T. Kappeler, P. Topalov
We prove that there exist potentials so that near them the focusing non-linear Schrödinger equation does not admit local Birkhoff coordinates. The proof is based on the construction of a local normal form of the linearization of the equation at such potentials.
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Transformation of the Gibbs measure of the cubic NLS and fractional NLS under an approximated Birkhoff map
该研究针对周期三次非线性薛定谔方程的吉布斯测度，推导其在Birkhoff范式约化第一步下的变量替换公式，并拓展至分数阶色散情形。
On the Darboux and Birkhoff steps in the asymptotic stability of solitons
本文给出了在证明NLS和非线性Dirac方程特殊情况的孤波的渐近稳定性过程中，寻找Darboux坐标的步骤和随后的Birkhoff规范形步骤的统一证明
We prove an infinite dimensional version of the Arnold-Liouville theorem for integrable non-linear PDEs: In a case study we consider the {\em focusing} NLS equation with periodic boundary conditions.
New estimates of the nonlinear Fourier transform for the defocusing NLS equation
圆上的散焦NLS方程$\mathrm{i} u_t = -u_{xx} + 2|u|^2u$允许全局非线性傅里叶变换，也称为Birkhoff映射，线性化NLS流。已知$u$的正则性与相应的非线性傅里叶系数的衰减性质密切相关。在本文中，我们量化的关系，提供双边多项式估计的所有整数Sobolev范数$\|u\|_m$，$m\ge 0$，在非线性傅立叶变换的加权范数。
We consider the large data scattering problem for the 2D and 3D cubic-quintic NLS in the focusing-focusing regime. Our attention is firstly restricted to the 2D space, where the cubic nonlinearity is $L^2$-critical. We establish a new type of scattering criterion that is uniquely determined by the mass of the initial data, which differs from the classical setting based on the Lyapunov functional. At the end, we also formulate a solely mass-determining scattering threshold for the 3D cubic-quintic NLS in the focusing-focusing regime.
We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS. We also prove failure of uniform space-time bounds at the threshold.
We revisit the scattering result of Holmer and Roudenko on the radial focusing cubic NLS in three space dimensions. Using the radial Sobolev embedding and a virial/Morawetz estimate, we give a simple proof of scattering below the ground state that avoids the use of concentration compactness.
We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the non-radial $\dot H^{1/2}$-critical focusing NLS. By proving an interaction Morawetz inequality, we give a simple proof of scattering below the ground state in dimensions $d\geq 3$ that avoids the use of concentration compactness.
Variants of the focusing NLS equation. Derivation, justification and open problems related to filamentation
聚焦立方非线性最小二乘模型是研究激光束传输的正则模型。在2维和3维中，已知大类初始数据导致有限时间爆破。现在，物理实验表明，这种爆炸并不总是发生。这可能是由于标准NLS模型忽略的一些物理现象在大强度的光束下变得相关。已经提出了聚焦NLS方程的许多特别变体来捕获这种效应。在本文中，我们从麦克斯韦方程组推导出其中的一些变量，并提出了一些新的变量。我们还提供了严格的误差估计的所有模型考虑。最后，我们讨论了与这些修正的NLS方程有关的一些公开问题。
Approximate traveling waves for the focusing NLS with an external potential
针对受快速衰减外势扰动的三维聚焦三次非线性薛定谔方程构造近似行波解，以此证明在未受扰方程基态阈值以上的任意质量-能量值，都存在向前整体、H¹有界且非散射的解。
In this paper we study long time dynamics (i.e., scattering and blow-up) of solutions for the focusing NLS with a repulsive inverse-power potential and with initial data lying exactly at the mass-energy threshold, namely, when $E_{V}(u_{0})M(u_{0})=E_{0}(Q)M(Q)$. Moreover, we prove failure of the uniform space-time bounds at the mass-energy threshold.
Scattering for the focusing energy-subcritical NLS
对于三维聚焦立方非线性薛定谔方程，Holmer和Roudenko~ \cite{HR2}（径向情形）和Duyckaerts，Holmer和Roudenko ~ \cite{DHR}（一般情形）建立了$H^1$解在（尺度不变）势阱中的散射.本文采用~ \cite{DHR}方法，将这一结果推广到任意空间维数和聚焦、质量-超临界和能量-亚临界功率非线性。
On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions $d\ge 2$
对于聚焦质量临界NLS $iu_t+Δu=-|u|^{\frac 4d}u$，证明了在满足方程对称性的条件下，具有基态质量的唯一整体非散射解必定是一个孤波。本文解决了在具有球对称的$d=2,3$维和具有某些分裂球对称初值的$d\ge 4$维中关于$H^1_x$初值的猜想.
A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton
该研究针对聚焦五次非线性薛定谔方程，证明了质量与孤子质量相等的爆破解仅为两类：重标度孤子及其伪共形变换，明确了该方程对应质量条件下爆破解的形式。
The focusing cubic NLS on exterior domains in three dimensions
我们考虑三维光滑、紧凑、严格凸障碍物外$Ω$中的聚焦立方NLS。我们证明了全局存在和散射的阈值是相同的欧几里得空间上的问题。具体地说，我们证明了如果$E(u_0)M(u_0)<E(Q)M(Q)$和$\|\nabla u_0\|_2\|u_0\|_2<\|\nabla Q\|_2\|Q\|_2$，则Dirichlet边值条件下的初值问题的解是全局存在的，并且在未来和过去渐近地分散到线性演化中.这里，$Q(x)$表示$\mathbb{R}^3$中的聚焦立方NLS的基态。
In this paper, we consider a 3d cubic focusing nonlinear Schrödinger equation (NLS) with slowing decaying potentials. Adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a condition for scattering. It is actually sharp in some sense since the solution will blow up if it's false. The proof of blow-up part relies on the method of Du-Wu-Zhang \cite{DWZ}
Threshold scattering for the focusing NLS with a repulsive potential
本文借鉴近期工作，建立了三维聚焦三次NLS在排斥势下尖锐阈值的散射结果，涵盖短程势与逆平方势两种情形。
In this paper we prove rigidity for blowup solutions to the focusing, mass-critical nonlinear Schr{ö}dinger equation in dimensions $2 \leq d \leq 15$ with mass equal to the mass of the soliton. We prove that the only such solutions are the solitons and the pseudoconformal transformation of the solitons.
In this paper, we prove the global well-posedness for the focusing, cubic nonlinear Schrödinger equation on the product space $\mathbb{R} \times \mathbb{T}^3$ with initial data below the threshold that arises from the the ground state in the Euclidean setting. The defocusing analogue was discussed and proved in Ionescu-Pausader \cite{IPRT3} (Comm. Math. Phys. 312 (2012), no. 3, 781-831).
In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state of Killip-Visan-Zhang by utilizing Dodson and Murphy's argument and the dispersive estimate established by Ivanovici and Lebeau, which avoids using the concentration compactness. We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary. Based on this effect and the interaction Morawetz estimate…
This paper is concerned with the global well-posedness and finite time blowup problem for the 3D focusing energy-critical inhomogeneous NLS. In the previous results \cite{chkl2, chkl3} the authors considered the same problems with the spatial inhomogeneity coefficient $g$ such that $g(x) \sim |x|^{-b}$ for $0 \le b < \frac43$. Here we extend the inhomogeneous index $b$ up to $\frac32$. For this purpose, we improve the local theory and develop a new profile decomposition based on weighted space.
Scattering for the focusing, $L^2$-supercritical fourth-order NLS in one dimension
本文证明一维聚焦 $L^2$ 超临界四阶 NLS 在偶性假设下基态阈值以下解的全局存在性与散射，推广了 Guo 和 Dinh 的高维径向结果。
We study the focusing energy-critical NLS \begin{align}\label{nls_abstract} i\partial_t u+Δ_{x,y} u=-|u|^{\frac{4}{d-1}} u\tag{NLS} \end{align} on the waveguide manifold $\mathbb{R}_x^d\times\mathbb{T}_y$ with $d\geq 2$. We reveal the somewhat counterintuitive phenomenon that despite the energy-criticality of the nonlinear potential, the long time dynamics of \eqref{nls_abstract} are purely determined by the semivirial-vanishing geometry which possesses an energy-subcritical characteristic. As a starting point, we consider a minimization problem $m_c$ defined on the semivirial-vanishing manifo…
In this paper, we utilize the method in Dodson-Murphy [4] to establish the radial scattering result for the focusing nonlinear Schrödinger equation with inverse square potential $i\pa_tu-\la u=-|u|^{p-1}u$ in the energy space $H^1_a(\R^d)$ in dimensions $d\geq3$, which extends the result of [10,11] to higher dimensions cases but with radial initial data. The new ingredient is to establish the dispersive estimate for radial function and overcome the weak dispersive estimate when $a<0$.
Going beyond the threshold: scattering and blow-up in the focusing NLS equation
研究了$L^2$ -超临界区域中有限能量、有限方差初始数据的聚焦非线性薛定谔方程。我们调查的解决方案以上的能量（或质能）阈值。在我们的第一个结果中，我们扩展了已知的散射与爆破二分法以上的能量亚临界和能量临界制度的有限方差解决方案的阈值，获得散射和爆破标准的解决方案具有任意大的质量和能量。因此，我们描述了由二次相位调制的基态初始数据的行为。我们的第二个结果给出了两个爆破准则，这也适用于能量超临界NLS设置。最后，我们用各种例子来说明我们的结果。
We establish the scattering of solutions to the focusing mass supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potential \[ i\partial_{t}u+\partial^{2}_{x}u+γδ(x)u+|u|^{p-1}u=0, \quad (t,x)\in {\mathbb R}\times{\mathbb R}, \] at the mass-energy threshold, namely, when $E_γ(u_{0})[M(u_{0})]^σ=E_{0}(Q)[M(Q)]^σ$ where $u_{0}\in H^{1}({\mathbb R})$ is the initial data, $Q$ is the ground state of the free NLS on the real line ${\mathbb R}$, $E_γ$ is the energy, $M$ is the mass and $σ=(p+3)/(p-5)$. We also prove failure of the uniform space-time bounds at the mass-energy thr…
Scattering for the focusing $L^{2}$ -supercritical and $\dot H^2$-subcritical biharmonic NLS Equations
考虑了聚焦$\dot H^{s_c}$ -临界双调和薛定谔方程，证明了满足$ M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E(Q) $和$ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{2}\|Δu_{0}\|_{2}<\|Q\|^{\frac{2-s_c}{s_c}}_{2}\|ΔQ\|_{2}, $的径向数据$u_0\in H^2(\mathbb R^N)$的整体适定性和散射结果，其中$s_c\in(0,2)$和$Q$是$Δ^2Q+(2-s_c)Q-|Q|^{p-1}Q=0$的基态.
Gibbs measure for the focusing fractional NLS on the torus
本文研究多维环面上聚焦质量临界分数阶非线性薛定谔方程的吉布斯测度构造，确定了其可正规化与不可正规化的尖锐质量阈值，并建立了几乎尖锐的分数阶 Gagliardo-Nirenberg-Sobolev 不等式作为关键工具。
On the Birkhoff factorization problem for the Heisenberg magnet and nonlinear Schroedinger equations
给出了具有经典自旋的Heisenberg磁体方程在商空间$G/H_+$上的流几何描述，其中$G$是无限维李群，$H_+$是$G$的子群.结果表明，HM流是由$\mathbb{R}^2$对$G/H_+$的作用引起的，HM方程可以通过求解$G$的Birkhoff分解问题来积分.对于谱变量为Laurent多项式的HM流，我们导出了非线性Schroedinger（NLS）和Heisenberg磁体方程的解之间的代数变换。$G$的Birkhoff分解是根据Segal-Wilson Grassmannian $Gr(H)$的几何来处理的。该问题的解用$Gr(H)$的特殊子空间的一对Baker函数给出。贝克函数的子空间，产生多孤子解的NLS和HM方程的显式构造。
Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS
令$d\geq 4$和$u$是具有球对称$H_x^1$初始数据和质量等于基态$Q$的聚焦质量临界非线性薛定谔方程$iu_t+Δu=-|u|^{\frac 4d}u$的整体解。我们证明，如果$u$不散射，然后，相位旋转和缩放，$u$是孤波$e^{it}Q$。结合Merle \cite{merle2}的结果，我们得到在$d\geq 4$维中，在相位旋转和标度下，唯一的球对称最小质量爆破解是赝共形基态和孤波.
Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$
本文通过将斯特里查茨估计整合进伯克霍夫正规形，首次在色散背景下证明了圆上五次NLS方程对 $s>2/5$ 的微小不规则初值的长时间行为。
We study the focusing intercritical NLS \begin{align}\label{abstract_nls} i\pt_t u+Δ_{x,y}u=-|u|^αu\tag{NLS} \end{align} on the semiperiodic waveguide manifold $\R^d_x\times \T_y$ with $d\geq 5$ and $α\in(\frac{4}{d},\frac{4}{d-1})$. In the case $d\leq 4$, with the aid of the semivirial vanishing theory \cite{Luo_inter}, the author was able to construct a sharp threshold, which being uniquely characterized by the ground state solutions, that sharply determines the bifurcation of global scattering and finite time blow-up solutions in dependence of the sign of the semivirial functional. As the d…
Global existence, scattering and blow-up for the focusing NLS on the hyperbolic space
证明了双曲空间上能量亚临界聚焦非线性薛定谔方程的整体适定性、散射和爆破结果。我们特别表明存在一个关键的元素散射的所有能量亚临界功率非线性。对于质量超临界非线性，我们显示了散射与爆破二分法的径向解决方案的方程在低维，低于自然的质量和能量阈值的基态方程。证明是基于质量和能量，紧性和刚性的陷阱，并类似于欧几里得空间上的，一个新的论点，基于广义Pohozaev恒等式，以获得适当的单调性公式。
Threshold solutions for the focusing $L^{2}$ -supercritical NLS Equations
研究了$H^1$中的$L^2$ -超临界和$\dot{H}^1$ -亚临界非线性薛定谔方程.在\cite{G1}和\cite{yuan}中，质能量$M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$已被证明是方程解的动力学行为的阈值。本文研究了$M(u)^{\frac{1-s_{c}}{s_{c}}}E(u)=M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$的临界态动力学，并利用调制理论对相应的解进行了分类，非平凡地推广了三维立方薛定谔方程的\cite{holmer3}的结果.
We consider the nonlinear Schrödinger equation of degree five on the circle $\mathbb{S}^1 = \mathbb{R}/2π$. We prove the existence of quasi-periodic solutions which bifurcate from "resonant" solutions (studied in [14]) of the system obtained by truncating the Hamiltonian after one step of Birkhoff normal form, exhibiting recurrent exchange of energy between some Fourier modes. The existence of these quasi-periodic solutions is a purely nonlinear effect.
In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus. We show that for any initial condition even in $x$, regular enough and of size $\varepsilon$ sufficiently small, the lifespan of the solution is of order $\varepsilon^{-N}$ for any $N\in\mathbb{N}$ if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonali…
Long-time limit studies of an obstruction in the g-function mechanism for semiclassical focusing NLS
本文研究了一维非线性薛定谔方程在半经典极限下的Riemann-Hilbert聚焦方法中的障碍物的长时间性质。对于某些值的参数的大量的孤子在系统中的干扰与$g$ -函数机制在最速下降到振荡的黎曼-希尔伯特问题。该障碍阻碍了在$(x,t)$平面中的区域中的Riemann-Hilbert分析。我们得到了区域边界（阻塞曲线）的长时间渐近性。作为$t\to\infty$，阻塞曲线具有垂直渐近线$x=\pm \ln 2$。数值结果支持渐近分析。
We study the focusing NLS \begin{align}\label{nls_abstract} i\partial_t u+Δ_{x,y} u=-|u|^αu\tag{NLS} \end{align} on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$ in the intercritical regime $α\in(\frac{4}{d},\frac{4}{d-1})$. By assuming that the \eqref{nls_abstract} is independent of $y$, it reduces to the focusing intercritical NLS on $\mathbb{R}^d$, which is known to have standing wave and finite time blow-up solutions. Naturally, we ask whether these special solutions with non-trivial $y$-dependence exist. In this paper we give an affirmative answer to this question. To that end, we…
We study the influence of Szeg{ö} projector $Π$ on the L 2 --critical one-dimensional non linear focusing Schr{ö}dinger equation, leading to the quintic focusing NLS-Szeg{ö} equation i$\partial$ t u + $\partial$ 2 x u + $Π$(|u| 4 u) = 0, (t, x) $\in$ R x R, u(0, $\times$) = u 0. This equation is globally well-posed in H 1 + = $Π$(H 1 (R)), for every initial datum u 0. The solution L 2-scatters both forward and backward in time if u 0 has sufficiently small mass. We prove the orbital stability with scaling of the traveling wave : u $ω$,c (t, x) = e i$ω$t Q(x + ct), for some $ω$, c > 0, where Q …
We consider the focusing nonlinear Schrödinger equation $i u_t + Δu + |u|^{p-1}u=0$, $p>1,$ and the generalized Hartree equation $iv_t + Δv + (|x|^{-(N-γ)}\ast |v|^p)|v|^{p-2}u=0$, $p\geq2$, $γ<N$, in the mass-supercritical and energy-subcritical setting. With the initial data $u_0\in H^1(\mathbb{R}^N)$ the characterization of solutions behavior under the mass-energy threshold is known for the NLS case from the works of Holmer and Roudenko in the radial [16] and Duyckaerts, Holmer and Roudenko in the nonradial setting [10] and further generalizations (see [1,11,14]); for the generalized Hartre…
Blow-up or grow-up for the focusing 3D cubic NLS with a repulsive inverse-power potential at the mass--energy threshold
本文研究带排斥逆幂势的聚焦三次非线性薛定谔方程在质量-能量阈值下的动力学，证明在负virial区域中解在每个时间方向要么有限时间爆破要么增长。
A solution to the focusing 3d NLS that blows up on a contracting sphere
我们严格构造径向$H^1$解决方案的三维立方聚焦NLS方程$i\partial_t ψ+ Δψ+ 2 |ψ|^2ψ=0$，吹沿收缩球。当爆破时间设置为$t=0$时，解集中在半径为$\sim t^{1/3}$的球体上，但以更快的速率$\sim t^{2/3}$集中在该球体上。这种动力学最初由Degtyarev-Zakharov-Rudakov（1975）提出，后来由Holmer-Roudenko（2007）独立提出，在那里它被证明与该方程的所有守恒律一致。在后一篇论文中，它被提出作为一种解，该解将在“宽”半径$\sim |\nabla u(t)|_{L_x^2}^{-1/2}$内产生$L_x^3$范数的发散，但不在“紧”半径$\sim |\nabla u(t)|_{L_x^2}^{-2}$内，第二个是数值观察到的自相似爆破解的收缩率，并在Sulem-Sulem（1998）的第7章中详细描述。
The aim of this note is to adapt the strategy in [4][See,B.Dodson, J.Murphy, a new proof of scattering below the ground state for the 3D radial focusing cubic NLS, arXiv:1611.04195 ] to prove the scattering of radial solutions below sharp threshold for certain focusing fractional NLS with cubic nonlinearity. The main ingredient is to apply the fractional virial identity proved in [11][See,T.Boulenger, D.Himmelsbach,E.Lenzmann, Blow up for fractional NLS,J.Func.Anal,271(2016),2569-2603] to exclude the concentration of mass near the origin.
Geometric structure of NLS evolution
我们澄清了非线性发展方程的哈密顿方法和拉格朗日方法之间的关系，特别关注非线性薛定谔方程。特别是，我们解释了最小作用原理和诺特定理在这方面。
We consider the focussing energy-critical inhomogeneous nonlinear Schrödinger equation: $$ iu_t + Δu + g|u|^2u = 0, u(0)= φ\in \dot{H}^1,\;\; 0 \le g_i \le |x|g \le g_s.$$ On the road map of Kenig-Merle \cite{km} we show the global well-posedness and scattering of radial solutions under energy condition $$E_g(φ) < E_g(Q),\;\;\mbox{and}\;\; g_s\|φ\|_{\dot{H}^1}^2 < \|Q\|_{\dot{H}^1}^2,$$ where $Q$ is the solution of $ΔQ + |x|^{-1}Q^3 = 0$, together with scaling condition $|g(x)| + |x||\nabla g(x)| \lesssim |x|^{-1}$, variational condition $g_s(2-g_i) \le 1$, and rigidity condition $-g(x) \le x\…
We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.
We consider a perturbed energy critical focusing Nonlinear Schrödinger Equation in three dimensions. We construct solitary wave solutions for focusing subcritical perturbations as well as defocusing supercritical perturbations. The construction relies on the resolvent expansion, which is singular due to the presence of a resonance. Specializing to pure power focusing subcritical perturbations we demonstrate, via variational arguments, and for a certain range of powers, the existence of a ground state solitary wave, which is then shown to be the previously constructed solution. Finally, we pres…
In this article we consider the Cauchy problem for the cubic focusing nonlinear Schrö\-dinger (NLS) equation on the line with initial datum close to a particular $N$-soliton. Using inverse scattering and the $\bar{\partial}$ method we establish the decay of the $L^{\infty}$ norm of the residual term in time.
In this paper, we prove a refined version of a compactness lemma and we use it to establish mass-concentration for the focusing nonlinear Schrödinger equation with an inverse-square potential.
Spectra of Linearized Operators for NLS Solitary Waves
具有聚焦幂非线性项的非线性薛定谔方程存在孤波解。这些孤波周围的线性化算子的谱与孤波的稳定性以及（NLS）解的长时间动力学密切相关。我们研究这些光谱的详细，分析和数值。
Instability of vortex solitons for 2D focusing NLS
研究了涡旋孤子$e^{i(mθ+ωt)}ϕ_{ω,m}(r)$到$$iu_t+Δu+|u|^{p-1}u=0,\quad\text{for $x\in\R^n$, $t>0$,}$$的不稳定性，其中$n=2$，$m\in\N$和$(r,θ)$是$\R^2$中的极坐标. Grillakis \cite{Gr}证明了如果$p>1+4/n$，则每个径向驻波解都是不稳定的.然而，我们没有任何例子不稳定的驻波解决方案在亚临界情况下$(p<1+n/4)$。假设$ϕ_{ω,m}$是非负的。我们研究了$ϕ_{ω,m}$的极限轮廓为$m\to\infty$，并证明了对于每个$p>1$，存在一个$m_*\in \N$，使得对于$m\ge m_*$，涡旋孤子$e^{i(mθ+ωt)}ϕ_{ω,m}(r)$对于具有$1\ll j\ll m$的$e^{i(m+j)θ}v(r)$形式的扰动变得不稳定.
Smooth parametric dependence of asymptotics of the semiclassical focusing NLS
本文研究了具有指数衰减复值初值的一维聚焦（立方）非线性薛定谔方程（NLS）的半经典极限解，其相位乘以一个实参数。我们证明了渐近解对参数的光滑依赖性。数值结果支持我们的重要数量的估计。
Sharp mass-threshold for Dancer-type solutions of the focusing mass-critical NLS on $\Bbb R^d\times\Bbb T$
通过引入新的严格单调性机制和半消失几何框架，证明了在质量$c\in(0,2\pi\widehat{M}(Q))$范围内，半消失变分问题$m_c$存在归一化的Dancer型优化子，从而解决了聚焦质量临界NLS在$\Bbb R^d\times\Bbb T$上的解的存在性。
We study the dynamics of the focusing $3d$ cubic nonlinear Schrödinger equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when $ E_Ω[u_0] M_Ω[u_0] = E_{\R^3}[Q] M_{\R^3}[Q] $ and $ \left\| \nabla u_0 \right\|_{L^{2}(Ω)} \left\|u_0\right\|_{L^{2}(Ω)}< \left\| \nabla Q \right\|_{L^2(\R^3)} \left\| Q \right\|_{L^2(\R^3)} ,$ where $u_0 \in H^1_0(Ω)$ is the initial data, $Q$ is the ground state on the Euclidean space, $E$ is the energy and $M$ is the mass. In the whole Euclidean space Duyckaerts and Roudenko (following the work of Duyckaerts and Merle on th…
The energy-critical NLS with inverse-square potential
考虑三维空间中带平方反比势$iu_t = -Δu + a|x|^{-2}u + |u|^4u$的离焦能量临界非线性薛定谔方程.我们证明了$a>-\frac14 +\frac1{25}$的整体适定性和散射。我们还进行了变分分析需要治疗的聚焦情况。
The semiclassical limit of focusing NLS for a family of non-analytic initial data
聚焦非线性薛定谔方程（NLS）的小色散极限表现出丰富的结构，在微观尺度上表现出不同的快速振荡的急剧分离的区域。聚焦NLS的非自伴散射问题和不适定的极限Whitham方程使得严格的渐近结果变得困难。以往的研究[KMM 03，TVZ 04，TVZ 06]主要集中在一类特殊的解析初值问题上，即极限椭圆型Whitham方程是适定的。在本文中，我们考虑另一个精确可解的初始数据族，平方障碍族，$ψ_0(x) = q χ_[-L,L]$实振幅q。利用Riemann-Hilbert技巧，我们得到了在空间中全局聚焦NLS的半激光极限的严格逐点渐近性，并得到了最大时间为O（1）.特别是，我们表明，初始数据中的不连续性通过立即生成发射到初始数据支持中的亏格一振荡来规则化。据我们所知，这是第一次在非解析初始数据下计算fNLS的半经典渐近的亏格结构。
Scattering below the ground state for the three-dimensional focusing energy-critical NLS
该论文证明了三维聚焦能量临界NLS在基态能量和梯度阈值以下的全局适定性与散射，证实了阈值猜想，并通过改进的Morawetz恒等式和频率局域化估计实现了这一结果。
We derive rigorously the 2D periodic focusing cubic NLS as the mean-field limit of the 3D focusing quantum many-body dynamics describing a dilute Bose gas with periodic boundary condition in the $x$-direction and a well of infinite-depth in the $z$-direction. Physical experiments for these systems are scarce. We find that, to fulfill the empirical requirement for observing NLS dynamics in experiments, namely, the kinetic energy dominates the potential energy, it is necessary to impose an extra restriction on the system parameters. This restriction gives rises to an unusual coupling constant.
Global Behavior Of Finite Energy Solutions To The $d$-Dimensional Focusing Nonlinear Schrödinger Equation
研究了具有初始数据$u_0\in H^1,\; x \in R^n$的$d$维聚焦非线性薛定谔方程（$i \partial_t u+Δu+ |u|^{p-1}u=0, $）有限能量解的整体行为.非线性功率$p$和维度$d$使得标度指数$s=\frac{d}2-\frac2{p-1}$在0和1之间，因此，NLS是质量超临界$(s>0)$和能量亚临界$(s<1).$对于具有$\ME[u_0]<1$的解（$\ME[u_0]$代表$u_0$质量和能量方面的不变量和守恒量），给出了散射和爆破的尖锐阈值。即，如果解$u$与NLS的重正化梯度$\g_u$最初小于1，即，$\g_u(0)<1,$则解在时间上全局存在，并在$H^1$中散射（以$t\to\pm\infty$的形式接近某种线性薛定谔演化）;如果重整化梯度$\g_u(0)>1,$则解表现出爆破行为，即要么发生有限时间爆破，要么在无限时间内$H^1$范数发散。本文推广了Holmer-Roudenko和Duyckaerts-Holmer-Roudenko在能量临界NLS和非线性波动方程的背景下发展起来的3d立方NLS的结果。
We consider the mass-subcritical NLS in dimensions $d\geq 3$ with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.
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