Related papers: Scattering for a mass critical NLS system below th…

We consider the quadratic nonlinear Schr\"{o}dinger system (NLS system) \begin{align*}\begin{cases} i\partial_t u + \Delta u = v \overline{u}, \\ i\partial_t v+\kappa \Delta v = u^2, \end{cases} \text{ on } I \times \mathbb{R}^5,…

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^\alpha u = 0\quad\text{on}\quad\mathbb{R}\times\mathbb{R}^N, \] with $N\geq 2$, $0<b<\min\{\tfrac{N}{2},2\}$, and…

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…

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…

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…

We prove scattering below the mass-energy threshold for the focusing inhomogeneous nonlinear Schr\"odinger equation \begin{equation} iu_t + \Delta u + |x|^{-b}|u|^{p-1}u=0, \end{equation} when $b \geq 0$ and $N > 2$ in the intercritical…

We consider the mass-subcritical nonlinear Schr\"odinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity $s_c\in(\max\{-1,-\frac{d}{2}\},0)$, we prove that any solution…

In this article, we consider the dynamics of the energy-critical quadratic nonlinear Schr\"odinger system $\[ \left\{ \begin{aligned} & i u^1_t + \kappa_1 \Delta u^1 = -\overline{u^2}u^3, \\ & i u^2_t + \kappa_2 \Delta u^2 =…

In this paper, we simplify the proof of M. Hamano in \cite{Hamano2018}, scattering theory of the solution to \eqref{NLS system}, by using the method from B. Dodson and J. Murphy in \cite{Dodson2018}. Firstly, we establish a criterion to…

We consider the three-dimensional cubic nonlinear Schr\"odinger system \begin{equation*} \begin{cases} i\partial_tu+\Delta u+(|u|^2+\beta |v|^2)u=0,\\ i\partial_tv+\Delta v+(|v|^2+\beta |u|^2)v=0. \end{cases} \end{equation*} Let $(P,Q)$ be…

In this paper, we study the long-time behavior of global solutions to the Schr\"odinger-Choquard equation $$i\partial_tu+\Delta u=-(I_\alpha\ast|\cdot|^b|u|^{p})|\cdot|^b|u|^{p-2}u.$$ Inspired by Murphy, who gave a simple proof of…

We consider the focusing nonlinear Schr\"odinger equation $i u_t + \Delta u + |u|^{p-1}u=0$, $p>1,$ and the generalized Hartree equation $iv_t + \Delta v + (|x|^{-(N-\gamma)}\ast |v|^p)|v|^{p-2}u=0$, $p\geq2$, $\gamma<N$, in the…

We investigate a class of nonlinear equations of Schr\"odinger type with competing inhomogeneous nonlinearities in the non-radial inter-critical regime, \begin{align*} i \partial_t u +\Delta u &=|x|^{-b_1} |u|^{p_1-2} u - |x|^{-b_2}…

In this paper, we consider the scattering problem for a class of $N$-coupled systems of the cubic nonlinear Schr\"odinger equations in three space dimensions. We prove the scattering of solutions that have a mass-energy quantity less than…

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…

This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schr\"odinger equation of power type with data in a critical homogeneous weighted $L^2$ space. We give a sharp sufficient condition for…

In this paper we study the scattering of non-radial solutions in the energy space to coupled system of nonlinear Schr\"{o}dinger equations with quadratic-type growth interactions in dimension five without the mass-resonance condition. Our…

We study the scattering for the energy-subcritical stochastic nonlinear Schr\"odinger equation (SNLS) with additive noise. In particular, we examine the long-time behavior of solutions associated with the noise…

We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…

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…