Related papers: Classification of Asymptotic Profiles for Nonlinea…
We investigate the asymptotic behavior at time infinity of solutions close to a non-zero constant equilibrium for the Gross-Pitaevskii (or Ginzburg-Landau Schroedinger) equation. We prove that, in dimensions larger than 3, small…
We consider large time asymptotics for damped nonlinear Schr\"{o}dinger equations. It is known that the nonlinear solution asymptotically behaves like a linear solution when time $t$ tends to infinity in the energy space. We prove that its…
A nonlocal-in-time problem for the abstract Schr\"odinger equation is considered. By exploiting the linear nature of nonlocal condition we derive an exact representation of the solution operator under assumptions that the spectrum of…
We study the nonlinear Schr\"odinger equation with initial data in $\mathcal{Z}^s_p(\mathbb{R}^d)=\dot{H}^s(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, where $0<s<\min\{d/2,1\}$ and $2<p<2d/(d-2s)$. After showing that the linear Schr\"odinger…
The simplest nonlinear Schrodinger equation that contains the time derivative of the probability density is investigated. This equation has the same stationary solutions as its linear counterpart, and these solutions are the eigenstates of…
We consider the Schr\"odinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +\Delta u=\lambda|u|^{\alpha}u \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $\lambda\in {\mathbb C} $ with $\Im\lambda<0$. Assuming…
We consider a nonlinear semi-classical Schrodinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we…
In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schr\"odinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same…
In this paper we announce the result of asymptotic dynamics of solitons of nonlinear Schrodinger equations with external potentials. To each local minima of the potential there is a soliton centered around it. Under some conditions on the…
We are concerned with the existence of solutions to the following nonlinear Schr\"odinger system in $\mathbb{R}^3$: \begin{equation*} \left\{ \begin{aligned} -\Delta u_1 + (x_1^2+x_2^2)u_1&= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta…
In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Schr"odinger equations in one space dimension. It turns out that for a system there exists a small solution of which asymptotic…
We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schr\"odinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction…
The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schr\"odinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group. For initial states…
We consider the cubic Schr\"odinger equation on the euclidean space perturbed by a short-range potential $V$. The presence of a negative simple eigenvalue for $-\Delta+V$ gives rise to a curve of small and localized nonlinear ground states…
Using the recent analysis of the output of the low-energy resolvent of Schr\"odinger operators on asymptotically conic manifolds (including Euclidean space) when the potential is short-range, we produce detailed asymptotic expansions for…
We revisit aspects of dynamics and stability of localized states in the deterministic and stochastic discrete nonlinear Schr\"odinger equation. By a combination of analytic and numerical techniques, we show that localized initial conditions…
In numerical experiments involving nonlinear solitary waves propagating through nonhomogeneous media one observes "breathing" in the sense of the amplitude of the wave going up and down on a much faster scale than the motion of the wave. In…
It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for…
We consider the large time behavior of solutions to defocusing nonlinear Schrodinger equation in the presence of a time dependent external potential. The main assumption on the potential is that it grows at most quadratically in space,…
We show global asymptotic stability of solitary waves of the nonlinear Schr\"odinger equation in space dimension 1. Furthermore, the radiation is shown to exhibit long range scattering if the nonlinearity is cubic at the origin, or standard…