Rémi Carles
We review some results concerning the semi-classical limit for the nonlinear Schrodinger equation, with or without an external potential. We consider initial data which are either of the WKB type, or very concentrated as the semi-classical parameter goes to zero. We sketch the techniques used according to various frameworks, and point out some open problems.
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We prove the existence of ground state solutions for the nonlinear Schrodinger-Maxwell equations.
The initial value problem for some defocusing coupled nonlinear Schrodinger equations is investigated. Global well-posedness and scattering are established.
A regularized $α-$system of the Nonlinear Schrödinger Equation (NLS) with $2σ$ nonlinear power in dimension $N$ is studied. We prove existence and uniqueness of local solution in the case $1 \le σ<\frac{4}{N-2}$ and existence and uniqueness of global solution in the case $1 \le σ< \frac{4}{N}$. When $α\to 0^+$, this regularized system will converge to the classical NLS in the appropriate range. In particular, the purpose of this numerical regularization is to shed light on the profile of the blow up solutions of the original Nonlinear Schrödinger Equation in the range $\frac{2}{N}\le σ<\frac{4…
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.
We prove that for the mass critical nonlinear Schrodinger equations, the wave operators and their inverse are related explicitly in terms of the Fourier transform. We discuss some consequences of this property. In the one-dimensional case, we show a precise similarity between the mass critical nonlinear Schrodinger equation and a nonlinear Schrodinger equation of derivative type.
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We prove the existence of ground state solutions for the nonlinear Schrodinger-Maxwell equations with a singular potential.
We prove asymptotic stability of trapped solitons in the generalized nonlinear Schrödinger equation with a potential in dimension 1 and for even potential and even initial conditions.
Using a sharp Gagliardo-Nirenberg type inequality, well-posedness issues of the initial value problem for a fractional inhomogeneous Schrodinger equation are investigated.
In this paper we prove the existence of a nontrivial solution to the nonlinear Schrodinger-Maxwell equations in $\R^3,$ assuming on the nonlinearity the general hypotheses introduced by Berestycki & Lions.
We prove the existence of ground state in a multidimensional nonlinear Schrodinger model of paraxial beam propagation in isotropic local media with saturable nonlinearity. Such ground states exist in the form of bright counterpropagating solitons. From the proof, a general threshold condition on the beam coupling constant for the existence of such fundamental solitons follows.
Coupled nonlinear Schrodinger systems describe some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optics and Bose-Einstein condensates. In this paper, we study the existence of concentrating solutions of a singularly perturbed coupled nonlinear Schrodinger system, in presence of potentials. We show how the location of the concentration points depends strictly on the potentials.
Well-posedness and global attractor are established for 2D damped driven nonlinear Schrödinger equation with almost periodic pumping in a bounded region. The key role is played by a novel application of the energy equation.
A generalized nonlinear Schrödinger equation admits WTC formal solutions as was shown by Clarkson. We show that they are convergent and determine real-analytic solutions near a noncharacteristic, movable singularity manifold.
We consider a system of nonlinear Schrodinger equations with three waves interaction studying the existence of ground state solutions. In particular, we find a vector ground state, namely a ground state with the three components all different from zero.
Well-posedness of the periodic nonlinear Schrödinger equation with concentrated nonlinearity
研究环面上具有集中非线性的非线性薛定谔方程的解理论，通过基于紧致性结果和利用沃尔泰拉积分方程结构的方法，建立全局能量守恒解存在唯一性及局部适定性，是该方程首个严格解理论。
We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schrödinger equations $$ iu_t+Δu=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}^2\times\mathbb{R}$,}$$ in the energy class.
In this paper we consider generalized nonlinear Schrödinger equations with external potentials. we compute the forth and the sixed order Fermi Golden Rules (FGR), conjectured in our previous papers, which is used in a study of the asymptotic dynamics of trapped solitons.
We construct space quasi-periodic standing wave solutions to the nonlinear Schrödinger equations on R^d for arbitrary d. This is a type of quasi-periodic nonlinear Bloch-Floquet waves.
We show that the thermodynamics of the focusing cubic discrete nonlinear Schrodinger equation are exactly solvable in dimensions three and higher. A number of explicit formulas are derived.
We consider construction of ansatzes for nonlinear Schrodinger equations in three space dimensions and arbitrary nonlinearity, and conditions of their reduction to ordinary differential equations. Complete description of ansatzes of certain types is presented. We also discuss the relationship between solutions, and both Lie and conditional symmetry of these equations.
The nonlinear Schrödinger equation with sprinkled nonlinearity
本文证明了非线性项集中在齐次泊松过程上的三次非线性薛定谔方程的全局适定性，建立了该随机非线性的解的存在性与唯一性。
We prove strong instability (instability by blowup) of standing waves for some nonlinear Schrödinger equations with double power nonlinearity.
Homogenization for the nonlinear Schrödinger equation with sprinkled nonlinearity
本文研究了非均匀非线性薛定谔方程的均质化问题，并探讨了解在均质化解附近的波动特性。
In this paper we use a concentration and compactness argument to prove the existence of a nontrivial nonradial solution to the nonlinear Schrodinger-Poisson equations in R3, assuming on the nonlinearity the general hypotheses introduced by Berestycki & Lions
The authors suggest a new powerful tool for solving group classification problems, that is applied to obtaining the complete group classification in the class of nonlinear Schrödinger equations of the form $iψ_t+Δψ+F(ψ,ψ^*)=0$.
We analize the Nonlinear Schrödinger Equation.
On recovering the nonlinearity for generalized higher-order Schrödinger equations
本文将经典三次非线性Schrödinger方程的非线性恢复结果推广至广义高阶Schrödinger方程，通过小振幅波包数据解恢复空间局部化系数。
In this paper, we introduce some new ideas to study Schrodinger equations in RN with power-type nonlinearities.
We prove the existence of non-trivial solutions to a system of coupled, nonlinear, Schroedinger equations with general nonlinearity.
In this paper, regularity properties of Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained.
In this article, we consider nonlinear Schrödinger equation with nonlocal nonlinearity which is a generalized model of the Schrödinger-Poisson system (Schrödinger-Newton equations) in low dimensions. We first prove the global well-posedness in wider space than in previous result and show the stability of standing waves including excites states. It turns out that an example of stable excite states with high Morse index is contained. Several examples of traveling-wave-type solutions are also given.
We give short survey on the question of asymptotic stability of ground states of nonlinear Schrödinger equations, focusing primarily on the so called nonlinear Fermi Golden Rule.
Local well-posedness for the nonlinear heat and Schrödinger equations with nonlinear Wentzell boundary conditions on time-dependent compact Riemannian manifolds
该研究针对时间依赖紧黎曼流形上带Wentzell型非线性动力边界条件的次临界半线性热方程与薛定谔方程，证明了其局部适定性，丰富了相关偏微分方程边界问题的适定性理论。
We consider the Cauchy problem for the $L^{2}$-critical nonlinear Schrödinger equation with a nonlinear damping. According to the power of the damping term, we prove the global existence or the existence of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.
In this paper, we consider in $R^n$ the Cauchy problem for nonlinear Schrödinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in $W^{s,p}$ for $\frac{2n}{n+1}<p<2$ and $s>n(1-\frac{1}{p})$. Moreover, we show that in one space dimension, the problem is locally well posed in $L^p$ for any $1<p<2$.
Large deviation principle by the weak convergence approach is established for the stochastic nonlinear Schrodinger equation in one-dimension and as an application the exit problem is investigated.
We study the derivative nonlinear Schrödinger equation on the real line and obtain global-in-time bounds on high order Sobolev norms.
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.
In this paper, we prove the scattering for radial solutions to energy-critical nonlinear Schrödinger equations with regular potentials in defocusing case.
In this paper we consider the transverse instability for a nonlinear Schrödinger equation with a linear potential on $\mathbb{R} \times \mathbb{T}_L$, where $2πL$ is the period of the torus $\mathbb{T}_L$. Rose and Weinstein showed the existence of a stable standing wave for a nonlinear Schrödinger equation with a linear potential. We regard the standing wave of nonlinear Schrödinger equation on $\mathbb{R}$ as a line standing wave of nonlinear Schrödinger equation on $\mathbb{R} \times \mathbb{T}_L$. We show the stability of line standing waves for all $L>0$.
Applying an Abstract Interpolation Lemma, we can show persistence of solutions of the initial value problem to higher order nonlinear Schrödinger equation, also called Airy-Schrödinger equation, in weighted Sobolev spaces $\mathcal{X}^{2,θ}$, for $0 \le θ\le 1$.
Traveling waves for nonlinear Schrödinger equations
该研究针对亚声速非线性薛定谔方程的行波解问题，采用含动量的Sobolev型不等式变体方法，证明了对应方程极小值子的存在性，覆盖多种含Sobolev次临界非线性效应的物理模型。
Continuum limit of 3D fractional nonlinear Schrödinger equation
该研究针对三维分数阶非线性薛定谔方程的连续极限问题，通过振荡积分一致估计与牛顿多面体技巧证明离散方程的一致Strichartz估计，得出离散解强收敛于连续解。
This paper is devoted to a comprehensive study of the nonlinear Schrödinger equations with combined nonlinearities of the power-type and Hartree-type in any dimension n\ge3. With some structural conditions, a nearly whole picture of the interactions of these nonlinearities in the energy space is given. The method is based on the Morawetz estimates and perturbation principles.
We consider the initial value problem for the fractional nonlinear Schrödinger equation with a fractional dissipation. Global existence and scattering are proved depending on the order of the fractional dissipation.
In this paper we investigate the existence of positive solutions and ground state solution for a class of fractional Schrödinger-Poisson equations in $\mathbb R^3$ with general nonlinearities.
We study the orbital stablity and instability of solitary wave solutions for nonlinear Schrödinger equations of derivative type.
We study the instability of standing waves for nonlinear Schrödinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a Strichartz type estimate for the propagator generated by the linearized operator around standing wave.
We justify supercritical geometric optics in small time for the defocusing semiclassical Nonlinear Schrodinger Equation for a large class of non-necessarily homogeneous nonlinearities. The case of a half-space with Neumann boundary condition is also studied.
Quantized Vortex Dynamics of the Nonlinear Schrödinger Equation with Wave Operator on the Torus
本文严格推导环面上带波算子的非线性 Schrödinger 方程在涡旋核心趋于零时的约化涡旋动力学，证明其为两类方程运动定律的混合，并通过数值模拟考察其收敛性。
In this short note, we present some decay estimates for nonlinear solutions of 3d quintic, 3d cubic and 2d quintic NLS (nonlinear Schrödinger equations).
On standing waves of 1D nonlinear Schrödinger equation with triple power nonlinearity
该研究解析与数值分析了一维三次幂非线性薛定谔方程驻波的存在性与稳定性，重点考察参数平面上的不存在性与稳定性变化曲线。
The initial value problem is considered for a higher order nonlinear Schrödinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional damping large-time decay of solutions without any smallness assumptions is also established. The main difficulty of the study is the non-smooth character of the nonlinearity.
We look for positive solutions to the nonlinear Schrodinger equation with a potential, under the hypothesis of zero mass on the nonlinearity, in a particular situation. Existence and multiplicity results are provided.
We prove the existence of global analytic solutions to the nonlinear Schrödinger equation in one dimension for a certain type of analytic initial data in $L^2$.
We investigate the initial value problem for some defocusing coupled nonlinear fourth-order Schrodinger equations. Global well-posedness and scattering in the energy space are obtained.
This paper is concerned with the Cauchy problem for an inhomogeneous nonlinear Schrodinger equation with exponential growth nonlinearity and harmonic potential in two space dimensions. We prove global well-posedness, existence of the associated ground state and instability of the standing wave.
We prove wellposedness of the Cauchy problem for the nonlinear Schrodinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.
Given $p$ points in a bounded domain of $\R^d$, with $d=2,3$, we show the existence of solutions of the $L^2$-critical focusing nonlinear Schrödinger equation blowing up exactly in these points.
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