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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…

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require,…

We prove that the initial value problem for the equation \[ - i\partial_t u + \sqrt{m^2-\Delta} \, u= (\frac{e^{-\mu_0 |x|}}{|x|} \ast |u|^2)u \ \text{in} \ \mathbb R^{1+3}, \quad m\ge 0, \ \mu_0 >0\] is globally well-posed and the solution…

In this paper, we consider the following three dimensional defocusing cubic nonlinear Schr\"odinger equation (NLS) with partial harmonic potential \begin{equation*}\tag{NLS} i\partial_t u + \left(\Delta_{\mathbb{R}^3 }-x^2 \right) u = |u|^2…

In this paper, we consider global solutions of the following nonlinear Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u = 0,$ in $\R^N,$ with $\lambda\in\R,$ $\alpha\in(0,\frac{4}{N-2})$ $(\alpha\in(0,\infty)$ if $N=1)$ and…

This paper is a continuation of our previous study on the long time behavior of solution to the nonlinear Schr"odinger equation with higher order anisotropic dispersion (4NLS). We prove the long range scattering for (4NLS) with the…

We investigate the scattering theory for the nonlinear Schr\"{o}dinger equation $i \partial_{t}u+ \Delta u+\lambda|u|^\alpha u=0$ in $\Sigma=H^{1}(\mathbb{R}^{d})\cap L^{2}(|x|^{2};dx)$. We show that scattering states $u^{\pm}$ exist in…

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 study the solutions to the energy-critical quadratic nonlinear Schr\"odinger system in ${\dot H}^1\times{\dot H}^1$, where the sign of its potential energy can not be determined directly. If the initial data ${\rm u}_0$ is…

We investigate the cubic defocusing fourth order Schr\"odinger equation $iu_t + \Delta^2u + |u|^2u=0$ in arbitrary space dimension $\mathbb{R}^n$ for arbitrary $H^2$ initial data. We prove that the equation is globally well-posed when $n…

We consider a class of nonlinear Schr\"odinger equations with potential \[ i\partial_t u +\Delta u - Vu = \pm |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^3, \] where $\frac{4}{3}<\alpha<4$ and $V$ is a Kato-type potential. We…

In this paper, we study the defocusing energy-critical nonlinear Schr\"odinger equations $$ i\partial_t u + \Delta u = |u|^{\frac{4}{d-2}} u. $$ When $d=3,4$, we prove the almost sure scattering for the equations with non-radial data in…

We obtain global well-posedness, scattering, and global $L_t^4H_{x}^{1,4}$ spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schr\"odinger equation \[iu_t+\Delta u=u(e^{4\pi |u|^2}-1)\] in two spatial…

This paper deals with the 2-D Schr\"odinger equation with time-oscillating exponential nonlinearity $i\partial_t u+\Delta u= \theta(\omega t)\big(e^{4\pi|u|^2}-1\big)$, where $\theta$ is a periodic $C^1$-function. We prove that for a class…

For $n\geq 3$, we study the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equations, for which the existence of the scattering operators and the global well-posedness of solutions with small data in Besov spaces…

We consider the quintic generalized Benjamin-Bona-Mahony equation $$ u_t-u_{xxt} + \partial_x\big(u + u^{5}\big)= 0,\qquad (t,x)\in \mathbb{R}_+ \times \mathbb{R}. $$ Using the space-time resonance method, we prove that sufficiently small…

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schr\"odinger equation with a subcritical dissipative nonlinearity $\lambda |u|^\alpha u$, where $0<\alpha<2$, and $\lambda $ is a complex constant…

The fifth-order KP II equation $$ \partial_t u + \alpha \partial_x^3 u + \beta \partial_x^5 u + u \partial_x u + \partial_x^{-1} \partial_y^2u=0$$ ($\beta<0$, $\alpha>0$) is a nonlinear dispersive equation that models long dispersive waves…

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…

We study the scattering problem for the nonlinear Schr\"odinger equation $i\partial_t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that…