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The purpose of this work is to study the $3D$ energy-critical inhomogeneous generalized Hartree equation $$ i\pa_tu+\Delta u+|x|^{-b}(I_\alpha\ast|\cdot|^{-b}|u|^{p})|u|^{p-2}u=0,\;\ x\in\R^3, $$ where $p=3+\alpha-2b$. We establish global well-posedness and scattering below the ground state threshold with non-radial initial data in $\dot{H}^1$. To this end, we exploit the decay of the nonlinearity, which together with the Kenig-Merle roadmap, allows us to treat the non-radial case as the radial case. In this paper are introduced new techniques to overcome the challenges posed by the presence of the potential and the nonlocal nonlinear term of convolution type. In particular, we also show scattering for the classical generalized Hartree equation ($b=0$) assuming radial data. Additionally, in the defocusing case, we show scattering with general data. We believe that the ideas developed here are robust and can be applicable to other types of nonlinear Hartree equations. In the introduction, we discuss some open problems., Comment: 27 pages

In this paper, we study the dynamics behavior of the NLS system with three waves interaction in the energy space $H^1(\mathbb{R}^5) \times H^1(\mathbb{R}^5)\times H^1(\mathbb{R}^5) $. Inspired by B. Dodson and J. Murphy in \cite{Dodson2018}, we establish an interaction Morawetz estimate for the NLS system, together with the criterion which proved by Tao-Dodson--Murphy we can get the scattering under the ground state in energy space with mass-resonance. Under the radial assumption, we can remove the mass-resonance condition., Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:2001.06835

In this article, we consider the dynamics of the energy-critical quadratic nonlinear Schr\"odinger system $\[ \left\{ \begin{aligned} & i u^1_t + \kappa_1 \Delta u^1 = -\overline{u^2}u^3, \\ & i u^2_t + \kappa_2 \Delta u^2 = -\overline{u^1}u^3, \\ & i u^3_t + \kappa_3 \Delta u^3 = -u^1u^2, \\ \end{aligned} \right. \qquad (t, x) \in \R \times \R^6 \] in energy-space $ {\dot H}^1 \times {\dot H}^1\times{\dot H}^1 $, where the sign of potential energy can not be determined. We prove the scattering theory with mass-resonance (or with radial initial data) below ground state via concentration compactness method. We discover a family of new physically conserved quantities with mass-resonance which play an important role in the proof of scattering., Comment: In this version, we correct some mistakes, add some chapters and change the title

In this article, we aim to study the scattering of the solution to the focusing inhomogeneous nonlinear Schr\"odinger equation with a potential of form \begin{align*} i\partial_t u+\Delta u- Vu=-|x|^{-b}|u|^{p-1}u \end{align*} in the energy space $H^1(\R^3)$. We prove a scattering criterion, and then we use it together with Morawetz estimate to show the scattering theory, which generalizes the results of Dinh \cite{DD} to the non-radial symmetric case., Comment: In this version, we correct some mistakes and change the title

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 radial or non-radial but satisfies the mass-resonance condition, and its energy is below that of the ground state, using the compactness/rigidity method, we give a complete classification of scattering versus blowing-up dichotomies depending on whether the kinetic energy of ${\rm u}_0$ is below or above that of the ground state., Comment: 34 pages

In this paper, we prove a limiting absorption principle for high-order Schr\"odinger operators with a large class of potentials which generalize some results by A. Ionescu and W. Schlag. Our main idea is to handle the boundary operators by the restriction theorem of Fourier transform. Two key tools we use in this paper are the Stein--Tomas theorem in Lorentz spaces and a sharp trace lemma given by S. Agmon and L. H\"ormander, Comment: 15 pages. All comments are welcome

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 scattering for the non-radial inhomogeneous NLS, we prove scattering theory below the ground state for the intercritical case in energy space without radial assumption.

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 ensure the solution scatters in $ H^1(\mathbb{R}^5) \times H^1(\mathbb{R}^5) $. In order to verify the correctness of the condition in scattering criterion, we must exclude the concentration of mass near the origin. The interaction Morawetz estimate and Galilean transform characterize a decay estimate, which implies that the mass of the system cannot be concentrated.

In this paper, we study the long time behavior of the solution of nonlinear Schr\"odinger equation with a singular potential. We prove scattering below the ground state for the radial NLS with inverse-square potential in dimension two $$iu_t+\Delta u-\frac{a u}{|x|^2}= -|u|^pu$$ when $20$. This work extends the result in [13, 14, 16] to dimension 2D. The key point is a modified version of Arora-Dodson-Murphy's approach [2]., Comment: 14pages

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

In this article, we consider the focusing cubic nonlinear Schr\"odinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state of Killip-Visan-Zhang by utilizing Dodson and Murphy's argument and the dispersive estimate established by Ivanovici and Lebeau, which avoids using the concentration compactness. We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary. Based on this effect and the interaction Morawetz estimates, we prove the solution decays at a large time interval, which meets the scattering criterions., Comment: 15 pages, comments are welcome

In this paper, we give a simple proof of scattering result for the Schr\"odinger equation with combined term $i\pa_tu+\Delta u=|u|^2u-|u|^4u$ in dimension three, that avoids the concentrate compactness method. The main new ingredient is to extend the scattering criterion to energy-critical., Comment: 12 pages