Scipio Cuccagna
We consider radial solutions of a mass supercritical monic NLS and we prove the existence of a set, which looks like a hypersurface, in the space of finite energy functions, invariant for the flow and formed by solutions which converge to ground states.
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Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as $t\to +\infty$, were constructed in previous works for the L2 critical and subcritical (NLS) and (gKdV) equations. In this paper, we extend the construction of multi-soliton solutions to the L2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.
We consider two classes of defocusing energy-supercritical nonlinear Schrödinger equations in dimensions $d\geq 5$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot H^{s_c}_x$, then $u$ is global and scatters.
In this paper we study the existence and stability of normalized standing waves for the nonlinear Schrödinger equation on a general starlike graph with potentials. Under general assumptions on the graph and the potential, we show the existence of orbitally stable standing waves when the nonlinearity is $L^{2}$-critical and supercritical.
Infinitely many normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs with localized nonlinearities
该研究针对带局部非线性的非紧度量图上的非线性薛定谔方程，在$L^2$-超临界 regime 下证明了对任意给定质量都存在无穷多归一化解，拓展了相关方程解的存在性结论。
This paper is devoted to the existence of non-trivial bound states of prescribed mass for the mass-supercritical nonlinear Schrödinger equation on compact metric graphs. The investigation is based upon a general variational principle which combines the monotonicity trick and a min-max theorem with second order information, and upon the blow-up analysis of bound states with prescribed mass and bounded Morse index.
We consider the radial defocusing nonlinear Schrödinger equations $iu_t+Δu=|u|^{p}u$ with supercritical exponent $p>4$ in four space dimensions, and prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter.
We consider a class of defocusing energy-supercritical nonlinear Schrödinger equations in four space dimensions. Following a concentration-compactness approach, we show that for $1<s_c<3/2$, any solution that remains bounded in the critical Sobolev space $\dot{H}_x^{s_c}(\R^4)$ must be global and scatter. Key ingredients in the proof include a long-time Strichartz estimate and a frequency-localized interaction Morawetz inequality.
We investigate the existence of solutions to the fractional nonlinear Schrödinger equation $(-Δ)^s u = f(u)$ with prescribed $L^2$-norm $\int_{\mathbb{R}^N} |u|^2 \, dx =m$ in the Sobolev space $H^s(\mathbb{R}^N)$. Under fairly general assumptions on the nonlinearity $f$, we prove the existence of a ground state solution and a multiplicity result in the radially symmetric case.
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 consider the focusing nonlinear Schrödinger equations $i\partial_t u+Δu +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$ supercritical nonlinearities $p_c<p<(1+\e)p_c$ with $p_c=1+\frac{4}{N}$ and $0<\e\ll 1$. We prove the existence and stability in the energy space $H^1$ of a self similar finite time blow up dynamics and provide a qualitative description of the singularity formation near the blow up time
Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs with localized nonlinearities
本文针对非紧度量图上带局部非线性项的NLS方程，在$L^2$超临界情形下证明了任意给定质量的规范化解存在，所用方法可推广至更一般方程。
We investigate the $L^2$-supercritical and $\dot{H}^1$-subcritical nonlinear Schrödinger equation in $H^1$. In \cite{G1} and \cite{yuan}, the mass-energy quantity $M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$ has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present paper, we study the dynamics at the critical level $M(u)^{\frac{1-s_{c}}{s_{c}}}E(u)=M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$ and classify the corresponding solutions using modulation theory, non-trivially generalize the results obtained in \cite{holmer3} for the 3D cubic Schrödinger equation.
Sign-changing prescribed mass solutions for $L^2$-supercritical NLS on compact metric graphs
本文研究了在紧致度量图上临界质量非线性薛定谔方程的多变号解的存在性，首次在超临界质量范围内获得预设质量解的多重性结果，并证明了线性算子的特征值均为分支点。
We consider the focusing mass supercritical nonlinear Schrödinger equation with rotation \begin{equation*} iu_{t}=-\frac{1}{2}Δu+\frac{1}{2}V(x)u-|u|^{p-1}u+L_Ωu,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}, \end{equation*} where $N=2$ or $3$ and $V(x)$ is an anisotropic harmonic potential. Here $L_Ω$ is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stabl…
This paper studies the quintic nonlinear Schrödinger equation on $\mathbb{R}^d$ with randomized initial data below the critical regularity $H^{\frac{d-1}{2}}$. The main result is a proof of almost sure local well-posedness given a Wiener Randomization of the data in $H^s$ for $s \in (\frac{d-2}{2}, \frac{d-1}{2})$. The argument further develops the techniques introduced in the work of Á. Bényi, T. Oh and O. Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global well-posedness.
Normalized ground state solutions for the fractional Sobolev critical NLSE with an extra mass supercritical nonlinearity
针对含质量超临界项的分数阶 Sobolev 临界非线性 Schrödinger 方程，本文借助 Soave 的紧性技巧证明归一化基态解存在，并推广了局部与非局部情形的已有结果。
We consider the focusing $\dot H^{s_c}$-critical biharmonic Schrödinger equation, and prove a global wellposedness and scattering result for the radial data $u_0\in H^2(\mathbb R^N)$ satisfying $ M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E(Q) $ and $ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{2}\|Δu_{0}\|_{2}<\|Q\|^{\frac{2-s_c}{s_c}}_{2}\|ΔQ\|_{2}, $ where $s_c\in(0,2)$ and $Q$ is the ground state of $Δ^2Q+(2-s_c)Q-|Q|^{p-1}Q=0$.
We study solutions of a semilinear elliptic equation with prescribed mass and Dirichlet homogeneous boundary conditions in the unitary ball. Such problem arises in the search of solitary wave solutions for nonlinear Schrödinger equations (NLS) with Sobolev subcritical power nonlinearity on bounded domains. Necessary and sufficient conditions are provided for the existence of such solutions. Moreover, we show that standing waves associated to least energy solutions are always orbitally stable when the nonlinearity is L^2-critical and subcritical, while they are almost always stable in the L^2-s…
We investigate the initial value problem for a defocusing nonlinear Schrödingerequation with exponential nonlinearity. We identify subcritical, critical and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.
Multiplicity result for a mass supercritical NLS with a partial confinement
本文研究 $\n\mathbb{R}^3$ 中带部分约束的质量超临界 NLS，通过有界域解序列取极限并利用其对称性，非微扰地证明了山路能量水平上第二解的存在。
We show that all supercritical monic focusing NLS in one space dimension exhibit asymptotic stability of perturbed standing waves provided the perturbations are chosen on a small hypersuface in a suitable space.
We study the Cauchy problem for NLS with a class of $H^s$-super-critical data \begin{align} & {\rm i}u_t +Δu+ λ|u|^{2κ} u =0, \quad u(0)=u_0 \label{NLSabstract} \end{align} and show that \eqref{NLSabstract} is globally well-posed and scattering in $α$-modulation spaces $M^{s,α}_{2,1}$ ($α\in [0,1), \ s> dα/2-α/κ$, $κ\in \mathbb{N}$ and $κ\geq 2/d$) for the sufficiently small data. Moreover, NLS is ill-posed in $M^{s,α}_{2,1}$ if $s< dα/2-α/κ$. In particular, we obtain a class of initial data $u_0$ satisfying for any $M\gg 1$, \begin{align} \|u_0\|_2 \sim M^{1/κ-d/2 }, \ \ \|u_0\|_\infty \ =\in…
We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. "Point" means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of solution, blow-up occurrence and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e, we show that the global solution scatters…
Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs
本文基于 mountain pass 技术研究非紧度量图上 $L^2$-超临界非线性 Schrödinger 方程的归一化解，证明了小质量下正解的存在性并揭示了拓扑结构的作用。
On collapsing ring blow up solutions to the mass supercritical NLS
针对质量超临界且能量次临界范围的非线性Schrödinger方程，本文证明了径向对称初始数据下爆破速度的普适上界，并证明该上界精确且在塌缩环爆破解族上达到。
We investigate the nonlinear Schrödinger equation $iu_{t}+Δu+|u|^{p-1}u=0$ with $1+\frac{4}{N}<p<1+\frac{4}{N-2}$ (when $N=1, 2$, $1+\frac{4}{N}<p<\infty$) in energy space $H^1$ and study the divergent property of infinite-variance and nonradial solutions. If $M(u)^{\frac{1-s_{c}}{s_{c}}}E(u)<M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$ and $\|u_{0}\|_{2}^{\frac{1-s_{c}}{s_{c}}}\|\nabla u_{0}\|_{2}>\|Q\|_{2}^{\frac{1-s_{c}}{s_{c}}}\|\nabla Q\|_{2},$ then either $u(t)$~blows up in finite forward time, or $u(t)$ exists globally for positive time and there exists a time sequence $t_{n}\rightarrow+\infty$ su…
We study stable blow-up dynamics in the $L^2$-supercritical nonlinear Schrödinger equation in various dimensions. We first investigate the profile equation and extend the result of X.-P. Wang [38] and Budd et al. [4] on the existence and local uniqueness of solutions of the cubic profile equation to other $L^2$-supercritical nonlinearities and dimensions $d \geq 2$. We then numerically observe the multi-bump structure of such solutions, and in particular, exhibit the $Q_{1,0}$ solution, a candidate for the stable blow-up profile. Next, using the dynamic rescaling method, we investigate stable …
Unconditional Uniqueness for the Energy-critical and Energy-supercritical Quadratic NLS
研究能量临界和超临界二次非线性薛定谔方程，通过引入新的无限二次层级，利用其与双线性\(U\)-\(V\)估计兼容的组合结构，以二次层级方法在临界正则性下建立无条件唯一性，回答布尔甘的唯一性问题。
We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are $L^2$-supercritical, in particular we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC.
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。
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…
We consider the energy super critical nonlinear Schrödinger equation $$i\pa_tu+Δu+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical regime, we construct a family of smooth finite time blow up solutions which become singular via concentration of a universal profile with the so called type II quantized blow up rates. The essential feature of these solutions is that all norms below scaling remain bounded. Our analysis fully revisits the construction of type II blow up solutions for the corresponding he…
In this paper, we present two almost sure global well-posedness (GWP) results for the energy supercritical nonlinear Schrödinger equations (NLS) on the unit ball of $\Bbb R^3$ using two different approaches. First, for the NLS with algebraic nonlinearities with the subcritical initial data, we show the almost sure global well-posedness and the invariance of the underlying measures, and establish controls on the growth of Sobolev norms of the solutions. This global result is based on a deterministic local theory and a probabilistic globalization. Second, for the NLS with generic power nonlinear…
Global well-posedness and scattering for the defocusing energy supercritical NLS in high dimensions
针对高维散焦能量超临界非线性薛定谔方程，本文通过建立新的非线性估计改进了临界正则性指标s_c的上界，拓展了全局适定性与散射结果的适用范围。
We present numerical simulations of the defocusing nonlinear Schrodinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig-Merle and Kilip-Visan who considered some energy supercritical wave equations and proved that if the solution is {a priori} bounded in the critical Sobolev space (i.e. the space whose homogeneous norm is invariant under the scaling leaving the equation invariant), then it exists for all time and scatters. In this paper, we numerically investigate the boundedness of the $H^2$-critical Sobolev norm for soluti…
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 investigate the NLS equation with competing Hartree-type and power-type nonlinearities \begin{equation*} \begin{array}{ll} i\partial _{t}ψ+Δψ+γ(I_{α}\ast |ψ|^{p})|ψ|^{p-2}ψ+μ|ψ|^{q-2}ψ=0, & \text{ }\forall (t,x)\in \mathbb{R\times R}^{N},% \end{array}% \end{equation*}% where $γμ<0$. We establish conditions for the local well-posedness in the energy space. Under the double $L^{2}$-supercritical case, we prove the existence and multiplicity of standing waves with prescribed mass by developing a constraint method when $γ<0,μ>0$ and $γ>0,μ<0, $ respectively. Moreover, we prove weak orbital stab…
The defocusing energy-supercritical inhomogeneous NLS in four space dimension
本文研究四维空间中散焦能量超临界非齐次非线性薛定谔方程的全局适定性与散射，利用集中紧性/刚性框架及多种估计证明临界 Sobolev 空间中的先验界可推出解的全局性与散射。
Normalized Solutions of the $L^2$-Supercritical NLS Equation on Noncompact Metric Graphs with a Short-Range Potential
本文研究带短程势的非紧度量图上L²-超临界非线性薛定谔方程，结合山路几何等方法证明了归一化解的存在性，是该领域的首次相关结果。
On the formation of singularities for the slightly supercritical NLS equation with nonlinear damping
本文研究了带非线性阻尼的聚焦质量超临界NLS方程，给出了有限时间爆破的充分条件，证明了阻尼项的非正则化效应，并通过调制分析证明了自相似爆破结构的近似保留。
Numerical evidence for singularity formation in defocusing fractional NLS in one space dimension
针对含分数次拉普拉斯算子和散焦非线性项的薛定谔型方程，通过小能量超临界s的数值模拟，为解中新型高振荡奇性形成提供数值证据。
In this paper we study radial solutions of certain two-dimensional nonlinear Schrödinger equation with harmonic potential, which is supercritical with respect to the initial data. By combining the nonlinear smoothing effect of Schr ödinger equation with Lp estimates of Laguerre functions, we are able to prove an almost sure global well-posedness result and the invariance of the Gibbs measure. We also discuss an application to the NLS equation without harmonic potential.
For the 3D focusing cubic nonlinear Schrodinger equation, Scattering of $H^1$ solutions inside the (scale invariant) potential well was established by Holmer and Roudenko~\cite{HR2} (radial case) and Duyckaerts, Holmer and Roudenko~\cite{DHR} (general case). In this paper, we extend this result to arbitrary space dimensions and focusing, mass-supercritical and energy-subcritical power nonlinearities, by adapting the method of~\cite{DHR}.
Solutions with prescribed mass for $L^2$-supercritical NLS equations under Neumann boundary conditions
本文研究Neumann边界条件下带指定质量约束的$L^2$-超临界非线性薛定谔方程，通过结合极小极大原理、Morse指标与新型爆破分析，证明了山路型规范化解的存在性，统一处理了多种非线性项情形。
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…
We consider the defocusing energy-critical NLS in the exterior of the unit ball in three dimensions. For the initial value problem with Dirichlet boundary condition we prove global well-posedness and scattering with large radial initial data in the Sobolev space $\dot H_0^1$. We also point out that the same strategy can be used to treat the energy-supercritical NLS in the exterior of balls with Dirichlet boundary condition and radial $\dot H_0^1$ initial data.
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.
Normalized solutions of $L^2$-supercritical NLS equations with periodic potentials
本文研究带周期势与局域非线性的L²超临界非线性薛定谔方程的归一化解，结合单调性技巧等方法证明小μ>0时解存在，f在原点附近具质量超临界增长时所有μ>0时解存在。
We prove large-data scattering and existence of wave operators in the energy space for the systems of $N$ defocusing fourth-order Schrödinger equations with mass-supercritical and energy-subcritical power-type nonlinearity. In addition, new nonlinear interaction Morawetz identities and inequalities are given, suitable to shed lights on the decay of the solution with respect some Lebesgue norms when the space dimensions are $d=3,4$.
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。
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 …
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。
We study the existence and stability of standing waves associated to the Cauchy problem for the nonlinear Schrödinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an effective model describing the attractive Bose-Einstein condensation in a magnetic trap rotating with an angular velocity. By viewing the equation as NLS with a constant magnetic field and with (or without) a partial harmonic confinement, we establish the existence and orbital stability of prescribed mass standing waves for the equation with mass-subcritical, …
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 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 obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2, which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surpris…
We prove $H^1$ scattering for defocusing NLS with a delta potential and mass-supercritical nonlinearity, hence extending in an inhomogeneous setting the classical 1-D scattering results first proved by Nakanishi in the translation invariant case.
We consider the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b}|u|^αu = 0\quad\text{on}\quad\mathbb{R}\times\mathbb{R}^N, \] with $N\geq 2$, $0<b<\min\{\tfrac{N}{2},2\}$, and $\tfrac{4-2b}{N}<α<\tfrac{4-2b}{N-2}$. These constraints make the equation mass-supercritical and energy-subcritical. We extend the results of Farah-Guzmán and Miao-Murphy-Zheng and prove scattering below the ground state with general initial data.
Normalized solutions of $L^2$ supercritical NLS equations in exterior domains with inhomogeneous nonlinearities
本文研究了非均匀非线性情况下，$L^2$超临界非线性薛定谔方程在外部领域中归一化山峰解的存在性，通过新型极小-极大论证揭示了非自主非线性在非紧致领域中的新现象。
Global well-posedness for the defocusing 3D quadratic NLS in the sharp critical space
针对3D散焦二次非线性薛定谔方程，本文基于伪共形能量几乎守恒的方法，移除了径向初值下尖锐临界加权空间中全局适定性所需的先验有界假设，完善了相关理论结果。
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