Benjamin Dodson
In this paper we generalize a weak sequential result of \cite{fan20182} to any non-scattering solutions in one dimension. No symmetry assumptions are required for the initial data.
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In this paper we generalize a weak sequential result of \cite{fan20182} to a non-scattering solutions in dimension $d \geq 2$. No symmetry assumptions are required for the initial data. We build on a previous result of \cite{dodson20202} for one dimension.
We study the non-scattering $L^{2}$ solution $u$ to the radial mass critical nonlinear Schrödinger equation with mass just above the ground state, and show that there exists a time sequence $\{t_{n}\}_{n}$, such that $u(t_{n})$ weakly converges to the ground state $Q$ up to scaling and phase transformation. We also give some partial results on the mass concentration of the minimal mass blow up solution.
In this paper, we study the nonlinear Schrödinger equation with focusing point nonlinearity in dimension one. First, we establish a scattering criterion for the equation based on Kenig-Merle's compactness-rigidity argument. Then we prove the energy scattering below and above the mass-energy threshold. We also describe the dynamics of solutions with data at the ground state threshold. Finally, we prove a blow-up criteria for the equation with initial data with arbitrarily large energy.
We study the focusing nonlinear Schrödinger equation in the $L^2$-supercritical regime with finite energy and finite variance initial data. We investigate solutions above the energy (or mass-energy) threshold. In our first result, we extend the known scattering versus blow-up dichotomy above that threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, obtaining scattering and blow-up criteria for solutions with arbitrarily large mass and energy. As a consequence, we characterize the behavior of the ground state initial data modulated by a quadratic phase.…
We study the final-state problem for the mass-subcritical NLS above the Strauss exponent. For $u_+\in L^2$, we perform a physical-space randomization, yielding random final states $u_+^ω\in L^2$. We show that for almost every $ω$, there exists a unique, global solution to NLS that scatters to $u_+^ω$. This complements the deterministic result of Nakanishi, which proved the existence (but not necessarily uniqueness) of solutions scattering to prescribed $L^2$ final states.
We study the nature of the Nonlinear Schrödinger equation ground states on the product spaces $\R^n\times M^k$, where $M^k$ is a compact Riemannian manifold. We prove that for small $L^2$ masses the ground states coincide with the corresponding $\R^n$ ground states. We also prove that above a critical mass the ground states have nontrivial $M^k$ dependence. Finally, we address the Cauchy problem issue which transform the variational analysis to dynamical stability results.
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…
We extend our previous result on the nonlinear Klein-Gordon equation to the nonlinear Schrodinger equation with the focusing cubic nonlinearity in three dimensions, for radial data of energy at most slightly above that of the ground state. We prove that the initial data set splits into nine nonempty, pairwise disjoint regions which are characterized by the distinct behaviors of the solution for large time: blow-up, scattering to 0, or scattering to the family of ground states generated by the phase and scaling freedom. Solutions of this latter type form a smooth center-stable manifold, which c…
We consider the focusing mass-supercritical and energy-subcritical nonlinear Schrödinger equation (NLS). We are interested in the global behavior of the solutions to (NLS) with group invariance. By the group invariance, we can determine the global behavior of the solutions above the ground state standing waves.
We consider the nonlinear Schrödinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above proper…
We study the scattering problem for the nonlinear Schrödinger equation $i\partial_t u + Δu = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $Σ$ holds and the wave operator is well-defined on $Σ$. We show that there exists $0<β<p$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+β}$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in t…
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…
In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: \begin{equation*} \begin{cases} (-Δ)^{s}u=λu +μ|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u,&x\in\mathbb{R}^{N}, \int_{\mathbb{R}^{N}}u^{2}dx=a^{2},\\ \end{cases} \end{equation*} where $(-Δ)^{s}$ is the fractional Laplacian, $0<s<1$, $N>2s$, $2<q<2_{s}^{\ast}=2N/(N-2s)$ is a fractional critical Sobolev exponent, $a>0$, $μ\in \mathbb{R}$. By using Jeanjean's trick in \cite{Jeanjean}, and the standard method which can be found in \cite{Brezis} to overcome the lack of compactness, we firs…
We prove that if $u(t)$ is a log-log blow-up solution, of the type studied by Merle-Raphaël (2001-2005), to the $L^2$ critical focusing NLS equation $i\partial_t u +Δu + |u|^{4/d} u=0$ with initial data $u_0\in H^1(\mathbb{R}^d)$ in the cases $d=1, 2$, then $u(t)$ remains bounded in $H^1$ away from the blow-up point. This is obtained without assuming that the initial data $u_0$ has any regularity beyond $H^1(\mathbb{R}^d)$. As an application of the $d=1$ result, we construct an open subset of initial data in the radial energy space $H^1_{rad}(\mathbb{R}^3)$ with corresponding solutions that bl…
Blow-up of radially symmetric solutions for a cubic NLS type system in dimension 4
针对四维能量临界的三次非线性 Schrödinger 系统，本文利用凸性方法证明低初始能量、高动能的径向对称解有限时间爆破，并通过集中紧性与变分方法建立基态及最佳 Sobolev 常数。
Norm inflation for the cubic hyperbolic NLS on $\mathbb T^2$
研究 $\mathbb T^2$ 上三次双曲非线性薛定谔方程在 $H^s$ 空间中的范数膨胀，证明对 $s\in(-\infty,0)\cup(0,\frac12]$ 发生范数膨胀，揭示了标度临界点以下和以上两种不同机制导致的强不适定性。
We study the nonlinear Schrödinger equation with $δ'_s$ coupling of intensity $β\in\mathbb{R}\setminus\{0\}$ on the star graph $Γ$ consisting of $N$ half-lines. The nonlinearity has the form $g(u)=|u|^{p-1}u, p>1.$ In the first part of the paper, under certain restriction on $β$, we prove the existence of the ground state solution as a minimizer of the action functional $S_ω$ on the Nehari manifold. It appears that the family of critical points which contains a ground state consists of $N$ profiles (one symmetric and $N-1$ asymmetric). In particular, for the attractive $δ'_s$ coupling ($β<0$) …
On a star graph made of $N \geq 3$ halflines (edges) we consider a Schrödinger equation with a subcritical power-type nonlinearity and an attractive delta interaction located at the vertex. From previous works it is known that there exists a family of standing waves, symmetric with respect to the exchange of edges, that can be parametrized by the mass (or $L^2$-norm) of its elements. Furthermore, if the mass is small enough, then the corresponding symmetric standing wave is a ground state and, consequently, it is orbitally stable. On the other hand, if the mass is above a threshold value, then…
Scattering and Blow-Up in Both Time Directions Above the Ground State for the Focusing Nonlinear Schrödinger Equation
本文针对临界间聚焦NLS方程，构造了基态以上质量-能量任意大且不由伽利略变换产生的双时间方向散射解，还得到双时间爆破新判据与单时间散射、爆破的改进充分条件。
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。
Sharp bilinear eigenfunction estimate, $L^\infty_{x_2}L^p_{t,x_1}$-type Strichartz estimate, and energy-critical NLS
本文通过利用球面的表示论，建立了三维球面上的精确双线性特征函数估计，并改进了各向异性Strichartz估计，从而获得了能量临界NLS在能量空间中的小数据全局适定性。
Global dynamics above the ground state for the energy-critical Hartree equation with radial data
本文研究径向数据下能量临界 Hartree 方程在能量略高于基态时的全局动力学，通过分析基态附近的双曲动力学行为和单次通过引理，将 Nakanishi 和 Schlag 在次临界非线性 Klein-Gordon 方程中的方法推广到能量临界情形，从而对径向解进行完整分类。
A General and Unified Method to prove the Uniqueness of Ground State Solutions and the Existence/Non-existence, and Multiplicity of Normalized Solutions with applications to various NLS
本文提出了一般统一的抽象框架，证明了一大类PDE基态解的唯一性与非退化性，并研究了归一化解的存在性、非存在性及多重性，成功将其推广至具非自治非线性的分数阶非线性薛定谔方程，进而分析了驻波的轨道稳定性。
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.
Blowup rate for rotational NLS with a repulsive potential
本文研究了具有旋转和排斥势的临界NLS方程的爆破速率，通过virial恒等式和R_γ变换证明了『log-log』爆破速率，并发现增加排斥势强度可使方程具有全局解，同时提供了数值模拟验证。
Normalized solutions for the NLS equation with mixed fractional Laplacians and combined nonlinearities
本文研究了具有混合分数阶拉普拉斯算子和组合非线性项的NLS方程的归一化解，证明了在次临界和临界指数情形下基态解及山路型解的存在性，推广了已有结果。
We consider time global behavior of solutions to the focusing mass-subcritical NLS equation in weighted $L^2$ space. We prove that there exists a threshold solution such that (i) it does not scatter; (ii) with respect to a certain scale-invariant quantity, this solution attains minimum value in all non-scattering solutions. In the mass-critical case, it is known that ground states are this kind of threshold solution. However, in our case, it turns out that the above threshold solution is not a standing-wave solution.
Well-posedness and invariant measures for complex valued modified KdV equation
本文研究一维周期复值mKdV方程，构造了支撑在递增正则性Sobolev空间上的不变测度序列，并指出复值情形比实值情形更难分析。
We consider a nonlinear Schrödinger equation with focusing nonlinearity of power type on a star graph ${\mathcal G}$, written as $ i \partial_t Ψ(t) = H Ψ(t) - |Ψ(t)|^{2μ}Ψ(t)$, where $H$ is the selfadjoint operator which defines the linear dynamics on the graph with an attractive $δ$ interaction, with strength $α< 0$, at the vertex. The mass and energy functionals are conserved by the flow. We show that for $0<μ<2$ the energy at fixed mass is bounded from below and that for every mass $m$ below a critical mass $m^*$ it attains its minimum value at a certain $\hat Ψ_m \in H^1(\GG) $, while for…
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