The nonlinear wave and Schrodinger equations on Euclidean space of any dimension, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space of index s whenever the exponent s is lower than that predicted by scaling or Galilean invariances, or when the regularity is too low to support distributional solutions. This extends previous work of the authors, which treated the one-dimensional cubic nonlinear Schrodinger equation. In the defocusing case soliton or blowup examples are unavailable, and a proof of ill-posedness requires the construction of other solutions. In earlier work this was achieved using certain long-time asymptotic behavior which occurs only for low power nonlinearities. Here we analyze instead a class of solutions for which the zero-dispersion limit provides a good approximation.
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
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
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 $\|\nabla u(t)\|_{L^2}$.
Cauchy problem of nonlinear Schrödinger equation with Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$
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 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 $0 \le θ\le 1$.
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 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.
In this short note, we present some decay estimates for nonlinear solutions of 3d quintic, 3d cubic and 2d quintic NLS (nonlinear Schrödinger equations).
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.
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.
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.
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.
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 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 $H^1(\mathbb T^3)$.
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.
