A. Pankov
We present basic results, known and new, on nontrivial solutions of periodic stationary nonlinear Schrödinger equations. We also sketch an application to nonlinear optics and discuss some open problems.
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We prove exponential decay for a system of two Schr{ö}dinger equations in a wave guide, with coupling and damping at the boundary. This relies on the spectral analysis of the corresponding coupled Schr{ö}dinger operator on the one-dimensional cross section. We show in particular that we have a spectral gap and that the corresponding generalized eigenfunctions form a Riesz basis.
I review an information-theoretic variant of the quantum de Finetti theorem due to Brand{ã}o and Harrow and discuss its applications to the topic of bosonic mean-field limits. This leads to slightly improved methods for the derivation of the local non-linear Schr{ö}dinger energy functional from many-body quantum mechanics.
We prove the local energy decay and the smoothing effect for the damped Schr{ö}dinger equation on R^d. The self-adjoint part is a Laplacian associated to a long-range perturbation of the flat metric. The proofs are based on uniform resolvent estimates obtained by the dissipative Mourre method. All the results depend on the strength of the dissipation which we consider.
We prove low frequency resolvent estimates and local energy decay for the Schr{ö}dinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed Schr{ö}dinger operator with the resolvent of the free Laplacian. This gives the leading term for the developpement of this resolvent when the spectral parameter is close to 0. For this, we show in particular how we can apply the usual commutators method for generalized resolvents and simultaneously for different operators. Finally, we deduce similar results for the …
Exponential decay for the Schr{ö}dinger equation on a dissipative waveguide
针对边界处处吸收却不满足几何控制条件的耗散波导，本文利用变量分离与横向算子特征函数的 Riesz 基性质，证明 Schrödinger 方程解呈指数衰减，并讨论负吸收指数情形。
We prove a Limiting Absorption Principle for Schr{ö}dinger operators in tubes about infinite curves embedded in the Euclidian space with different types of boundary conditions. The argument is based on the Mourre theory with conjugate operators different from the generator of dilations which is usually used in this case, and permits to prove a Limiting Absorption Principle for Schr{ö}dinger operators in singular waveguides.
On the ground state of the nonlinear Schr{ö}dinger equation: asymptotic behavior at the endpoint powers
本文研究了非线性薛定谔方程基态在临界幂次下的渐进行为，通过缩放分析得出了高斯函数和Aubin-Talenti代数孤子的极限，并给出了强收敛性证明及数值验证。
Dependence of the nonlinear Schr{ö}dinger flow upon the nonlinearity
研究非聚焦非线性Schrödinger方程在能量亚临界情况下的解对非线性幂次的依赖，探讨时间全局描述，重点分析总功率趋近于一的极限、长程效应的存在以及短程情况下散射算子的连续性。
We recall a Moure theory adapted to non self-adjoint operators and we apply this theory to Schr{ö}dinger operators with non real potentials, using different type of conjugate operators. We show that some conjugate operators permits to relax conditions on the derivatives of the potential that were required up to now.
In this paper we consider time dependent Schr{ö}dinger linear PDEs of the form i$\partial$t$ψ$ = L(t)$ψ$, where L(t) is a continuous family of self-adjoint operators. We give conditions for well-posedness and polynomial growth for the evolution in abstract Sobolev spaces. If L(t) = H + V (t) where V (t) is a perturbation smooth in time and H is a self-adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing, we prove that the Sobolev norms of the solution grow at most as t $ε$ when t $\rightarrow$ $\infty$, for any $ε$ \textgreater{} 0. If V (t…
The time local and global well-posedness for the Maxwell-Schr{ö}dinger equations is considered in Sobolev spaces in three spatial dimensions. The Strichartz estimates of Koch and Tzvetkov type are used for obtaining the solutions in the Sobolev spaces of low regularities. One of the main results is that the solutions exist time globally for large data.
Logarithmic Schr{ö}dinger equation with quadratic potential
本文研究二次势下对数薛定谔方程的动力学，证明谐振势产生的广义高斯子孤立波轨道稳定，并刻画了部分约束与排斥势情形下的色散行为。
Approximate controllability in small times of bilinear Schr{ö}dinger equations with magnetic drift
研究双线性薛定谔方程的小时间近似可控性，其中漂移为磁薛定谔算子，控制为电势，在两种情形下证明了该性质。
Global Energy Solutions for Normalized Linear Stochastic Schr{ö}dinger Equations
本文研究马尔可夫开放量子系统中的非线性薛定谔方程，在能量空间中证明线性方程的全局适定性，并在放宽单调性假设下证明非线性SPDE解的不可区分性，最终通过路径唯一性和归一化建立非线性随机方程的适定性。
The 2D Chern-Simons-Schr{ö}dinger system reduced to 1D
该研究通过引入强各向异性陷阱势，将二维Chern-Simons-薛定谔系统简化为一维五次非线性薛定谔方程，揭示了二维任意子系统在波导中的有效动力学行为。
Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{ö}dinger equation
本文通过线性化方法和几何光学解，利用输入输出映射重建准线性动力学薛定谔方程中的时变系数。
Logarithmic Schr{ö}dinger equation and isothermal fluids
本文研究对数薛定谔方程与可压缩等温 Euler、Korteweg 及量子 Navier-Stokes 方程在全空间上的长时间行为，揭示两族方程之间的联系并给出启发式论证。
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…
In this paper we prove the smoothing effect for solutions of Schr{ö}dinger equations with variable coefficients and in a non trapping exterior domain. We allow quadratic potentials at infinity.
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.
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.
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。
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 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.
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.
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。
We prove that if a solution of the time-dependent Schr{ö}dinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schr{ö}dinger operator, we use the spectral theory of the Laplacian and complex analysis and obtain a characterization of the initial conditions that lead to a sharp decay at any time. We then use the recent spectral decomposition of the Schr{ö}dinger operator with compactly supported potential due to Colin de Verdi{è}rre and Turc to extend our results in the presence of such potentials. Finally, we u…
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 linear and nonlinear Schr{ö}dinger equation with a spatial white noise as a potential in dimension 2. We prove existence and uniqueness of solutions thanks to a change of unknown originally used in [8] and conserved quantities.
Small-time approximate controllability of the logarithmic Schr\''dinger equation
该论文证明了对数非线性薛定谔方程在双线性控制下的小时间全局近似可控性，通过结合相位控制和梯度流方法，并借助WKB分析处理非线性估计，为首个此类结果。
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.
We develop an elementary and self-contained differential approach, in an $L^\infty$ setting, for well-posedness (existence, uniqueness and smooth dependence with respect to the data) for the multi-marginal Schr{ö}dinger system which arises in the entropic regularization of optimal transport problems.
In this paper we prove that the energy - critical nonlinear Schr{ö}dinger equation in the domain exterior to a convex obstacle is globally well - posed and scattering for initial data having finite energy. To prove this we utilize frequency localized Morawetz estimates adapted to an exterior domain.
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。
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.
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.
We study standing wave solutions to nonlinear Schr{ö}dinger equations, on a manifold with a rotational symmetry, which transform in a natural fashion under the group of rotations. We call these vortex solutions. They are higher dimensional versions of vortex standing waves that have been studied on the Euclidean plane. We focus on two types of vortex solutions, which we call spherical vortices and axial vortices.
Long-time dynamics for the magnetic Schr{ö}dinger equation on the torus
本文研究二维环面上磁薛定谔流的长时间半经典测度，在几何非消失条件下证明量子唯一遍历性的动力学形式，并给出构型与动量边际的收敛阈值。
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.
A Stochastic Porous Media Schr{ö}dinger Equation: Feynman-type Motivation, Well-Posedness and Control Interpretation
本文基于Feynman路径方法推导了带布朗扰动与平均场依赖的复多孔介质型Schrödinger方程，证明其适定性，并通过控制问题给出解的Brézis-Ekeland型刻画以克服复空间严格单调性限制。
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.
We consider the cubic nonlinear Schr{ö}dinger equation on the spatial domain $\mathbb{R}\times \mathbb{T}^d$, and we perturb it with a convolution potential. Using recent techniques of Hani-Pausader-Tzvetkov-Visciglia, we prove a modified scattering result and construct modified wave operators, under generic assumptions on the potential. In particular, this enables us to prove that the Sobolev norms of small solutions of this nonresonant cubic NLS are asymptotically constant.
We consider a system of coupled nonlinear Schr{ö}dinger equations in one space dimension. First, we prove the existence of multi-speed solitary waves, i.e solutions to the system with each component behaving at large times as a solitary wave. Then, we investigate numerically the interaction of two solitary waves supported each on one component. Among the possible outcomes, we find elastic and inelastic interactions, collision with mass extraction and reflexion.
In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{ö}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.
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.
In this paper we prove that the defocusing, $d$-dimensional mass critical nonlinear Schr{ö}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{d})$ and $d \geq 3$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [10]. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.
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.
In this paper, we consider the global well-posedness of the defocusing, $L^{2}$ - critical nonlinear Schr{ö}dinger equation in dimensions $n \geq 3$. Using the I-method, we show the problem is globally well-posed in $n = 3$ when $s > {2/5}$, and when $n \geq 4$, for $s > \frac{n - 2}{n}$. We combine energy increments for the I-method, interaction Morawetz estimates, and almost Morawetz estimates to prove the result.
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