Related papers: Scattering for quadratic-type Schr\"{o}dinger syst…
Consider the focusing inhomogeneous nonlinear Schr\"odinger equation in $H^1(\mathbb{R}^N)$, $$iu_t + \Delta u + |x|^{-b}|u|^{p-1}u=0,$$ when $b > 0$ and $N \geq 3$ in the intercritical case $0 < s_c <1$. In previous works, the second…
This work is concerned with a coupled system of focusing nonlinear Schr\"odinger equations involving general power-type nonlinearities in the energy-critical setting for dimensions $3\leq d\leq 5$ in the radial setting. Our aim is to…
We prove small data scattering for the fourth-order Schr\"odinger equation with quadratic nonlinearity \begin{equation*} i\partial_t u+\Delta^2 u+\alpha u^2 + \beta \bar{u}^2=0\qquad\text{in }\mathbb{R}^5 \end{equation*} for $\alpha, \beta…
We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the non-radial $\dot H^{1/2}$-critical focusing NLS. By proving an interaction Morawetz inequality, we give a simple proof of scattering below the ground state in…
Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schr\"odinger equation below a mass-energy threshold $M[u]E[u] < M[Q]E[Q]$ and satisfying an initial mass-gradient bound $\|u_0\|_{L^2} \|\nabla u_0 \|_{L^2} <…
We study the energy-critical focusing nonlinear Schr\"odinger equation with an energy- subcritical perturbation. We show the existence of a ground state in the four or higher dimensions. Moreover, we give a sufficient and necessary…
We revisit the problem of scattering below the ground state threshold for the mass-supercritical focusing nonlinear Schr\"odinger equation in two space dimensions. We present a simple new proof that treats the case of radial initial data.…
We prove decay with respect to some Lebesgue norms for a class of Schr\"odinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space…
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 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 =…
We investigate the large time behavior of the solutions to the nonlinear focusing Schr\"odinger equation with a time-dependent damping in the energy sub-critical regime. Under non classical assumptions on the unsteady damping term, we prove…
We consider the Zakharov system in the energy critical dimension $d=4$ with energy below the ground state. It is known that below the ground state solutions exist globally in time, and scatter in the radial case. Scattering below the ground…
We consider the scattering results of the radial solutions below the ground state to the focusing inhomogeneous nonlinear Schr\"odinger equation $$i\partial_tu+\Delta u +|x|^{-b}|u|^{p}u=0$$ in two dimension, where $0<b<1$ and…
In this paper we consider the real-valued mass-critical nonlinear Klein-Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the…
We show that the solutions of the three-dimensional critical defocusing nonlinear wave equation with Neumann boundary conditions outside a ball and radial initial data scatter. This is to our knowledge the first result of scattering for a…
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 investigate scattering properties of a Moyal deformed version of the nonlinear Schr\"odinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general…
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…
In this article, we consider the nonlinear Schr\"odinger equation on the cylinder $\mathbb{R}^d\times \mathbb{T}$. In the long range case, we show there is no linear scattering state of the nonlinear Schr\"odinger equation on $\mathbb{R}^d…
In this paper, we prove the scattering for radial solutions to energy-critical nonlinear Schr\"odinger equations with regular potentials in defocusing case.