相关论文: Modified scattering for the cubic Schr{\"o}dinger …
We study the long-time behavior of the (critical) Kohn--Sham equation in two and three dimensions, i.e.,\[ \mathrm{i} \partial_t {\gamma} = \Big[-\frac{1}{2}\Delta + \lambda \, |\cdot|^{-1} \ast \rho_{{\gamma}} + \mu \,…
We consider the Schroedinger equation with a general interaction term, which is localized in space. The interaction may be x, t dependent and non-linear. Purely non-linear parts of the interaction are localized via the radial Sobolev…
We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T^d, d \geq 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length…
We consider the small time semi-classical limit for nonlinear Schrodinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To…
We prove scattering for small solutions to of nonlinear Schroedinger equations in 1D with a space periodic potential
We develop a modified semi-classical approach to the approximate solution of Schrodinger's equation for certain nonlinear quantum oscillations problems. At lowest order, the Hamilton-Jacobi equation of the conventional semi-classical…
We use multiscale perturbation theory in conjunction with the inverse scattering transform to study the interaction of a number of solitons of the cubic nonlinear Schroedinger equation under the influence of a small correction to the…
We prove several scattering results for dispersion-managed nonlinear Schr\"odinger equations. In particular, we establish small-data scattering for both `intercritical' and `mass-subcritical' powers by suitable modifications of the standard…
The inverse scattering method for the Novikov-Veselov equation is studied for a larger class of Schr\"odinger potentials than could be handled previously. Previous work concerns so-called conductivity type potentials, which have a bounded…
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…
We study the scattering theory for the Schr\"odinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of the paper concerns with an inverse-square potential in both of subcritical and…
We establish a small-data modified scattering result for the $1d$ cubic dispersion-managed NLS (with time-dependent dispersion map) for initial data in a weighted space.
We consider the existence of a dynamically stable soliton in the one-dimensional cubic-quintic nonlinear Schr\"odinger model with strong cubic nonlinearity management for periodic and random modulations. We show that the predictions of the…
In this paper, we discuss a class of nonlinear Schr\"odinger equations with the power-type nonlinearity: $(\mathrm{i} \frac{\partial}{\partial t} + \Delta ) \psi = \lambda |\psi|^{2\eta}\psi$ in $\mathbf R^N \times \mathbf R^+$. Based on…
In this article, we investigate the semiclassical version of the wave equation for the discrete Schr\"{o}dinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where…
In this paper we study 3-d Hartree type fractional Schr\"odin-ger equations: \begin{equation} i\partial_{t}u-|\nabla|^{\alpha}u = \lambda\left(|x|^{-\gamma} *| u|^{2} \right)u,\;\;1 < \alpha < 2,\;\;0 < \gamma < 3,\;\; \lambda \in \mathbb R…
We present some physically interesting, in general non-stationary, one-dimensional solutions to the nonlinear phase modification of the Schr\"{o}dinger equation proposed recently. The solutions include a coherent state, a phase-modified…
Strichartz estimates, well-posedness theory and long time behavior for (nonlinear) Schr\"odinger equations on waveguide manifolds $\mathbb{R}^m \times \mathbb{T}^n$ are intensively studied in recent decades while the corresponding control…
In this paper, we prove the decay and scattering in the energy space for nonlinear Schr\"odinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + \Delta u - V(x)u + \lambda |u|^{p - 1}u = 0$. We will prove decay…
The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schr\"odinger equation $$ i u_t +\Delta u+|x|^{-b}|u|^2 u = 0, $$ where $0<b<1/2$. Let $Q$ be the ground state solution of $-Q+\Delta Q+ |x|^{-b}|Q|^{2}Q=0$ and…