Benjamin Dodson
We prove global well-posedness for the cubic, defocusing, nonlinear Schr{ö}dinger equation on $\mathbf{R}^{2}$ with data $u_{0} \in H^{s}(\mathbf{R}^{2})$, $s > 1/4$. We accomplish this by improving the almost Morawetz estimates in [9].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Bootstrapped Morawetz Estimates And Resonant Decomposition For Low Regularity Global Solutions Of Cubic NLS On R^{2}
利用数据u_{0} \in H^{s}（R^{2}）证明了R^{2}上L^{2}临界三次散焦非线性薛定谔方程的整体适定性，其中s > {1/3}.
We establish an interaction Morawetz estimate for the magnetic Schrödinger equation for $n\geq 3$ under certain smallness conditions on the gauge potentials, but with almost optimal decay. As an application, we prove global wellposedness and scattering in $H^1$ for the cubic defocusing magnetic Schrödinger equation for $n = 3.$
Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $\R^2$
证明了二维$L^2-critical$散焦非线性薛定谔方程（NLS）在低正则性数据下的整体适定性.更精确地说，我们表明，一个全球性的解决方案存在的Sobolev空间中的初始数据$H^{s}(\mathbb R^2)$和任何$s>{2/5}$。这改进了方和Grillakis以前的结果，其中建立了任何$s \geq {1/2}$的全局适定性。我们使用$I$-方法来利用方程的守恒律。新的成分是一个相互作用Morawetz估计类似的一个已被用来获得全球适定性和散射的立方NLS在3D。在我们的情况下，估计的推导是技术性的，因为解$Iu$的平滑版本在相互作用Morawetz不等式中引入了误差项。该方法的一个副产品是解的$H^{s}$范数服从多项式时间边界。
We study the defocusing nonlinear Schrödinger equation in three space dimensions. We prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter. In the energy-supercritical setting, we employ a space-localized Lin--Strauss Morawetz inequality of Bourgain. In the inter-critical regime, we prove long-time Strichartz estimates and frequency-localized Lin--Strauss Morawetz inequalities.
Decay of solutions to one-dimensional inhomogeneous nonlinear Schrödinger equations
研究一类一维非齐次非线性Schrödinger方程整体解的衰减估计，通过建立局部化Virial-Morawetz恒等式导出衰减结果并应用于能量散射准则，同时推广至含反幂型与Yukawa型外部势的方程。
In this paper, we study the scattering theory for the cubic inhomogeneous Schrödinger equations with inverse square potential $iu_t+Δu-\frac{a}{|x|^2}u=λ|x|^{-b}|u|^2u$ with $a>-\frac14$ and $0<b<1$ in dimension three. In the defocusing case (i.e. $λ=1$), we establish the global well-posedness and scattering for any initial data in the energy space $H^1_a(\mathbb R^3)$. While for the focusing case(i.e. $λ=-1$), we obtain the scattering for the initial data below the threshold of the ground state, by making use of the virial/Morawetz argument as in Dodson-Murphy [Proc. Amer. Math. Soc.,145(2017…
Global well-posedness and scattering for the defocusing septic one-dimensional NLS via new smoothing and almost Morawetz estimates
该研究通过证明新的光滑化估计和低正则性下的近Morawetz估计，结合线性-非线性分解，确立了一维散焦七次非线性薛定谔方程在$s>19/54$时的$H^s(\bR)$全局适定性与散射性。
{\bf Abstract} \,\, We consider the following nonlinear Schrödinger equation on exterior domain. \begin{equation} \begin{cases} iu_t+Δ_g u + ia(x)u - |u|^{p-1}u = 0 \qquad (x,t) \in Ω\times (0,+\infty), \qquad (1)\cr u\big|_Γ= 0\qquad t \in (0,+\infty), \cr u(x,0) = u_0(x)\qquad x \in Ω, \end{cases} \end{equation} where $1<p<\frac{n+2}{n-2}$, $Ω\subset\mathbb{R}^n$ ($n\ge3$) is an exterior domain and $(\mathbb{R}^n,g)$ is a complete Riemannian manifold. We establish Morawetz estimates for the system (1) without dissipation ($a(x)\equiv 0$ in (1)) and meanwhile prove exponential stability of th…
In this paper we study the scattering of radial solutions to a $l$-component system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five. Our approach is based on the recent technique introduced by Dodson and Murphy, which relies on the radial Sobolev embedding and a Morawetz estimate.
Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $\R$
本文考虑一维空间中一类具有单项非线性项的半线性散焦薛定谔方程的Cauchy问题。我们建立全球适定性和散射。我们的分析是基于四粒子相互作用Morawetz估计\emph{a priori} $L^8_{t,x}$时空控制的解决方案。
For the solution of the free Schrödinger equation, we obtain the optimal constants and characterise extremisers for forward and reverse smoothing estimates which are global in space and time, contain a homogeneous and radial weight in the space variable, and incorporate a certain angular regularity. This will follow from a more general result which permits analogous sharp forward and reverse smoothing estimates and a characterisation of extremisers for the solution of the free Klein-Gordon and wave equations. The nature of extremisers is shown to be sensitive to both the dimension and the size…
The nonlinear Schrödinger equations with combined nonlinearities of power-type and Hartree
本文研究了任意维数n \ge 3中具有幂型和Hartree型组合非线性项的非线性薛定谔方程.在一定的结构条件下，给出了这些非线性项在能量空间中相互作用的近似全貌。该方法是基于Morawetz估计和扰动原理。
Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R^3
我们获得了能量临界的五次散焦薛定谔方程在$\R^{1+3}$中的能量类解的整体适定性、散射和整体$L^{10}_{t,x}$时空界.特别地，这建立了整体存在的经典解。我们的工作扩展了Bourgain和Grillakis的结果，该结果处理了径向情况。该方法在精神上类似于Bourgain的能量感应策略，但我们同时在频率空间和物理空间进行感应分析，并用相互作用变量代替Morawetz不等式。相互作用Morawetz估计的主要优点是它不局限于空间原点，因此能够更好地处理非径向解。特别是，这种相互作用的估计，以及几乎守恒的参数控制运动的$L^2$质量在频率空间，排除了能量集中的可能性。
In this paper we prove that the defocusing, quintic nonlinear Schrödinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R})$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [11]. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.
In this paper, we prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schrödinger equation. We show that a unique solution exists for $u_{0} \in H^{s}(\mathbf{R})$, $s > {8/29}$. This improves the result in [13], which proved global well-posedness for $s > {1/3}$. The main new argument is that we obtain almost Morawetz estimates with improved error.
In this paper, we consider the defocusing mass-critical nonlinear fourth-order Schrödinger equation. Using the $I$-method combined with the interaction Morawetz estimate, we prove that the problem is globally well-posed in $H^γ(\mathbb{R}^d), 5\leq d\leq 7$ with $γ(d)<γ<2$, where $γ(5)=\frac{8}{5}, γ(6)=\frac{5}{3}$ and $γ(7)=\frac{13}{7}$.
We investigate the global existence and scattering for the cubic fourth-order Schrödinger equation $iu_t+Δ^2u+|u|^2u=0$ in the low regularity space $H^s(\R^n)$ with $s<2$. We provide an alternative approach to obtain a new interaction Morawetz estimate and extend the range of the dimension of the interactive estimate in Pausader \cite{P08} by modifying a tensor product method appeared in \cite{CGT}. We combine interaction Morawetz estimates, energy increments for the I-method to prove the result.
In this paper, we prove pointwise decay rates for cubic and higher order nonlinear wave equations, including quasilinear wave equations, on asymptotically flat and time-dependent spacetimes. We assume that the solution to the linear equation (rather than the nonlinear equation) satisfies a weaker form of the standard integrated local energy decay, or Morawetz, estimate. For nonlinearities with a total derivative structure, we prove better pointwise decay rates.
We prove weighted L^2 (Morawetz) estimates for the solutions of linear Schrodinger and wave equation with potentials that decay like |x|^{-2} for large x, by deducing them from estimates on the resolvent of the associated elliptic operator. We then deduce Strichartz estimates for these equations.
In this work, we consider the energy-supercritical defocusing cubic nonlinear wave equation in dimension d=5 for radially symmetric initial data. We prove that an a priori bound in the critical space implies global well-posedness and scattering. The main tool that we use is a frequency localized version of the classical Morawetz inequality, inspired by recent developments in the study of the mass and energy critical nonlinear Schrödinger equation.
We revisit the problem of scattering below the ground state threshold for the mass-supercritical focusing nonlinear Schrödinger equation in two space dimensions. We present a simple new proof that treats the case of radial initial data. The key ingredient is a localized virial/Morawetz estimate; the radial assumption aids in controlling the error terms resulting from the spatial localization.
In this paper we study the scattering of non-radial solutions in the energy space to coupled system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five without the mass-resonance condition. Our approach is based on the recent technique introduced by Dodson and Murphy, which relies on an interaction Morawetz estimate. It is proved that any solution below the ground states scatters in time.
Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\R^{1+4}$
在$\R\times\R^4$中，我们得到了散焦能量临界非线性薛定谔方程能量空间解的整体适定性、散射性、一致正则性和整体$L^6_{t,x}$时空界.我们的论点密切遵循Colliander-Keel-Staffilani-Takaoka-Tao的论点，尽管我们对频率定域相互作用Morawetz估计的推导稍微简单一些。因此，我们的方法产生一个更好的界上的$L^6_{t,x}$ -范数。
Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space
研究了双曲空间上非线性薛定谔方程径向解的长时间行为。我们表明，通常的区分短程和长程非线性修改：几何的双曲空间，使每一个权力的非线性短程。证明依赖于加权的Eschhartz估计，这意味着Eschhartz估计更广泛的家庭的容许对，和Morawetz型不等式。后者是在没有对称性假设的情况下建立的。
In this article, we consider the infinite dimensional vector-valued resonant nonlinear Schrödinger system, which arises from the study of the asymptotic behavior of the defocusing nonlinear Schrödinger equation on "wave guide" manifolds like $\mathbb{R}^2\times \mathbb{T}$ in [7]. We show global well-posedness and scattering for this system by long time Strichartz estimates and frequency localized interaction Morawetz estimates. As a by-product, our results make the arguments of scattering theory in [7] closed as crucial ingredients for compactness of the critical elements.
Global Well-Posedness for the $L^2$-critical nonlinear Schrödinger equation in higher dimensions
考虑了在$\R^n, n \geq 3$中$L^{2}$临界半线性薛定谔方程的初值问题.当$1>s>\frac{\sqrt{7}-1}{3}$为$n=3$，$1>s> \frac{-(n-2)+\sqrt{(n-2)^2+8(n-2)}}{4}$为$n \geq 4$时，问题在$H^{s}({\Bbb R^{n}})$中是全局适定性的.我们使用“PDMATHAMT方法”结合当地的时间Morawetz估计。
Global well-posedness and scattering for the mass-critical nonlinear Schrödinger equation for radial data in high dimensions
我们建立了大的球对称$L^2_x(\R^n)$初始数据在尺寸$n\geq 3$的散焦质量临界（伪共形）非线性薛定谔方程$iu_t + Δu = |u|^{4/n} u$的解的整体适定性和散射。在使用\cite{compact}中的约简来消除几乎是周期性模标度的爆破解之后，我们得到了频率局部化的Morawetz估计，并排除了大规模疏散场景（有点类似于\cite{ckstt:gwp}，\cite{RV}，\cite{thesis:art}），以得出结论。
Global existence and scattering for rough solutions of a nonlinear Schroedinger equation on R^3
证明了在$H^s(\rr^3)$中的散焦三次非线性薛定谔方程对$s > {4/5}$的整体解的存在性和散射性.主要的新估计的论点是一个Morawetz型不等式的解决方案$ϕ$。这个估计限制了$\|ϕ(x,t)\|_{L^4_{x,t}(\rr^3 \times \rr)}$，而著名的Lin-Strauss的Morawetz型估计控制了$ \int_0^{\infty} \int_{\rr^3} \frac{(ϕ(x,t))^4}{|x|} dx dt
In this paper, we simplify the proof of M. Hamano in \cite{Hamano2018}, scattering theory of the solution to \eqref{NLS system}, by using the method from B. Dodson and J. Murphy in \cite{Dodson2018}. Firstly, we establish a criterion to ensure the solution scatters in $ H^1(\mathbb{R}^5) \times H^1(\mathbb{R}^5) $. In order to verify the correctness of the condition in scattering criterion, we must exclude the concentration of mass near the origin. The interaction Morawetz estimate and Galilean transform characterize a decay estimate, which implies that the mass of the system cannot be concent…
We study the asymptotic dynamics for solutions to a system of nonlinear Schrödinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of Morawetz and interaction Morawetz estimates. The blow-up results are shown by combining variational analysis and an ODE argument, which overcomes the unavailability of the convexity argument based on virial-t…
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…
Semilinear Schrödinger Flows on Hyperbolic Spaces: Scattering in H^1
本文证明了双曲空间$\H^d$，$d\geq 2$上的散焦非线性薛定谔方程\begin{equation*} \begin{cases} &(i\partial_t+Δ_\g)u=u|u|^{2σ}; &u(0)=ϕ, \end{cases} \end{equation*}在$H^1$中的整体适定性和散射性，其中指数为$σ\in(0,2/(d-2))$ .主要的非预期结论是在小指数$σ$的情况下分散到线性解;为了比较，在欧几里得空间上，$H^1$中的分散对于任何指数$σ\in(1/d,2/d]$都是未知的，并且已知对于$σ\in(0,1/d]$是失败的。我们的主要成分是某些非欧整体的时间Escherichartz估计和非欧Morawetz不等式。
In this paper, we study the dynamics behavior of the NLS system with three waves interaction in the energy space $H^1(\mathbb{R}^5) \times H^1(\mathbb{R}^5)\times H^1(\mathbb{R}^5) $. Inspired by B. Dodson and J. Murphy in \cite{Dodson2018}, we establish an interaction Morawetz estimate for the NLS system, together with the criterion which proved by Tao-Dodson--Murphy we can get the scattering under the ground state in energy space with mass-resonance. Under the radial assumption, we can remove the mass-resonance condition.
In this paper, we consider a class of nonlinear fourth-order Schrödinger equation, namely \[ \left\{ \begin{array}{rcl} i\partial_t u +Δ^2 u &=&-|u|^{ν-1} u, \quad 1+ \frac{8}{d}<ν<1+\frac{8}{d-4},\\ u(0)&=&u_0 \in H^γ(\mathbb{R}^d), \quad 5 \leq d \leq 11. \end{array} \right. \] Using the $I$-method combined with the interaction Morawetz inequality, we establish the global well-posedness and scattering in $H^γ(\mathbb{R}^d)$ with $γ(d,ν)<γ<2$ for some value $γ(d,ν)>0$.
Bilinear virial identities and applications
证明了非线性薛定谔方程的双线性维里恒等式，它是Morawetz相互作用不等式的推广。我们恢复和扩展已知的双线性改进的Eschhartz不等式，并提供各种非线性问题的应用，最显着的域的边界。
In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global Morawetz inequality adapted to the geometry of the manifold (in other words we adapt the method of Kenig-Merle to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (in our case the main theorem of Colliander-Keel-Staffilani-Takaoka-Tao). As an application we prove global well-posedness and scattering in $H^1$ for the e…
Global well-posedness and polynomial bounds for the defocusing $L^{2}$-critical nonlinear Schrödinger equation in $\R$
证明了一维五次离焦非线性薛定谔方程在低正则性数据下的整体适定性。精确地说，我们表明，一个唯一的和全球性的解决方案存在于Sobolev空间中的初始数据$H^{s}(\mathbb R)$的任何$s>{1/3}$。这改进了\cite{tz}中的结果，其中为任何$s>{4/9}$建立了全局适定性。我们使用$I$方法来利用方程的守恒律。在我们的证明中的新成分是平滑的解决方案$Iu$的相互作用Morawetz估计。作为我们的证明的副产品，我们也得到了解决方案的$H^{s}$规范服从多项式的时间界限。
We obtain weighted $L^2$ Strichartz estimates for Schrödinger equations $i\partial_tu+(-Δ)^{a/2}u=F(x,t)$, $u(x,0)=f(x)$, of general orders $a>1$ with radial data $f,F$ with respect to the spatial variable $x$, whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in \cite{BBCRV} concerning weighted homogeneous St…
New invariant surface measures for the cubic Schrödinger equation
本文研究一维与二维三次散焦非线性 Schrödinger 方程，构造了支撑于质量水平集上的新不变测度。
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.
In this paper, we establish bilinear and gradient bilinear Strichartz estimates for Schrödinger operators in 2 dimensional compact manifolds with boundary. Using these estimates, we can infer the local well-posedness of cubic nonlinear Schrödinger equation in $H^s$ for every $s>{2/3}$ on such manifolds.
We review some recent results concerning the Initial Value Problem of 1d-cubic non-linear Schrödinger equation (NLS) and other related systems as the Schrödinger Map. For the latter we prove the existence of a cascade of energy. Finally, some new examples of the Talbot effect at the critical level of regularity are given.
We consider a system of coupled cubic Schrödinger equations. We prove that there exists a beating effect, i.e. an energy exchange between different modes. This construction may be transported to the linear time-dependent Schrödinger equation: we build solutions such that their Sobolev norms grow logarithmically. All these results are stated for large but finite times.
A decay estimate for cubic defocusing non-linear Schrödinger equation in three dimensions
本文研究了三维立方非线性薛定谔方程非线性解的衰减估计问题，通过数学分析给出了相关估计结果。
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.
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 $σ_0$.
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.
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。
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.
For the Schrödinger equation with a cubic-quintic, focusing-defocusing nonlinearity in one space dimension, we prove the asymptotic stability of solitary waves for a large range of admissible frequencies. For this model, the linearized problem around the solitary waves does not have internal mode nor resonance.
We consider the problem of recovering a spatially-localized cubic nonlinearity in a nonlinear Schrödinger equation in dimensions two and three. We prove that solutions with data given by small-amplitude wave packets accrue a nonlinear phase that determines the X-ray transform of the nonlinear coefficient.
The cubic fourth-order Schrodinger equation
研究了任意空间维数$\mathbb{R}^n$中立方离焦四阶薛定谔方程$iu_t + Δ^2u + |u|^2u=0$的解，对任意$H^2$初值.我们证明了该方程是全局适定时$n \le 8$和不适定时$n \ge 9$，与额外的重要信息，散射成立时，$5 \le n \le 8$。
Consider the initial value problem for systems of cubic derivative nonlinear Schrödinger equations in one space dimension with the masses satisfying a suitable resonance relation. We give structural conditions on the nonlinearity under which the small data solution gains an additional logarithmic decay as $t \to +\infty$ compared with the corresponding free evolution.
This is a sequel to the paper "Large time asymptotics for a cubic nonlinear Schrödinger system in one space dimension" by the same authors. We continue to study the Cauchy problem for the two-component system of cubic nonlinear Schrödinger equations in one space dimension. We provide criteria for large time decay or non-decay in $L^2$ of the small amplitude solutions in terms of the Fourier transforms of the initial data.
We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation $i \partial_t u + Δu = - |x|^b |u|^{p-1} u$ in the radial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$. We show the global existence and energy scattering in the inter-critical regime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. …
In this note, we consider the ill-posedness issue for the cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, adapting the argument by Christ-Colliander-Tao [14] to the periodic setting, we exhibit a norm inflation phenomenon for both the usual cubic NLS and the Wick ordered cubic NLS for $s \leq s_\text{crit} :=- \frac 12$. We also discuss norm inflation phenomena for general cubic fractional NLS on the circle.
Tensor products and Correlation Estimates with applications to Nonlinear Schrödinger equations
我们证明新的相互作用Morawetz型（相关性）估计在一个和两个维度。在第二个维度的估计对应于非线性对角模拟布尔增益的双线性细化的Eschenhartz。对于二维情形，我们用两种不同的方法给出一个证明。首先，我们按照原来的方法林和施特劳斯，但适用于张量积的解决方案。然后，我们证明证明使用换向器向量算子作用于方程的守恒律。这种方法可以推广到所有维度的相关估计。在一维中，我们使用高斯-维尔斯特拉斯求和方法作用于守恒律。然后，我们将二维估计应用到非线性薛定谔方程，并导出了Nakanishi的$H^1$散射结果的直接证明，对于每一个$L^{2}$ -超临界非线性。我们还证明了散射的能量空间以下的某一类$L^{2}$ -超临界方程。
Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation
本文研究一维三次-五次聚焦-聚焦非线性薛定谔方程，证明了能量空间中偶微扰下小驻波孤立子族的局部渐近完备性，并用费米黄金定则处理了线性化方程内模对动力学的贡献。
We obtain probabilistic local well-posedness in quasilinear regimes for the Schrödinger half-wave equation with a cubic nonlinearity. We need to use a refined ansatz because of the lack of probabilistic smoothing in the Picard's iterations, which is due to the high-low-low frequency interactions. The proof is an adaptation of the method of Bringmann on the derivative nonlinear wave equation to Schrödinger-type equations. In addition, we discuss ill-posedness results for this equation.
This paper addresses the focusing cubic-quintic nonlinear Schrodinger equation in three space dimensions. Especially, we study the global dynamics of solutions whose energy and mass equal to those of the ground state in the sprits of Duyckaerts and Merle (2009). When we try to obtain the corresponding result, we meet several difficulties due to the cubic-quintic nonlinearity. We overcome them by using the one-pass theorem (no return theorem) developed by Nakanishi and Schlag (2012).
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。