Existence and stability of solitons for the nonlinear Schrödinger equation on hyperbolic space
Hans Christianson, Jeremy Marzuola
研究了双曲空间上一类非线性定常方程基态解或孤子的存在性和稳定性。应用集中紧性方法，结果表明与欧几里得空间的结果密切相关。
We study the existence and stability of ground state solutions or solitons to a nonlinear stationary equation on hyperbolic space. The method of concentration compactness applies and shows that the results correlate strongly to those of Euclidean space.
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In [19] and [26], the authors proved the stability of multi-solitons for derivative nonlinear Schr{ö}dinger equations. Roughly speaking, sum of finite stable solitons is stable. We predict that if there is one unstable solition then multi-soliton is unstable. This prediction is proved in [7] for classical nonlinear Schr{ö}dinger equations. In this paper, we proved this prediction for derivative nonlinear Schr{ö}dinger equations by using the method of C{ô}te-Le Coz [7] with the help of Gauge transformation.
Rigidity for the non self-dual Chern--Simons--Schr{ö}dinger equation at the level of the soliton
本文在孤子层面为非自对偶 Chern–Simons–Schrödinger 方程的解建立了刚性结论。
The nonlinear Schr{ö}dinger equation with derivative cubic nonlinearity admits a family of solitons, which are orbitally stable in the energy space. In this work, we prove the orbital stability of multi-solitons configurations in the energy space, under suitable assumptions on the speeds and frequencies of the composing solitons. The main ingredients of the proof are modulation theory, energy coercivity and monotonicity properties.
We look for solutions to derivative nonlinear Schrodinger equations built upon solitons. We prove the existence of multi-solitons i.e. solutions behaving at large time as the sum of finite solitons. We also show that one can attach a kink at the begin of the sum of solitons i.e multi kink-solitons. Our proofs proceed by fixed point arguments around the desired profile, using Strichartz estimates.
We study pointwise spatial decay of multi-solitons of the generalized Korteweg-de Vries equations. We obtain that, uniformly in time, these solutions and their derivatives decay exponentially in space on the left of and in the solitons region, and prove rapid decay on the right of the solitons. We also prove the corresponding result for multi-solitons of the nonlinear Schr{ö}dinger equations, that is, exponential decay in the solitons region and rapid decay outside.
In this paper, we study some properties of multi-solitons for the non-linear Schr{ö}dinger equations in R^d with general non-linearities. Multi-solitons have already been constructed in H^1, successively by Merle, by Martel and Merle, and by C{ô}te, Martel and Merle. We show here that multi-solitons are smooth, depending on the regularity of the non-linearity. We obtain also a result of uniqueness in some class, either when the ground states are all stable, or in the mass-critical case.
In this note we prove global well-posedness for the defocusing, cubic nonlinear Schr{ö}dinger equation with initial data lying in a critical Sobolev space.
We are mainly interested in extending the known results on ob-servability inequalities and stabilization for the Schr{ö}dinger equation to the magnetic Schr{ö}dinger equation. That is in presence of a magnetic potential. We establish observability inequalities and exponential stabilization by extending the usual multiplier method, under the same geometric condition to that needed for the Schr{ö}dinger equation. We also prove, with the help of elliptic Carleman inequalities, logarithmic stabilization results through a resolvent estimate. Although the approach is classical, these results on loga…
The Kato smoothing effect for Schr{ö}dinger equations with unbounded potentials in exterior domains
本文证明了在非陷波外区域上变系数Schr{<$}dinger方程解的光滑效应。我们允许无穷远处的二次势。
We give an example of a linear, time-dependent, Schr{ö}dinger operator with optimal growth of Sobolev norms. The construction is explicit, and relies on a comprehensive study of the linear Lowest Landau Level equation with a time-dependent potential.
Existence of multi-solitons for logarithmic Schr{ö}dinger equation with a perturbation of a smooth $L^2$-subcritical function nonlinearity
本文针对含聚焦对数项和光滑 $L^2$-次临界函数非线性的 Schrödinger 方程，通过迭代构造与局部能量估计，证明了多孤子解的存在性，其渐近表现为多个不同速度孤子的叠加。
Schr{ö}dinger maps to a K{ä}hler manifold in two dimensions
本文研究了Schrödinger映射到二维Kähler流形的全局存在性和散射性质，采用小初始数据和Besov空间方法进行证明。
In this paper we prove a Liouville theorem for the Chern--Simons--Schr{ö}dinger equation. This result is consistent with the soliton resolution conjecture for initial data that does not lie in a weighted space. See [KKO22] for the soliton resolution result in a weighted space.
Existence of stationary solutions of supercritical nonlinear Schr{ö}dinger equations on some metric graphs
本文研究了非紧连通度量图上的超临界非线性Schr{ö}dinger方程，在无小质量假设下证明了具预定质量的驻波解的存在性。
In this note we prove scattering for a defocusing nonlinear Schr{ö}dinger equation with initial data lying in a critical Besov space. In addition, we obtain polynomial bounds on the scattering size as a function of the critical Besov norm.
Mass-energy Scattering Criterion For Double Power Schr{ö}dinger Equations
本文研究双幂次薛定谔方程，将散射结果推广至所有L2超临界幂次，并适用于能量超临界非线性情形。
The Schr{ö}dinger problem is an entropy minimisation problem on the space of probability measures. Its optimal value is a cost between two probability measures. In this article we investigate some regularity properties of this cost: continuity with respect to the marginals and time derivative of the cost along probability measures valued curves.
We prove existence results for a stationary Schr{ö}dinger equation with periodic magnetic potential satisfying a local integrability condition on the whole space using a critical value function.
We consider the logarithmic Schr{ö}dinger equations with damping, also called Schr{ö}dinger-Langevin equation. On a periodic domain, this equation possesses plane wave solutions that are explicit. We prove that these solutions are asymptotically stable in Sobolev regularity. In the case without damping, we prove that for almost all value of the nonlinear parameter, these solutions are stable in high Sobolev regularity for arbitrary long times when the solution is close to a plane wave. We also show and discuss numerical experiments illustrating our results.
Global well-posedness and scattering for the defocusing, energy -critical, nonlinear Schr{ö}dinger equation in the exterior of a convex obstacle when $d = 4$
本文证明了凸障碍物外区域上的能量临界非线性Schr{<$}dinger方程对于有限能量的初值是整体适定的和散射的.为了证明这一点，我们利用频率本地化的Morawetz估计适应外部域。
We consider the nonlinear Schr{ö}dinger-Langevin equation for both signs of the logarithmic nonlinearity. We explicitly compute the dynamics of Gaussian solutions for large times, which is obtained through the study of a particular nonlinear differential equation of order 2. We then give the asymptotic behavior of general energy weak solutions under some regularity assumptions. Some numerical simulations are performed in order to corroborate the theoretical results.
We pursue our work on the asymptotic regimes of the Landau-Lifshitz equation for bi-axial ferromagnets. We put the focus on the cubic Schr{ö}dinger equation, which is known to describe the dynamics in a regime of strong easy-axis anisotropy. In any dimension, we rigorously prove this claim for solutions with sufficient regularity. In this regime, we additionally classify the one-dimensional solitons of the Landau-Lifshitz equation and quantify their convergence towards the solitons of the one-dimensional cubic Schr{ö}dinger equation.
The problem we consider is given a mixed model $F(γ) \rightarrow γ\rightarrow \tildeγ$ and a potential $\widehat{V}$ such that $\log \widehat{V} \in L^{1}$, to determine a movement $U$ such that $F(U γ)=F(γ) + \widehat{V}$.
A quasi-linear Schr{ö}dinger equation for nucleons : existence, uniqueness and asymptotic behavior
本文研究原子核粒子动力学中的拟线性薛定谔方程，在参数假设下证明正径向解的存在性、唯一性与非退化性，并分析解对参数的依赖。
Evolution of Gaussian measures and application to the one dimensional nonlinear Schr{ö}dinger equation
本文概述文献[3]关于随机初值下一维非线性 Schrödinger 方程的研究，通过刻画高斯测度的非线性演化，得到方程的整体适定性与散射结论。
We study the Littlewood-Paley-Stein functions associated with Hodge-de Rham and Schr{ö}dinger operators on Riemannian manifolds. Under conditions on the Ricci curvature we prove their boundedness on L p for p in some interval (p 1 , 2] and make a link to the Riesz Transform. An important fact is that we do not make assumptions of doubling measure or estimates on the heat kernel in this case. For p > 2 we give a criterion to obtain the boundedness of the vertical Littlewood-Paley-Stein function associated with Schr{ö}dinger operators on L p .
We consider the Schrödinger equation with nonlinear derivative term on $[0,+\infty)$ under Robin boundary condition at $0$. Using a virial argument, we obtain the existence of blowing up solutions and using variational techniques, we obtain stability and instability by blow up results for standing waves.
Higher Dimensional Vortex Standing Waves for Nonlinear Schr{ö}dinger Equations
研究了旋转对称流形上非线性薛定谔方程的驻波解，它在旋转群下以自然的方式变换.我们称之为涡旋解。它们是在欧几里得平面上研究过的涡旋驻波的高维版本。我们专注于两种类型的涡的解决方案，我们称之为球形涡和轴向涡。
We show how the solutions to a $2\times2$ linear system involving Schr{ö}dinger operators blow up as the parameter $μ$ tends to some critical value which is the principal eigenvalue of the system; here the potential is continuous positive with superquadratic growth and the square matrix of the system is with constant coefficients and may have a double eigenvalue.
This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{ö}dinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.
By using the Wigner transform, it is shown that the nonlinear Schr$\ddot{\textmd{o}}$dinger equation can be described, in phase space, by a kinetic theory similar to the Vlasov equation which is used for describing a classical collisionless plasma. In this paper we mainly show Landau damping in the quantum sense, namely,quantum Landau damping exists for the Wigner-Poisson system. At the same time, we also prove the existence and the stability of the nonlinear Schr$\ddot{\textmd{o}}$dinger equation under the quantum stability assumption.
In this paper we consider the Schr{ö}dinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on Zhidkov spaces X k (R) and in $ϕ$ + H k (R). Moreover, we prove the existence of conservation laws by using localizing functions. Finally, we give explicit formulas for stationary solutions on Zhidkov spaces.
Modified scattering for the cubic Schr{ö}dinger equation on product spaces: the nonresonant case
我们考虑空间域$\mathbb{R}\times \mathbb{T}^d$上的三次非线性薛定谔方程，用卷积势对其进行扰动.利用最近的技术的Hani-Pausader-Tzvetkov-Visciglia，我们证明了修改的散射结果和构造修改的波算子，在一般的假设下的潜力。特别是，这使我们能够证明，这个非共振立方NLS的小解的Sobolev范数是渐近常数。
Multi-speed solitary waves of nonlinear Schr{ö}dinger systems: theoretical and numerical analysis
考虑一维空间中耦合的非线性薛定谔方程组.首先，我们证明了多速孤波的存在性，即系统的每个分量在很大程度上表现为一个孤波。然后，我们数值研究了两个孤波相互作用的支持每一个组件。在可能的结果中，我们发现弹性和非弹性相互作用，碰撞与质量提取和反射。
Profile decomposition and scattering for general nonlinear Schr{ö}dinger equations
本文研究一般扰动非线性的 Schrödinger 方程，构建了相应的轮廓分解，并在能量超临界且临界 Sobolev 范数有界的假设下证明了全局存在性与散射，推广了双幂非线性的已有成果。
A sharper energy method for the localization of the support to some stationary Schr{ö}dinger equations with a singular nonlinearity
本文针对具有奇异非线性项的稳态 Schrödinger 方程，提出了一种更敏锐的能量方法，并据此证明了解的支集紧致性。
In this paper we continue our study [DSS20] of the nonlinear Schrödinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on $\mathbb{R}$ was proved for real analytic data. Here we prove global well-posedness for the 1D NLS with initial data lying in $L^{p}$ for any $2 < p < \infty$, provided the initial data is sufficiently smooth. We do not use the complete integrability of the cubic nonlinear Schr{ö}dinger equation.
Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schr{ö}dinger equation when $d \geq 3$
本文证明了$d$维散焦质量临界非线性Schr{<$}dinger初值问题对于$u_{0} \in L^{2}(\mathbf{R}^{d})$和$d \geq 3$是整体适定的和散射的.为了做到这一点，我们将证明一个类似于[10]中估计的频率局部相互作用Morawetz估计。由于我们考虑的是一个$L^{2}$临界初值问题，我们将局限于低频。
We study the stability issue in the inverse problem of determining the magnetic field and the time-dependent electric potential appearing in the Schrödinger equation, from boundary observations. We prove in dimension 3 or greater, that the knowledge of the Dicrichlet-to-Neumann map stably determines the magnetic field and the electric potential.
Almost Morawetz estimates and global well-posedness for the defocusing $L^2$-critical nonlinear Schr{ö}dinger equation in higher dimensions
本文研究了$L^{2}$维下的散焦$L^{2}$临界非线性Schr{<$}dinger方程的整体适定性.使用I-方法，我们证明了问题是全局适定的$n = 3$时，$s > {2/5}$，和$n \geq 4$时，为$s > \frac{n - 2}{n}$。我们结合能量增量的I-方法，相互作用Morawetz估计，几乎Morawetz估计来证明结果。
In this paper we prove that the defocusing, cubic nonlinear Schr{ö}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{2})$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in \cite{CKSTT4}. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.
Smoothing effect and quantum-classical correspondence for the Schr{ö}dinger equation with confining potential
本文研究具次二次约束势的 Schrödinger 方程，证明其解的局部平滑效应等价于对应经典流的逃逸率估计，并借助 Egorov 定理建立量子-经典对应。
It was proved by H. Bahouri, P. G{é}rard and C.-J. Xu in [9] that the Schr{ö}dinger equation on the Heisenberg group $\mathbb{H}^d$, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schr{ö}dinger equation, by a refined study of the Schr{ö}dinger kernel $S_t$ on $\mathbb{H}^d$. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kern…
Exponential stability of solutions to the Schr{ö}dinger-Poisson equation
本研究证明了无外参数圆周上Gevrey类薛定谔-泊松方程小解的指数稳定性，给出了最优时间量级，验证了Bourgain的相关猜想。
Bilinear Strichartz estimates for the Schr{ö}dinger map problem
本文在临界Schihartz空间$| |\nabla|^{\frac{d - 2}{2}} ψ_{x} |_{L_{t,x}^{\frac{2(d + 2)}{d}}}$中证明了小尺寸Schr{<$}dinger映射问题解的双线性Schihartz估计.这些估计将有助于在即将发表的论文中证明局部适定性结果。双线性估计使用了一个类似于Planchon和Vega（2009）中的论点。我们使用与Bejenaru，Ionescu和Kenig（2007），Bejenaru等人（2011）和Smith相同的量规。
We consider a nonlinear Schr{ö}dinger equation set in the whole space with a single power of interaction and an external source. We first establish existence and uniqueness of the solutions and then show, in low space dimension, that the solutions vanish at a finite time. Under a smallness hypothesis of the initial data and some suitable additional assumptions on the external source, we also show that we can choose the upper bound on which time the solutions vanish.
In this paper we deal with the so-called "spectral inequalities", which yield a sharp quantification of the unique continuation for the spectral family associated with the Schrödinger operator in $ \mathbb{R}^d$ \begin{equation*} H_{g,V} = Δ_g + V(x), \end{equation*} where $Δ_g$ is the Laplace-Beltrami operator with respect to an analytic metric $g$, which is a perturbation of the Euclidean metric, and $V(x)$ a real valued analytic potential vanishing at infinity.
For the double power one dimensional nonlinear Schr{ö}dinger equation, we establish a complete classification of the stability or instability of standing waves with positive frequencies. In particular, we fill out the gaps left open by previous studies. Stability or instability follows from the analysis of the slope criterion of Grillakis, Shatah and Strauss. The main new ingredients in our approach are a reformulation of the slope and the explicit calculation of the slope value in the zero-frequency case. Our theoretical results are complemented with numerical experiments.
In this paper, we consider the Cauchy problem in $\mathbb{R}^N$, $N\geq1$, for semi-linear Schrödinger equations with space-time fractional derivatives. We discuss the nonexistence of global $L^1$ or $L^2$ weak solutions in the subcritical and critical cases under some conditions on the initial data and the nonlinear term. Furthermore, the nonexistence of local $L^1$ or $L^2$ weak solutions in the supercritical case are studied.
Long time well-posedness and wave turbulence limit for a quadratic non-lienar Schr{ö}dinger equation with well prepared random initial data
本文证明二次非线性薛定谔方程在随机初始数据下的长时适定性，并在高维情形推导波动力学方程，方法基于Deng和Hani的思想。
Low-complexity approximations with least-squares formulation of the time-dependent Schr{ö}dinger equation
本文提出基于张量和高斯波包线性组合逼近含时薛定{ö}dinger方程解的新方法，将其转化为受限优化问题，分别采用Alternating Least Square算法和贪婪算法求解，并讨论了两者效率。
We consider an anisotropic model case for a strictly convex domain of dimension $d\geq 2$ with smoothboundary and we describe dispersion forthe semi-classical Schr{ö}dinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the linear semi-classical flow : a loss of $(\frac ht)^{1/4}$ occurs with respect to the boundary less case due to repeated swallowtail type singularities, and is proven optimal. Corresponding Strichartz estimates allow to solve the cubic nonlinear Schödinger equation on such a 3D model convex domain, hence match…
Global well--posedness for the mass--critical nonlinear Schr{ö}dinger equation on $\mathbb{T}$
本文证明环面上五次非线性薛定谔方程在 $H^s(\mathbb{T})$ ($s>1/3$) 初值下的全局适定性，改进了此前 $s>2/5$ 的最佳结果。
The Cauchy problem for logarithmic Schr{ö}dinger equation with a perturbation of power law nonlinearity
研究带对数与多项式增长非线性项的薛定谔方程柯西问题，在次临界情形构造能量空间解，并讨论临界情形的全局存在条件及更一般的柯西理论。
Observability of the Schr{ö}dinger equation with subquadratic confining potential in the Euclidean space
本文研究至多二次增长约束势下 Schrödinger 方程的能观性，利用半经典分析刻画满足正则性的观测开集，并估计最优观测时间。
This paper deals with the following fractional Schr$ \ddot{\textrm{o}}$dinger equations with Choquard-type nonlinearities \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} (-Δ)^{\fracα{2}}u + u - C_{n,-β} \,(|x|^{β-n}\ast u^{p})\, u^{p-1}& = &0 & \mbox{in}\ \ \mathbb{R}^{n}\,, \\[0.05cm] u & > & 0 & \mbox{on}\ \ \mathbb{R}^{n}, \end{array}\right. \end{equation*} where $ 0< α,β< 2, 1\leq p <\infty \,\,and\,\, n\geq 2. $ First we construct a decay result at infinity and a narrow region principle for related equations. Then we establish the radial symmetry of positive solutions for the a…
In this paper we discuss global well - posedness and scattering for some initial value problems that are $L^{2}$ supercritical and $\dot{H}^{1}$ subcritical, with radial data. We prove global well - posedness and scattering for radial data in $H^{s}$, $s > s_{c}$, where the problem is $\dot{H}^{s_{c}}$ - critical. We make use of the long time Strichartz estimates of \cite{D2} to do this.
Solutions with expanding compact support of saturated Schr{ö}dinger equations: self-similar solutions
本文研究带饱和非线性项的薛定谔方程，利用自相似解方法证明了存在具有紧支集且按 \\(C\sqrt{t}\\) 规律扩张的解。
Global well-posedness of quadratic and subquadratic half wave Schr{ö}dinger equations
针对平面及波导上1<p≤2阶非线性半波薛定谔方程因各向异性导致的全局适定性难题，本文在特定函数空间中证明了其全局适定性，并结合已有孤波研究完成了基态集稳定性的证明。
Mass Concentration Phenomena for the L^2-Critical Nonlinear Schr{ö}dinger Equation
本文证明了非线性薛定谔方程$iu\_t+Δu\pm|u|^\frac{4}{N}u=0,$的任意解在有限时间内爆破，在爆破时刻附近满足质量集中现象.我们的证明基本上是基于Bourgain的一个~ \cite{MR99f:35184}，它已经建立了这个结果在二维空间的情况下，和推广的Eschenhartz不等式，其中二维空间的情况下证明了Moyua，Vargas和Vega~ \cite{MR1671214}。我们还将Keraani~ \cite{MR2216444}和Merle及Vega~ \cite{MR1628235}中的结果推广到高维.
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