$L^2$ blowup solutions of cubic NLS on $\R^2$ concentrate a fixed amount of mass

A determination of the blowup solutions to the focusing NLS with mass equal to the mass of the soliton

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.

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…

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。

色散管理 NLS 的全局存在与有限时间爆破二分法

Global existence versus finite time blowup dichotomy for the dispersion managed NLS

本文研究色散管理非线性 Schrödinger 方程（Gabitov-Turitsyn 方程），针对质量超临界情形（$p>9$），证明了其全局存在与有限时间爆破的二分法。

双调和 NLS 柱对称解的爆破

Blowup of cylindrically symmetric solutions for biharmonic NLS

针对含 Δ² 与 −μΔ 项的双调和非线性 Schrödinger 方程，本文证明质量临界及超临界情形下柱对称初值存在爆破解，将径向对称结果推广至柱对称。

Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS

Let $d\geq 4$ and let $u$ be a global solution to the focusing mass-critical nonlinear Schrödinger equation $iu_t+Δu=-|u|^{\frac 4d}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state $Q$. We prove that if $u$ does not scatter then, up to phase rotation and scaling, $u$ is the solitary wave $e^{it}Q$. Combining this result with that of Merle \cite{merle2}, we obtain that in dimensions $d\geq 4$, the only spherically symmetric minimal-mass blowup solutions are, up to phase rotation and scaling, the pseudo-conformal ground state and the solitary wave.

Minimal-mass blowup solutions of the mass-critical NLS

We consider the minimal mass $m_0$ required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation $iu_t + Δu = μ|u|^{4/d} u$ to blow up. If $m_0$ is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in $L^2_x(\R^d)$ is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, \cite{keraani}, in dimensions 1, 2 and Begout and Vargas, \cite{begout}, in di…

Local Dynamics Near Unstable Branches of NLS Solitons

Consider a branch of unstable solitons of NLS whose linearized operators have one pair of simple real eigenvalues in addition to the zero eigenvalue. Under radial symmetry and standard assumptions, solutions to initial data from a neighbourhood of the branch either converge to a soliton, or exit a larger neighbourhood of the branch transversally. The qualitative dynamic near a branch of unstable solitons is irrespective of whether blowup eventually occurs, which has practical implications for the description of blowup of NLS with supercritical nonlinearity.

关于具有点集中非线性的NLS方程的综述

A general review on the NLS equation with point-concentrated nonlinearity

该论文全面综述了具有点集中非线性的NLS方程的研究，涵盖模型定义、适定性、驻波稳定性、爆破分析及从标准NLS方程的推导。

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

We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrödinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e. the ones describing self-action and non-linear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonli…

高维 $d\geq 3$ 中质量临界 NLS 的粗糙对数-对数爆破解

Rough log-log blowup solutions to mass-critical NLS in higher dimensions $d\geq 3$

研究聚焦质量临界非线性薛定谔方程在小扰动下对数-对数爆破解的稳定性，移除先前维数限制，将稳定性推广到所有次临界指数范围。

The unconditional uniqueness for the energy-supercritical NLS

We consider the cubic and quintic nonlinear Schrödinger equations (NLS) under the $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [18,19], the unconditional uniqueness problems for $H^{1}$-critical and $H^{1}$-supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [54…

色散或非线性管理下NLS方程的爆破解

On blowup solution in NLS equation under dispersion or nonlinearity management

研究色散或非线性管理下NLS方程的爆破解，证明DM-NLS在T>1时全局适定，NM-NLS在T∈(1,2)全局适定且可构造任意聚焦层爆破解，辅以数值探索。

略超临界 NLS 自相似爆破的模态稳定性：I. 低能谱

Mode stability for self-similar blowup of slightly supercritical NLS: I. low-energy spectrum

本文研究了略质量超临界 NLS 自相似爆破的渐近稳定性，将其归结为矩阵线性化算子的模态稳定性，并利用 Jost 函数论证、WKB 分析和匹配渐近方法证明了低能谱的模态稳定性。

略超临界NLS自相似爆破的模态稳定性：II. 高能谱

Mode stability for self-similar blowup of slightly supercritical NLS: II. high-energy spectrum

本文利用线性Liouville论证证明了略超临界NLS在1至10维下的高能模态稳定性，完成了自相似爆破的渐近稳定性证明，并刻画了质量临界基态附近的线性化算子谱。

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.

阈值动力学对于四维质量-能量双重临界NLS

Threshold dynamics for the 4$d$ mass-energy double critical NLS

本文研究了四维质量-能量双重临界非线性薛定谔方程的阈值动力学，证明了在能量阈值处散射/破裂二元选择性依然成立。

关于质量超临界 NLS 自相似爆破的稳定性

On stability of self-similar blowup for mass supercritical NLS

本文证明了质量超临界 NLS 中一大类自相似爆破剖面的有限余维非线性渐近稳定性，核心是通过推导变形拉普拉斯算子的 Strichartz 色散估计来实现。

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.

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

We consider the focusing mass-critical nonlinear Schrödinger equation and prove that blowup solutions to this equation with initial data in $H^s(\R^d)$, $s > s_0(d)$ and $d\geq 3$, concentrate at least the mass of the ground state at the blowup time. This extends recent work by J. Colliander, S. Raynor, C. Sulem, and J. D. Wright, \cite{crsw}, T. Hmidi and S. Keraani, \cite{hk}, and N. Tzirakis, \cite{tzirakis}, on the blowup of the two-dimensional and one-dimensional mass-critical focusing NLS below the energy space to all dimensions $d\ge 3$.

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

We consider finite time blowup solutions of the $L^2$-critical cubic focusing nonlinear Schrödinger equation on $\R^2$. Such functions, when in $H^1$, are known to concentrate a fixed $L^2$-mass (the mass of the ground state) at the point of blowup. Blowup solutions from initial data that is only in $L^2$ are known to concentrate at least a small amount of mass. In this paper we consider the intermediate case of blowup solutions from initial data in $H^s$, with $1 > s > s_Q$, where $s_Q \le \sQ$. Our main result is that such solutions, when radially symmetric, concentrate at least the mass of …

Global Behavior Of Finite Energy Solutions To The $d$-Dimensional Focusing Nonlinear Schrödinger Equation

We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation (NLS), $i \partial_t u+Δu+ |u|^{p-1}u=0, $ with initial data $u_0\in H^1,\; x \in R^n$. The nonlinearity power $p$ and the dimension $d$ are such that the scaling index $s=\frac{d}2-\frac2{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with $\ME[u_0]<1$ ($\ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Na…

On mass - critical NLS with local and non-local nonlinearities

We consider the following nonlinear Schrödinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+Δu+|u|^\frac{4}{3}u+μ\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where $μ>0$ is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state $Q_μ$. In particular, we develop an appropriate perturbation approach to prove the radial non-degeneracy property and then obtain the general non-degeneracy of the ground state $Q_μ$. We then show the existence of finite time blowup solution with minimal mass $\|u_0\|…

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。

On the focusing energy-critical inhomogeneous NLS: weighted space approach

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 and blowup for $L^{2}$-supercritical and $\dot{H}^{2}$-subcritical biharmonic NLS with potentials

We mainly consider the focusing biharmonic Schrödinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(Δ^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {\bf{R}\times{\bf{R}}^{N}}, u(0, x)=u_{0}(x)\in H^{2}({\bf{R}}^{N}), \end{aligned}\right. \end{equation*} If $N>8$, \ $1+\frac{8}{N}<p<1+\frac{8}{N-4}$ (i.e. the $L^{2}$-supercritical and $\dot{H}^{2}$-subcritical case ), and $\langle x\rangle^β\big(|V(x)|+|\nabla V(x)|\big)\in L^\infty$ for some $β>N+4$, then we firstly prove a global well-posedness and scattering result for the radial data $u_0\in …

具有斯塔克效应的阻尼非线性薛定谔方程

Damped nonlinear Schrödinger equation with Stark effect

本研究针对带斯塔克势的$L^2$-临界阻尼非线性薛定谔方程，确定了其全局存在与有限时间爆破的阈值，证明了对数-对数爆破速度的结构稳定性，证明基于Avron-Herbst公式与无扰动阻尼NLS的相关结果。

Well-posedness and scattering of inhomogeneous cubic-quintic NLS

In this paper we consider inhomogeneous cubic-quintic NLS in space dimension $d = 3$: $$ iu_t = -Δu + K_1(x)|u|^2u + K_2(x)|u|^4u. $$ We study local well-posedness, finite time blowup, and small data scattering and non-scattering for the ICQNLS when $K_1, K_2 \in C^4(\mathbb R^3 \setminus \{0\})$ satisfy growth condition $|\partial^j K_i(x)| \lesssim |x|^{b_i-j}\, (j = 0, 1, 2, 3, 4)$ for some $b_i \ge 0$ and for $x \neq 0$. To this end we use the Sobolev inequality for the functions $f \in H^n \,(n = 1, 2)$ such that $\||\mathbf L|^\ell f\|_{H^n} < \infty \,(\ell = 1, 2)$, where $\mathbf L$ i…

Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus

We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schrödinger equation on the one-dimensional torus. In an influential paper, Lebowitz, Rose, and Speer (1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit disc. In this case, we answer a question posed by Bourgain and Bulut (2014) on the optimal mass threshold. Furthermore, …

Strongly interacting blow up bubbles for the mass critical NLS

We construct a new class of multi-solitary wave solutions for the mass critical two dimensional nonlinear Schrodinger equation (NLS). Given any integer K>1, there exists a global (for positive time) solution of (NLS) that decomposes asymptotically into a sum of solitary waves centered at the vertices of a K-sided regular polygon and concentrating at a logarithmic rate in large time. This solution blows up in infinite time with logarithmic rate. Using the pseudo-conformal transform, this yields the first example of solution blowing up in finite time with a rate strictly above the pseudo-conform…

临界间非均匀NLS的阈值解

Threshold solutions for the intercritical inhomogeneous NLS

本文研究三维空间中聚焦非均匀NLS方程的质能阈值解，证明了趋近驻波且负时间方向爆破或散射的特解$Q^\pm$的存在性，并完成了所有阈值解的行为分类。

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

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation $iu_t + Δu = \pm |u|^{4/d} u$ for large spherically symmetric L^2_x(R^d) initial data in dimensions $d\geq 3$. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.

The cubic nonlinear Schrödinger equation in two dimensions with radial data

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation $iu_t + Δu = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of sphe…

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 …

Invariant measures and the soliton resolution conjecture

The soliton resolution conjecture for the focusing nonlinear Schrodinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental open problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. This paper proves a "statistical version" of this conjecture at mass-subcr…

NLS方程在L²约束下的渐近球体集中于无穷远

Asymptotic Sphere Concentration at Infinity for NLS with L^2 Constraint

研究NLS方程在L²约束下的高维球体在无穷远处的集中现象，通过定制约简和吹胀分析扩展了二维质量临界结果。

空间维度 $n\ge 3$ 下抛物-椭圆型 Keller-Segel 系统的爆破轮廓

Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $n\ge 3$

本文研究了空间维度 $n\ge 3$ 下抛物-椭圆型 Keller-Segel 系统径向递减解的爆破渐近行为，证明了最终轮廓满足 $C_1|x|^{-2}\le u(x,T)\le C_2|x|^{-2}$，并获得了 I 型爆破解的精细时空估计，揭示了该现象的普遍性。