Related papers: Stabilization of small solutions of discrete NLS w…
We consider nonlinear Schr\"odinger equations in $\R^3$. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states…
Using the Fermi Golden Rule analysis developed in several results by the first author, we prove asymptotic stability of asymmetric nonlinear bound states bifurcating from linear bound states for a quintic nonlinear Schr\"odinger operator…
We consider perturbations of the one-dimensional cubic Schr\"odinger equation, of the form $i \, \partial_t \psi + \partial_x^2 \psi + |\psi|^2 \psi + g( |\psi|^2 ) \psi = 0$. Under hypotheses on the function $g$ that can be easily verified…
The long-time asymptotics is analyzed for finite energy solutions of the 1D Schr\"odinger equation coupled to a nonlinear oscillator; mathematically the system under study is a nonlinear Schr\"odinger equation, whose nonlinear term includes…
For the Schr\"odinger equation with a cubic-quintic, focusing-focusing nonlinearity in one space dimension, this article proves the local asymptotic completeness of the family of small standing solitary waves under even perturbations in the…
In this paper, we analyze the long-time dynamics of small solutions to the $1d$ cubic nonlinear Schr\"odinger equation (NLS) with a trapping potential. We show that every small solution will decompose into a small solitary wave and a…
We prove the existence of a 2-parameter family of small quasi-periodic in time solutions of discrete nonlinear Schr\"odinger equation (DNLS). We further show that all small solutions of DNLS decouples to this quasi-periodic solution and…
In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schr\"odinger equations (NLS) that we gave in \cite{CM15APDE}. We consider a NLS with a Schr\"odinger…
We investigate the asymptotic stability of standing waves for a model of Schr\"odinger equation with spatially concentrated nonlinearity in space dimension three. The nonlinearity studied is a power nonlinearity concentrated at the point…
We consider the initial-value problem for the one-dimensional nonlinear Schr\"odinger equation in the presence of an attractive delta potential. We show that for sufficiently small initial data, the corresponding global solution decomposes…
Consider a nonlinear Schr\"odinger equation in $R^3$ whose linear part has three or more eigenvalues satisfying some resonance conditions. Solutions which are initially small in $H^1 \cap L^1(R^3)$ and inside a neighborhood of the first…
We give short survey on the question of asymptotic stability of ground states of nonlinear Schr\"odinger equations, focusing primarily on the so called nonlinear Fermi Golden Rule.
We consider nonlinear Schr\"{o}dinger equations, $i\partial_t \psi = H_0 \psi + \lambda |\psi|^2\psi$ in $\mathbb{R}^3 \times [0,\infty)$, where $H_0 = -\Delta + V$, $\lambda=\pm 1$, the potential $V$ is radial and spatially decaying, and…
We consider a class of nonlinear Schr\"odinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$)…
We consider Schr\"odinger operators $H=- \d^2/\d r^2+V$ on $L^2([0,\infty))$ with the Dirichlet boundary condition. The potential $V$ may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum of $H$ is…
We consider the nonlinear Schr\"odinger equation on a unit ball in one and two dimensions with Dirichlet boundary conditions, which have stabilizing effect on solutions behavior. In particular, we confirm that the ground state solutions are…
We consider the existence and stability of real-valued, spatially antiperiodic standing wave solutions to a family of nonlinear Schr\"odinger equations with fractional dispersion and power-law nonlinearity. As a key technical result, we…
We prove for a class of nonlinear Schr\"odinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and…
We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schr\"odinger equation with a point interaction and a focusing power nonlinearity. The Schr\"odinger operator with a point interaction…
We explain how spectrally stable vortices of the Nonlinear Schr\"odinger Equation in the plane can be orbitally unstable. This relates to the nonlinear Fermi golden rule, a mechanism which exploits the nonlinear interaction between discrete…