Related papers: Threshold scattering for the focusing NLS with a r…
We investigate the $L^2$-supercritical and $\dot{H}^1$-subcritical nonlinear Schr\"{o}dinger equation in $H^1$. In \cite{G1} and \cite{yuan}, the mass-energy quantity $M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$ has been shown to be a threshold for…
The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space ($3\leq p<5$) whose initial data are radial and come with a finite energy. In…
We consider the $L^{2}$-critical quintic focusing nonlinear Schr\"odinger equation (NLS) on ${\bf R}$. It is well known that $H^{1}$ solutions of the aforementioned equation blow up in finite time. In higher dimensions, for $H^{1}$…
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 present paper is concerned with the large data scattering problem for the mass-energy double critical NLS \begin{align} i\partial_t u+\Delta u\pm |u|^{\frac{4}{d}}u\pm |u|^{\frac{4}{d-2}}u=0\tag{DCNLS} \end{align} in $H^1(\mathbb{R}^d)$…
In this paper, we study the scattering theory for the defocusing energy-critical Klein-Gordon equation with a cubic convolution $u_{tt}-\Delta u+u+(|x|^{-4}\ast|u|^2)u=0$ in the spatial dimension $d \geq 5$. We utilize the strategy in [S.…
In this paper, we consider the following inhomogeneous nonlinear Schr\"odinger equation (INLS) \[ i\partial_t u + \Delta u + \mu |x|^{-b} |u|^\alpha u = 0, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d \] with $b, \alpha>0$. First, we…
We show the $H^{1}$ scattering for a one dimensional nonlinear Schr\"odinger equation with a non-negative, repulsive potential $V$ such that $V,xV\in W^{1,1}$, and a mass-supercritical non-linearity. We follow the approach of…
We study the nonlinear Schr\"odinger equation posed on product spaces $\mathbf R^n\times \mathcal M^k$, for $n\geq 1$ and $k\geq1$, with $\mathcal M^k$ any $k$-dimensional compact Riemaniann manifold. The main results concern global…
We consider the Dirac $\delta$-function potential problem in general case of deformed Heisenberg algebra leading to the minimal length. Exact bound and scattering solutions of the problem in quasiposition representation are presented. We…
We consider the minimal mass $m_0$ required for solutions to the mass-critical nonlinear Schr\"odinger (NLS) equation $iu_t + \Delta u = \mu |u|^{4/d} u$ to blow up. If $m_0$ is finite, we show that there exists a minimal-mass solution…
In this paper, we consider the focusing mass-critical nonlinear fourth-order Schr\"odinger equation. We prove that blowup solutions to this equation with initial data in $H^\gamma(\mathbb{R}^d), 5\leq d \leq 7,…
We consider the nonlinear Schr\"odinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer…
We study the $d$-dimensional discrete nonlinear Schr\"odinger equation with general power nonlinearity and a delta potential. Our interest lies in the interplay between two localization mechanisms. On the one hand, the attractive…
In this paper we consider a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $p \in [3,5)$. We prove that if initial data $(u_0, u_1)$ are radial so that…
We consider the mass-critical focusing nonlinear Schrodinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently…
In this paper, we investigate the global well-posedness and $H^{1}$ scattering theory for a 3d energy-critical Schr\"odinger equation under the influence of magnetic dipole interaction $\lambda_{1}|u|^{2}u+\lambda_{2}(K\ast|u|^{2})u$, where…
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…
In this paper, we consider the Cauchy problem of Nonlinear Schr\"{o}dinger equation \begin{align*} \left\{\begin{array}{ll}&i u_t+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N…
In this article we will study the initial value problem for some Schr\"odinger equations with Diraclike initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in…