，基于Bourgain-Demeter标准估计，是Hong结果的周期类比。
Critical local well-posedness of the nonlinear Schrödinger equation on the torus
本文针对环面$\mathbb{T}^{d}$上非线性薛定谔方程，在临界正则性下聚焦小$a>0$的非代数非线性项$|u|^{a}u$，通过改进双线性估计并设计适配的函数空间，证明了含质量超临界区域的局部适定性，还研究了解映射的正则性。
Let $\mathcal{L}$ be a fixed $d$-dimensional lattice. We study the localization properties of solutions of the stationary Schrödinger equation with a positive $L^\infty$ potential on tori $\mathbb{R}^d/L\mathcal{L}$ in the limit, as $L\to\infty$, for dimension $d \leq 3$. We show that the probability measures associated with $L^2$-normalized solutions, with eigenvalue $E$ near the bottom of the spectrum, satisfy an algebraic delocalization theorem which states that these probability measures cannot be localized inside a ball of radius $r = o(E^{-1/4+ε})$, unless localization occurs with a suff…
Well-posedness for the periodic Hyperbolic nonlinear Schrödinger equations
本研究针对周期双曲型非线性薛定谔方程，借鉴Killip和Visan方法推导有理与无理环面上的尺度不变Strichartz估计，建立临界空间局部适定性，消除了已有估计中的导数ε损失。
In this work, we study the Gross-Pitaevskii hierarchy on general --rational and irrational-- rectangular tori of dimension two and three. This is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear Schrödinger equation. We prove a conditional uniqueness result for the hierarchy. In two dimensions, this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation from the dynamics of many-body quantum systems. On irrational tori, this question was posed as an open problem in previou…
We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schrödinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on $x$ in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is based on the integrability of the dNLS equation, in particular the fact…
Local well-posedness of the periodic nonlinear Schrödinger equation with a quadratic nonlinearity $\overline{u}^2$ in negative Sobolev spaces
针对一维和二维环面上带二次非线性 $\overline{u}^2$ 的 NLS，本文通过修正 $X^{s,b}$ 空间，在常规双线性估计失效的负 Sobolev 正则性下证明了局部适定性。
Sharp Strichartz estimates on quadratic irrational tori
本文证明二维周期薛定谔流在积分对角符号下的尖锐L^4 Strichartz估计，揭示算术二分性，并应用于三次双曲NLS的适定性转变。
In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain's Lemma which provides a partition of the "resonant sites" of the Laplace operator on irrational tori.
We consider the mass critical NLS on $\mathbb{T}$ and $\mathbb{T}^2$. In the $\mathbb{R}^d$ case the Strichartz estimates enable us to show well-posedness of the IVP in $L^2$ (at least for small data) via the Picard iteration method. However, counterexamples to the $L^6$ Strichartz on $\mathbb{T}$ and the $L^4$ Strichartz on $\mathbb{T}^2$ were given by Bourgain (1993) and Takaoka-Tzvetkov (2001), respectively, which means that the Strichartz spaces are not suitable for iteration in these problems. In this note, we show a slightly stronger result, namely, that the IVP on $\mathbb{T}$ and $\mat…
We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long timescales, they exhibit a strong form of transverse instability in Sobolev spaces $H^s(\mathbb{T}^2)$ ($0<s<1$). …
We consider the cubic nonlinear Schrödinger equation on $2$-dimensional irrational tori. We construct solutions which undergo growth of Sobolev norms. More concretely, for every $s>0$, $s\neq 1$ and almost every choice of spatial periods we construct solutions whose $H^s$ Sobolev norms grow by any prescribed factor. Moreover, for a set of spatial periods with positive Hausdorff dimension we construct solutions whose Sobolev norms go from arbitrarily small to arbitrarily large. We also provide estimates for the time needed to undergo the norm explosion. Note that the irrationality of the space …
Nonpersistence of full-dimensional tori with polynomial decay in nonlinear Schrödinger equations
该研究针对周期五次非线性薛定谔方程，证明了具有多项式衰减作用量的满维不变环面的局部非持久性，推进了Kuksin提出的一维哈密顿PDE的KAM理论开放问题研究，其证明方法可推广至其他哈密顿PDE。
Geometry-Conditioned Fourier Neural Operators for Cubic Nonlinear Schrodinger Dynamics on Periodic Domains
针对不同长宽比的二维平环面上的三次非线性薛定谔方程，提出几何条件傅里叶神经算子，学习一步解算子，并捕获有理与无理环面上不同的Sobolev范数行为。
On almost periodic solutions to NLS without external parameters
本文概述 [BGR24] 对无外部参数圆周非线性 Schrödinger 方程的证明，说明其存在大量无限维非共振不变环面及等价殆周期解，并突出证明中的新技术要点。
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