三阶导数非线性薛定谔方程存在爆破解

具有非齐次阻尼项的非线性薛定谔方程在$L^2$临界情形下的爆破

Blowup for the nonlinear Schrödinger equation with an inhomogeneous damping term in the $L^2$ critical case

考虑具有$L^2$ -临界指数和非齐次阻尼项的非线性薛定谔方程.利用Merle和Raphael发展的工具，证明了能量空间$H^1(\mathbb{R})$中爆破现象的存在性.

阻尼$L^{2}$ -临界非线性薛定谔方程解的爆破

Blowup for the Damped $L^{2}$-Critical Nonlinear Schrödinger Equation

本文考虑$L^{2}$ -临界阻尼非线性薛定谔方程的Cauchy问题.我们证明了有限时间爆破动力学的存在性和稳定性与对数-对数爆破速度$\|\nabla u(t)\|_{L^2}$。

On Blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation

In this paper we study dynamical properties of blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we obtain the $L^2$-concentration and the limiting profile with minimal mass of blowup solutions.

Blowup rate for mass critical rotational nonlinear Schrödinger equations

We consider the blowup rate for blowup solutions to $L^2$-critical, focusing NLS with a harmonic potential and a rotation term. Under a suitable spectral condition we prove that there holds the "$\log$-$\log$ law" when the initial data is slightly above the ground state. We also construct minimal mass blowup solutions near the ground state level with distinct blowup rates.

Universal Upper Bound on The Blowup Rate of Nonlinear Schrödinger Equation with Rotation

In this paper, we prove a universal upper bound on the blowup rate of a focusing nonlinear Schrödinger equation with an angular momentum under a trapping harmonic potential, assuming that the initial data is radially symmetric in the weighted Sobolev space. The nonlinearity is in the mass supercritical and energy subcritical regime. Numerical simulations are also presented.

Threshold for blowup for the supercritical cubic wave equation

We consider the focusing cubic wave equation in the energy supercritical case, i.e., in dimensions $d \geq 5$. For this model an explicit nontrivial self-similar blowup solution was recently found by the first and third author in \cite{GlogicSchoerkhuber}. Furthermore, the solution is proven to be co-dimension one stable in $d=7$. In this paper, we study the equation from a numerical point of view. For $d=5$ and $d=7$ in the radial case, we provide evidence that this solution is at the threshold between generic ODE blowup and dispersion. In addition, we investigate the spectral problem that un…

带旋转的非线性 Schrödinger 方程的爆破阈值与稳定性

Threshold for Blowup and Stability for Nonlinear Schrödinger Equation with Rotation

针对旋转磁阱中 Bose-Einstein 凝聚体的聚焦 NLS，本文利用变分法、基态与 virial 恒等式，建立质量临界情形下全局存在、有限时间爆破及轨道不稳定性的尖锐阈值。

Construction of $L^2$ log-log blowup solutions for the mass critical nonlinear Schrödinger equation

In this article, we study the log-log blowup dynamics for the mass critical nonlinear Schrödinger equation on $\mathbb{R}^{2}$ under rough but structured random perturbations at $L^{2}(\mathbb{R}^2)$ regularity. In particular, by employing probabilistic methods, we provide a construction of a family of $L^{2}(\mathbb{R}^2)$ regularity solutions which do not lie in any $H^{s}(\mathbb{R}^2)$ for any $s>0$, and which blowup according to the log-log dynamics.

关于$L^2$ -临界聚焦非线性薛定谔方程在能量级以下高维的爆破

On the blowup for the $L^2$-critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class

我们考虑了聚焦质量临界非线性薛定谔方程，证明了在$H^s(\R^d)$，$s > s_0(d)$和$d\geq 3$中初始数据下该方程的爆破解在爆破时刻至少集中基态的质量.这扩展了J. Colliander，S。雷诺角Sulem和J.D. Wright，\cite{crsw}，T. Hmidi和S. Keraani、\cite{hk}和N. Tzirakis，\cite{tzirakis}，关于能量空间以下的二维和一维质量临界聚焦NLS到所有维度$d\ge 3$的爆炸。

On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation

In this paper we study dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schrödinger equation. We firstly establish the profile decomposition of bounded sequences in $\dot{H}^{γ_{\text{c}}} \cap \dot{H}^2$. We also prove a compactness lemma and a variational characterization of ground states related to the equation. As a result, we obtain the $\dot{H}^{γ_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{γ_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{…

色散管理非线性薛定谔方程的适定性与爆破解

Well-posedness and blowup for the dispersion-managed nonlinear Schrödinger equation

本文研究周期色散管理非线性薛定谔方程，建立线性方程的全局Strichartz估计，由此得到三维三次方程的小数据散射结果，并利用virial论证证明爆破解存在。

三次wave equation系统整体动力学的普适性

Universality of global dynamics for the cubic wave equation

考虑三维空间中球对称聚焦三次波动方程的初值问题。我们给出了数值和分析的证据，一个普遍的吸引子，其中包括全球和爆破解决方案的存在。作为一个副产品，我们得到一个明确的描述临界行为的阈值爆破。

Quasiperiodicity and blowup in integrable subsystems of nonconservative nonlinear Schrödinger equations

In this paper, we study the dynamics of a class of nonlinear Schrödinger equation $ i u_t = \triangle u + u^p $ for $ x \in \mathbb{T}^d$. We prove that the PDE is integrable on the space of non-negative Fourier coefficients, in particular that each Fourier coefficient of a solution can be explicitly solved by quadrature. Within this subspace we demonstrate a large class of (quasi)periodic solutions all with the same frequency, as well as solutions which blowup in finite time in the $L^2$ norm.

非线性薛定谔方程的爆破与散射问题

Blowup and Scattering problems for the Nonlinear Schrödinger equations

我们考虑$L^{2}$-超临界和$H^{1}$-亚临界聚焦非线性薛定谔方程。我们为$d\ge 1$引入了$H^{1}(\mathbb{R}^{d})$的一个子集$PW$，并研究了初始数据在这个集合中的解的行为.为此，我们将$PW$分为两个不相交的组件$PW_{+}$和$PW_{-}$。然后，证明了从$PW_{+}$中的一个数据点出发的解是渐近自由的，从$PW_{-}$中的一个数据点出发的解是爆破的或增长的，由此我们发现基态有两个不稳定的方向.我们还研究了一般的整体解和爆破解的一些性质。

Blowup solutions for the nonlinear Schrödinger equation with complex coefficient

We construct a finite time blow up solution for the nonlinear Schrödinger equation with the power nonlinearity whose coefficient is complex number. We generalize the range of both the power and the complex coefficient for the result of Cazenave, Martel and Zhao \cite{CMZ}. As a bonus, we may consider the space dimension $5$. We show a sequence of solutions closes to the blow up profile which is a blow up solution of ODE. We apply the Aubin-Lions lemma for the compactness argument for its convergence.

Global existence versus finite time blowup dichotomy for the system of nonlinear Schrödinger equations

We construct an extremizer for the kinetic energy inequality (except the endpoint cases) developing the concentration-compactness technique for operator valued inequality in the formulation of the profile decomposition. Moreover, we investigate the properties of the extremizer, such as the system of Euler-Lagrange equations, regularity and summability. As an application, we study a dynamical consequence of a system of nonlinear Schrödinger equations with focusing cubic nonlinearities in three dimension when each wave function is restricted to be orthogonal. Using the critical element of the ki…

聚焦三次非线性Klein-Gordon方程的爆破/整体存在二象性的数值研究

Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation

通过数值模拟研究三维径向聚焦三次非线性Klein-Gordon方程中爆破与整体存在二象性的边界结构，发现了比预期更复杂的精细结构。

On blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation

We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation \[ i\partial_t u -(-Δ)^s u = -|u|^αu, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times \mathbb{R}^d, \] where $d \geq 2, \frac{d}{2d-1} \leq s <1$, $\frac{4s}{d}<α<\frac{4s}{d-2s}$ and $u_0 \in \dot{H}^{s_{\text{c}}} \cap \dot{H}^s$ is radial with the critical Sobolev exponent $s_{\text{c}}$. To this end, we establish a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in $\dot{H}^{s_{\text{c}}} \cap \dot{H}^s$. …

低于H^1的L^2临界非线性薛定谔方程的基态质量浓度

Ground state mass concentration in the L^2-critical nonlinear Schrodinger equation below H^1

考虑了$L^2$ -临界立方聚焦非线性薛定谔方程在$\R^2$上的有限时间爆破解.这样的函数，当在$H^1$中时，已知在爆破点集中固定的$L^2$ -质量（基态的质量）。已知来自仅在$L^2$中的初始数据的爆破解至少集中少量质量。在本文中，我们考虑的中间情况下，爆破解从初始数据的$H^s$，与$1 > s > s_Q$，其中$s_Q \le \sQ$。我们的主要结果是，这样的解决方案，径向对称时，集中至少在原点的基态在爆破时间的质量。

Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential

We consider a class of the focusing nonlinear Schrödinger equation with inverse-square potential \[ i\partial_t u + Δu -c|x|^{-2}u = - |u|^αu, \quad u(0)=u_0 \in H^1, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $\frac{4}{d}\leq α\leq \frac{4}{d-2}$ and $c\ne 0$ satisfies $c>-λ(d):=-\left(\frac{d-2}{2}\right)^2$. In the mass-critical case $α=\frac{4}{d}$, we prove the global existence and blowup below ground states for the equation with $d\geq 3$ and $c>-λ(d)$. In the mass and energy intercritical case $\frac{4}{d}<α<\frac{4}{d-2}$, we prove the global existence and blowu…

耦合宇称时间对称非线性薛定谔方程的超临界爆破

Supercritical blowup in coupled parity-time-symmetric nonlinear Schrödinger equations

证明了两个耦合的非线性薛定谔（NLS）方程的宇称时间对称系统的有限时间超临界爆破。其中一个方程包含增益，另一个方程包含耗散，使得增益和耗散的强度相等。我们处理两种情况：在第一种模型中，所有非线性系数（即描述自作用和非线性耦合的系数）对应于吸引（聚焦）非线性，在第二种情况下，具有增益的NLS方程具有吸引非线性，而具有耗散的NLS方程具有排斥（散焦）非线性，非线性耦合也是排斥的。证明是基于维里技术参数。几个特殊的情况下，也说明了数字。

Davey-Stewartson系统爆破的数值研究

Numerical Study of Blowup in the Davey-Stewartson System

非线性色散偏微分方程，如非线性薛定谔方程，可以有解爆破。通过对集中解和Ozawa精确解的扰动分析，数值研究了聚焦Davey-Stewartson II方程解的长时间行为和可能的爆破.它表明，在这种方式下，块是不稳定的爆破和分散，爆破在小泽解决方案是通用的。

Chern–Simons–Schrödinger 方程解的序列收敛性

Sequential convergence of a solution to the Chern--Simons--Schrodinger equation

本文研究 $m$-等变自对偶 Chern–Simons–Schrödinger 方程，证明质量小于孤子质量两倍的爆破解沿趋于爆破时间的子序列收敛到孤子。

二维聚焦立方非线性Schrödinger系统极小质量爆破解的分类

Classification of the minimal-mass blowup solutions to the two dimensional focusing cubic nonlinear Schrödinger system

本文研究了质量等于散射阈值的二维聚焦有限和无限耦合立方非线性Schrödinger系统，通过发展多分量调制分析和向量值长时间Strichartz估计，完整分类了有限耦合系统的极小质量爆破解，并证明无限耦合系统解的唯一动力学行为是散射。

Finite-time blowup for a Schrödinger equation with nonlinear source term

We consider the nonlinear Schrödinger equation \[ u_t = i Δu + | u |^αu \quad \mbox{on ${\mathbb R}^N $, $α>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) α\le 4$. Under the additional technical assumptions $α\geq 2$ (and thus $N\leq 4$), we construct $H^1$ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ${\mathbb R}^N$. The construction involves explicit functions $U$, solutions of the ordinary differential equation $U_t=|U|^αU$. In the simplest case, $U(t,x)=(|x|^k-α…

Nonexistence of self-similar blowup for the nonlinear Dirac equations in (1+1) dimensions

We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of classical self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of finite-time blowup, it still suggests that smooth solutions to the nonlinear Dirac equations in (1+1) dimensions do not develop self-similar singularities in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.

聚焦Gross-Pitaevskii方程组解的能量守恒和爆破

Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies

我们认为解决方案的重点三次和五次Gross-Pitaevskii（GP）的层次。我们确定了一个可观察到的对应于每个粒子的平均能量，我们证明了它是一个守恒量。证明了当初始条件下每个粒子的能量为负时，在$L^2$ -临界或$L^2$ -超临界水平上聚焦GP族的所有解在有限时间内爆破.我们的结果不假设任何分解的初始数据。

Blowup of $H^1$ solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation

In this paper, we consider a class of the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b} |u|^αu = 0, \quad u(0)=u_0 \in H^1(\mathbb{R}^d), \] with $0<b<\min\{2,d\}$ and $α_\star\leq α<α^\star$ where $α_\star =\frac{4-2b}{d}$ and $α^\star=\frac{4-2b}{d-2}$ if $d\geq 3$ and $α^\star = \infty$ if $d=1,2$. In the mass-critical case $α=α_\star$, we prove that if $u_0$ has negative energy and satisfies either $xu_0 \in L^2$ with $d\geq 1$ or $u_0$ is radial with $d\geq 2$, then the corresponding solution blows up in finite time. Moreover, when $d=1$, we prove …

二次非线性薛定谔方程中的不稳定爆破机制

Mechanisms of unstable blowup in a quadratic nonlinear Schrödinger equation

该研究针对二次非线性薛定谔方程实初值全局适定的猜想，找到会引发爆破的反例初值，提出爆破初值集为对应非线性热方程零平衡点稳定流形的猜想，并用参数化方法给出启发性论证。

具非零边界条件的Gros-Pitaevskii和五阶非线性薛定谔方程的整体适定性

Global well-posedness of the Gross--Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions

考虑了空间无穷远处具有非零边界条件的$\R^4$上的Gros-Pitaevskii方程和空间无穷远处具有非零边界条件的$\R^3$上的五阶非线性薛定谔方程.通过将这些方程看作对能量临界NLS的扰动，我们证明了它们在能量空间中是全局适定的。特别地，我们证明了这些方程在能量空间中的无条件唯一性。

Low regularity blowup solutions for the mass-critical NLS in higher dimensions

In this paper, we study the $H^s$-stability of the log-log blowup regime (which has been completely described in a series of recent works by Merle and Raphael) for the focusing mass-critical nonlinear Schrödinger equations $i\partial_tu+Δu+|u|^\frac4du=0$ in $\mathbb{R}^d$ with $d\geq3$. We aim to extend the result in [Colliander and Raphael, Rough blowup solutions to the $L^2$ critical NLS, Math. Anna., 345(2009), 307-366.] for dimension two to the higher dimensions cases $d\geq3$, where we use the bootstrap argument in the above paper and the commutator estimates in [M. Visan and X. Zhang, O…

三维及更高维径向数据的质量临界非线性薛定谔方程

The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher

我们建立了质量临界非线性薛定谔方程$iu_t + Δu = \pm |u|^{4/d} u$解的整体适定性和分散性，其中初始数据为大的球对称L^2_x（R^d），维数为$d\geq 3$。在聚焦的情况下，我们要求质量严格地小于基态的质量。因此，我们得到，在聚焦的情况下，任何球对称爆破解决方案必须集中至少在爆破时的基态的质量。

R^2上三次非线性薛定谔方程的共振分解和I-方法

Resonant decompositions and the I-method for cubic nonlinear Schrodinger on R^2

在$s>1/2$的条件下，证明了平面上三次离焦非线性薛定谔方程$i \partial_t u + Δu = |u|^2 u$的初值问题对于$H^s (\R^2)$中的初值是整体适定的.证明依赖于一个几乎守恒量构造使用多线性校正项。新的主要困难是控制共振相互作用对这些校正项的贡献。共振相互作用是显着的，由于多维设置的问题和一些正交性问题出现。

混合态三次非线性薛定谔方程的半经典极限

Semiclassical limit of cubic nonlinear Schrödinger equations for mixed states

本文证明混合态三次非线性薛定谔方程在半经典极限下收敛到奇异Vlasov方程，并给出量子Penrose稳定性条件，该条件对小数据及散焦情形下的物理相关扰动均成立。

具有双幂非线性项的非线性薛定谔方程驻波的强不稳定性

Strong instability of standing waves for nonlinear Schrödinger equations with double power nonlinearity

证明了一类具有双幂非线性项的非线性薛定谔方程驻波的强不稳定性（爆破不稳定性）。

苏莱曼诺夫-塔拉诺夫自聚焦与聚焦非线性薛定谔方程的层级结构

Suleimanov-Talanov self-focusing and the hierarchy of the focusing nonlinear Schrödinger equation

本文利用半经典孤子系综严格证明了苏莱曼诺夫对塔拉诺夫自聚焦爆破的色散正则化，并将其推广到高阶聚焦非线性薛定谔方程层级，同时分析了扰动导致的梯度突变正则化机制。

具有径向数据的二维立方非线性薛定谔方程

The cubic nonlinear Schrödinger equation in two dimensions with radial data

对于大的球对称L^2_x（\R^2）初始数据，我们建立了质量临界非线性薛定谔方程$iu_t + Δu = \pm |u|^2 u$解的整体适定性和散射;当然，在聚焦的情况下，我们要求质量严格小于基态的质量。因此，我们推断，在聚焦的情况下，任何球对称爆破解决方案必须集中至少在爆破时间的基态的质量。我们还建立了一些部分结果，对类似的索赔在其他方面，没有假设的球对称。

Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators

In this paper, we study the local well-posedness of the cubic Schrödinger equation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I \times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ being an operator of degree $σ\geq 2$. Using estimates in directional spaces, we improve and extend known results for the standard Schrödinger equation (i.e. $\mathscr{L} = Δ$) to any dimension and obtain results under natural assumptions for general $\mathscr{L}$.

双调和非线性Schrodinger方程的奇异解

Singular solutions of the biharmonic Nonlinear Schrodinger equation

我们考虑双调和NLS的奇异解。在L^2-临界情形下，爆破速率受四次方根幂律约束，解接近自相似轮廓，且有限量的L^2-范数（不小于小于临界幂）集中于奇点（“强坍缩”）。在L^2-临界和超临界情形下，我们利用渐近分析和数值模拟来刻画具有峰值型自相似坍缩核的奇异解。在临界情况下，爆破速率略快于四次根，并且自相似轮廓由驻波基态给出。在超临界情况下，爆破速率是一个四次根，自相似轮廓是一个非线性本征值问题的零Hamilton解。使用自适应网格方法对这些发现进行了数值验证（直到10^8的聚焦水平）。我们还计算了二维和三维驻波方程的基态和临界崩溃功率。

Finite time blowup for a supercritical defocusing nonlinear Schrödinger system

We consider the global regularity problem for defocusing nonlinear Schrödinger systems $$ i \partial_t + Δu = (\nabla_{{\bf R}^m} F)(u) + G $$ on Galilean spacetime ${\bf R} \times {\bf R}^d$, where the field $u\colon {\bf R}^{1+d} \to {\bf C}^m$ is vector-valued, $F\colon {\bf C}^m \to {\bf R}$ is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order $p+1$ outside of the unit ball for some exponent $p >1$, and $G: {\bf R} \times {\bf R}^d \to {\bf C}^m$ is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schrödinge…

一维三次耦合薛定谔方程组解的渐近性态

Asymptotic behavior of solutions to the cubic coupled Schrödinger systems in one space dimension

本文研究了一维欧几里得空间中具有小初值的耦合非线性Schr{<$}dinger系统.这样的系统出现在两个不同的光波导之间的耦合的上下文中。我们建立了该系统解的渐近非线性行为和衰减估计。证明使用加藤和Pusateri最近的工作。

L^2-超临界双调和非线性Schrodinger方程的奇异解

Singular solutions of the L^2-supercritical biharmonic Nonlinear Schrodinger equation

利用渐近分析和数值模拟研究了超临界双调和NLS的峰值型奇异解。这些解具有四次方根爆破速率，并以准自相似的普适轮廓坍缩，这是一个四阶非线性本征值问题的零Hamilton解.

短脉冲初始数据对三维半线性Klein-Gordon方程光滑解整体存在性或爆破的临界幂

The critical power of short pulse initial data on the global existence or blowup of smooth solutions to 3-D semilinear Klein-Gordon equations

本文研究三维半线性Klein-Gordon方程短脉冲大初值问题，确定临界幂ν=-1/2：ν≤-1/2时解有限时间爆破，ν>-1/2时整体存在。

The focusing cubic NLS with inverse-square potential in three space dimensions

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.

带δ势的非线性薛定谔方程驻波的强不稳定性

Strong instability of standing waves for nonlinear Schrödinger equations with a delta potential

研究了一维空间中具有吸引δ势和$L^2$ -超临界幂非线性项的非线性薛定谔方程驻波解的强不稳定性（爆破不稳定性）.我们也比较了我们关于强不稳定性的充分条件和一些已知的关于轨道不稳定性的结果。

Modified Scattering of Cubic Nonlinear Schrödinger Equation on Rescaled Waveguide Manifolds

We use modified scattering theory to demonstrate that small-data solutions to the cubic nonlinear Schrödinger equation on rescaled waveguide manifolds, $\mathbb{R} \times \mathbb{T}^d$ for $d\geq 2$, demonstrate boundedness of Sobolev norms as well as weak instability.

$d$ -维聚焦非线性薛定谔方程有限能量解的整体行为

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。

On the focusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space

In this paper, we consider the focusing mass-critical nonlinear fourth-order Schrödinger equation. We prove that blowup solutions to this equation with initial data in $H^γ(\mathbb{R}^d), 5\leq d \leq 7, \frac{56-3d+\sqrt{137d^2+1712d+3136}}{2(2d+32)} <γ<2$ concentrate at least the mass of the ground state at the blowup time. This extends the work in \cite{ZhuYangZhang11} where Zhu-Yang-Zhang studied the formation of singularity for the equation with rough initial data in $\mathbb{R}^4$. We also prove that the equation is globally well-posed with initial data $u_0 \in H^γ(\mathbb{R}^d), 5\leq …

A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces

In this note, we consider the ill-posedness issue for the cubic nonlinear Schrödinger equation. In particular, we prove norm inflation based at every initial condition in negative Sobolev spaces below or at the scaling critical regularity.

On the radius of spatial analyticity for cubic nonlinear Schrödinger equation

It is shown that the uniform radius of spatial analyticity $σ(t)$ of solutions at time $t$ to the 1d, 2d and 3d cubic nonlinear Schrödinger equations cannot decay faster than $1/|t|$ as $|t| \to \infty$, given initial data that is analytic with fixed radius $σ_0$.

On a mixed cubic-superlinear non radially symmetric Schrödinger system - Part I: Existence and uniqueness of solutions

In this paper a nonlinear coupled Schrodinger system in the presence of mixed cubic and superlinear power laws is considered. Focus are made on the steady state solutions of the continuous system for existence and uniqueness by minimizing problem on some associated Nehari manifold.

三维球上具有径向数据的三次非线性薛定谔方程

Cubic nonlinear Schrodinger equation on three dimensional balls with radial data

本文证明了三维球上三次非线性Schrodinger方程在Dirichlet边界条件和径向数据下Cauchy问题的适定性。主要参数是基于双线性特征函数估计和使用$X^{s,b}$空间。最后一部分首次尝试研究非径向情形。我们证明了双线性估计的线性薛定谔流与特定的初始数据。

Modified scattering for the cubic Schrödinger equation small data solution on product space

In this paper we consider the long time behavior of solutions to the cubic nonlinear Schrödinger equation posed on the spatial domain $\mathbb{R}\times\mathbb{T}^{d}$, $1\leq d\leq4$. For sufficiently small, smooth, decaying data we prove global existence and derive modified asymptotic dynamics. We also consider the asymptotic completeness problem.

三次非线性Schrodinger方程的改进的几乎Morawetz估计

Improved almost Morawetz estimates for the cubic nonlinear Schrodinger equation

利用$u_{0} \in H^{s}(\mathbf{R}^{2})$，$s > 1/4$数据，在$\mathbf{R}^{2}$上证明了三次、散焦、非线性Schr{<$}dinger方程的整体适定性.我们通过改进[9]中的几乎Morawetz估计来实现这一点。