Antonio Azzollini, Alessio Pomponio
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.
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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
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 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.
We present a numerical study of a derivative nonlinear Schrödinger equation with a general power nonlinearity, $|ψ|^{2σ}ψ_x$. In the $L^2$-supercritical regime, $σ>1$, our simulations indicate that there is a finite time singularity. We obtain a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schrödinger equation with supercritical power law nonlinearity.
This paper is concerned with the quantum Zakharov system. We prove that when the ionic speed of sound goes to infinity, the solution to the fourth order Schrodinger part of the quantum Zakharov system converges to the solution to quantum modified nonlinear Schrodinger eqaution.
Dispersive decay for the mass-critical nonlinear Schrödinger equation
该研究证明了在空间维度1、2、3中，质量临界非线性薛定谔方程的解具有逐点时间上的色散衰减性质。
Recent advances on Schrödinger equations with dissipative nonlinearities
本文综述了带耗散结构的非线性薛定谔方程的最新进展，基于作者近期工作总结了该领域的研究成果。
We prove existence and multiplicity of bound and ground state solutions, under appropriate conditions on the parameters, for a bi-harmonic stationary system of coupled nonlinear Schrödinger--Korteweg-de Vries equations.
We obtain global well-posedness, scattering, and global $L^{\frac{2(n+2)}{n-2}}_{t,x}$ spacetime bounds for energy-space solutions to the energy-critical nonlinear Schrödinger equation in $\R_t\times \R^n_x$, $n\geq 5$.
Higher order asymptotics for the nonlinear Schrödinger equation
本文利用∂̄-非线性最速下降法，在加权Sobolev空间的初始条件假设下，推导散焦非线性薛定谔方程的高阶长时间渐近行为。
Scattering for the nonlinear Schrödinger equation with concentrated nonlinearity
本文研究一维立方散焦非线性薛定谔方程在单点集中非线性下的全局适定性与散射，并证明该现象对空间充分集中的非线性效应普遍成立。
In this paper, we prove the exponential stability of the solution of the nonlinear dissipative Schrödinger equation on a star-shaped network and where the damping is localized on one branch and at the infinity.
We study a system of nonlinear Schrödinger equations with cubic interactions in one space dimension. The orbital stability and instability of semitrivial standing wave solutions are studied for both non-degenerate and degenerate cases.
We prove that the weakly damped nonlinear Schrödinger flow in $L^2(\mathbb{R})$ provides a dynamical system which possesses a global attractor. The proof relies on the continuity of the Schrödinger flow for the weak topology in $L^2(\R)$.
We prove new well-posedness results for energy-critical nonlinear Schrödinger equations in modulation spaces. This covers initial data with infinite mass and energy. The proof is carried out via bilinear refinements and adapted function spaces.
An explicit lifespan estimate is presented for the derivative Schrödinger equations with periodic boundary condition.
We construct solutions to a class of Schrödinger equations involving the fractional laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.
In this paper, we consider the energy critical nonlinear Schrödinger equation with a repulsive inverse square potential. In particular, we deal with radial initial data, whose energy is equal to the energy of static solution to the corresponding nonlinear Schrödinger equation without a potential. We investigate time behavior of the radial solutions with such initial data.
In this work, we explain in what sense the generic level set of the constants of motion for the periodic nonlinear Schrodinger equation is an infinite dimensional torus on which each generalized nonlinear Schrodinger flow is reduced to straight line almost periodic motion, and describe how neighboring generic infinite dimensional tori are connected.
In this paper we prove an almost sure local well-posedness result for the periodic 3D quintic nonlinear Schrödinger equation in the supercritical regime, that is below the critical space $H^1(\mathbb T^3)$.
The present paper is devoted to weighted Nonlinear Schrödinger- Poisson systems with potentials possibly unbounded and vanishing at infinity. Using a purely variational approach, we prove the existence of solutions concentrating on a circle.
We show that the knowledge of the Dirichlet--to--Neumann map for a nonlinear magnetic Schrödinger operator on the boundary of a compact complex manifold, equipped with a Kähler metric and admitting sufficiently many global holomorphic functions, determines the nonlinear magnetic and electric potentials uniquely.
We study the instability of standing wave solutions for nonlinear Schrödinger equations with a one-dimensional harmonic potential in dimension $N\ge 2$. We prove that if the nonlinearity is $L^2$-critical or supercritical in dimension $N-1$, then any ground states are strongly unstable by blowup.
We consider the nonlinear Schrödinger equation with a repulsive Dirac delta potential in one dimensional Euclidean space. We classify the global dynamics of even solutions with the same action as the high-frequency ground state standing wave solutions.
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.
We revisit the work of Bourgain on the invariance of the Gibbs measure for the cubic, defocusing nonlinear Schrödinger equation in 2D on a square torus, and we prove the equivalent result on any tori.
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