Related papers: On defocusing fourth-order coupled nonlinear Schro…
We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. We study in detail…
We consider the defocusing energy-critical nonlinear Schr\"odinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions. For the initial-value problem with Dirichlet boundary condition we prove global…
The local and global well-posedness for the one dimensional fourth-order nonlinear Schr\"odinger equation are established in the modulation space $M^{s}_{2,q}$ for $s\geq \frac12$ and $2\leq q <\infty$. The local result is based on the…
In this paper, we consider the defocusing nonlinear Schr\"odinger equation in space dimensions $d\geq 4$. We prove that if $u$ is a radial solution which is \emph{priori} bounded in the critical Sobolev space, that is, $u\in L_t^\infty…
In this paper we prove that the defocusing, mass - critical generalized KdV initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R})$. We prove this via a concentration compactness argument.
In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u-V(x)u…
This note studies the asymptotic behavior of global solutions to the fourth-order Schr\"odinger equation $$i\dot u+\Delta^2 u+F(x,u)=0 .$$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing…
We consider a class of defocusing energy-supercritical nonlinear Schr\"odinger equations in four space dimensions. Following a concentration-compactness approach, we show that for $1<s_c<3/2$, any solution that remains bounded in the…
In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= |u|^p u, \quad (t,x)\in \mathbb R^{d+1}, $$ with $p\in (\frac4d,\frac4{d-2})$. We prove that…
We study the scattering behavior of global solutions to stochastic nonlinear Schr\"odinger equations with linear multiplicative noise. In the case where the quadratic variation of the noise is globally finite and the nonlinearity is…
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…
We consider the Cauchy problem for the defocusing cubic nonlinear Schr\"odinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in…
We investigate an energy-subcritical defocusing nonlinear Schr\"odinger equation in $\mathbb R^3$ subject to a lower order nonlinear trapping potential and a spatially dependent nonlinear damping: \begin{equation*} i\partial_t u + \Delta u…
We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schr\"odinger equation $iu_t + \Delta u = |u|^{4/n} u$ for large spherically symmetric $L^2_x(\R^n)$ initial data in…
We investigate the global well-posedness and modified scattering for the one-dimensional Schr\"odinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we…
We consider the Cauchy problem for the energy-critical nonlinear Schr\"odinger equation with fractional Laplacian (fNLS) in the radial case. We obtain global well-posedness and scattering in the energy space in the defocusing case, and in…
In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with general nonlinearity \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u + f(u)=0,\ (x, t)…
The initial value problem for the cubic defocusing nonlinear Schr\"odinger equation $i \partial_t u + \Delta u = |u|^2 u$ on the plane is shown to be globally well-posed for initial data in $H^s (\R^2)$ provided $s>1/2$. The proof relies…
In this paper, we consider the initial value problem for the quintic, defocusing nonlinear Schr\"odinger equation on $\Bbb T^2$ with general data in the critical Sobolev space $H^{\frac{1}{2}} (\Bbb T^2)$. We show that if a solution remains…
In this paper, we study the global well-posedness and scattering problem in the energy space for both focusing and defocusing the Klein-Gordon-Hartree equation in the spatial dimension $d \geq 3$. The main difficulties are the absence of an…