Related papers: Focusing NLS with inverse square potential

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^\alpha u = 0\qtq{on}\R\times\R^N, \] with $\alpha=\tfrac{4-2b}{N-2}$, $N=\{3,4,5\}$ and $0<b\leq…

In this paper, we consider the 3d cubic focusing inhomogeneous nonlinear Schr\"{o}dinger equation with a potential $$ iu_{t}+\Delta u-Vu+|x|^{-b}|u|^{2}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{3}}, $$ where $0<b<1$. We first establish…

In this paper, we study the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-power potential \[iu_{t} +\Delta u-c|x|^{-a}u=\pm |x|^{-b} |u|^{\sigma } u,\;\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where…

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} <…

In this paper, we prove the scattering for radial solutions to energy-critical nonlinear Schr\"odinger equations with regular potentials in defocusing case.

In this paper, we explore the integrable fractional derivative nonlinear Schr\"odinger (fDNLS) equation by using the inverse scattering transform. Firstly, we start from the recursion operator and obtain a formal fDNLS equation. Then the…

We consider a class of biharmonic nonlinear Schr\"odinger equations with a focusing inhomogeneous power-type nonlinearity \[ i\partial_t u -\Delta^2 u+\mu\Delta u +|x|^{-b} |u|^\alpha u=0, \quad \left. u\right|_{t=0}=u_0 \in…

We consider the $L^2$-critical NLS with inverse-square potential $$ i \partial_t u +\Delta u + c|x|^{-2} u = -|u|^{\frac{4}{d}} u, \quad u(0) = u_0, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, $$ where $d\geq 3$ and $c\ne 0$ satisfies…

In this paper, we first provide an alternative proof of the uniqueness of the ground state solution for NLS with inverse square potential and power nonlinearity $|u|^pu$ for all $0<p<\frac 4{d-2}$ in dimensions $d\ge 3$. While the…

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 consider the energy critical nonlinear Schr\"odinger equation with a repulsive inverse square potential. In particular, we deal with radial initial data, whose energy is equal to the energy of static solution to the…

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…

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…

In this paper, we focus on the inverse scattering problem for the nonlinear Schrodinger equation with magnetic potentials. Specifically, we investigate whether the scattering operator associated with the nonlinear Schrodinger equation can…

This paper is mainly concerned with the inverse scattering problem of determining the unknown potential for the classical Schr\"odinger equation in two and three dimensions. We establish the increasing stability of the inverse scattering…

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 consider the defocusing $\dot H^1$-critical nonlinear Schr\"odinger equation in all dimensions ($n\geq 3$) with a quadratic potential $V(x)=\pm \tfrac12 |x|^2$. We show global well-posedness for radial initial data obeying $\nabla…

We consider the one-dimensional Schr\"odinger equation with a potential satisfying the standard assumptions of the inverse scattering theory and supported on the half-line $x\ge 0$. For this equation at fixed positive energy we give…

A one-dimensional generalized nonlinear Schroedinger equation is considered, and the corresponding inverse scattering problem is analyzed when the potential is compactly supported and depends on the wave function. The unique recovery of the…

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…