Related papers: On the inhomogeneous NLS with inverse-square poten…

We consider the defocusing fourth-order nonlinear Schr\"{o}dinger equation with potential \[ i\partial_t u + \Delta^2 u + Vu + \lambda |u|^{p-1}u = 0 \qquad (x \in \mathbb{R}^n,\ t \in \mathbb{R}), \] in dimensions $n \ge 5$. In the…

This paper is dedicated to the blow-up solution for the divergence Schr\"{o}dinger equations with inhomogeneous nonlinearity (dINLS for short) \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)=-|x|^c|u|^pu,\quad\quad u(x,0)=u_0(x),\] where…

We consider the nonlinear Schr\"odinger equation $iu_t + \Delta u= \lambda |u|^{\frac {2} {N}} u $ in all dimensions $N\ge 1$, where $\lambda \in {\mathbb C}$ and $\Im \lambda \le 0$. We construct a class of initial values for which the…

In this paper, we construct for every $\alpha >0$ and $\lambda \in {\mathbb C}$ a space of initial values for which there exists a local solution of the nonlinear Schr\"odinger equation \begin{equation*} \begin{cases} iu_t + \Delta u +…

We consider the 1D nonlinear Schr\"odinger equation with focusing point nonlinearity. "Point" means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This…

We investigate the initial value problem for a defocusing nonlinear Schr\"odinger equation with weighted exponential nonlinearity $$ i\partial_t u+\Delta u=\frac{u}{|x|^b}(e^{\alpha|u|^2}-1); \qquad (t,x) \in \mathbb{R}\times\mathbb{R}^2,…

We study the defocusing energy-critical inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_tu+\Delta u=|x|^{-b}|u|^{\frac{4-2b}{d-2}}u, \qquad (t,x)\in\R\times\R^d, \] with initial data $u_0\in\dot H_x^1(\R^d)$, where $d\ge 3$ and…

\We consider an inverse scattering problem for Schr\"odinger operators with energy dependent potentials. The inverse problem is formulated as a Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for two distinct…

We study inhomogeneous heat equation with inverse square potential, namely, \[\partial_tu + \mathcal{L}_a u= \pm |\cdot|^{-b} |u|^{\alpha}u,\] where $\mathcal{L}_a=-\Delta + a |x|^{-2}.$ We establish some fixed-time decay estimate for…

For the stationary nonlinear Schr\"odinger equation $-\Delta u+ V(x)u- f(u) = \lambda u$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^2$-norm. To this end we introduce…

In this paper we consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation \begin{align}\label{inls} i \partial_t u +\Delta u +|x|^{-b} |u|^{2\sigma}u = 0, \,\,\, x \in \mathbb{R}^N \end{align} with $N\geq 3$. We focus on the…

We consider the following nonlinear Schr\"{o}dinger equation with an inverse potential: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2\sigma}}u=0 \] in $\mathbb{R}^N$. From the classical argument, the…

In this paper we study radial solutions of certain two-dimensional nonlinear Schr\"odinger equation with harmonic potential, which is supercritical with respect to the initial data. By combining the nonlinear smoothing effect of Schr…

In this paper, we study the stability and instability of the ground states for the focusing inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-square potential (for short, INLS$_c$ equation): \[iu_{t} +\Delta…

An inverse problem is considered for an inhomogeneous Schr\"odinger equation. Assuming that the potential vanishes outside a finite interval and satisfies some other technical assumptions, one proves the uniqueness of the recovery of this…

Thanks to an approach inspired from Burq-Lebeau \cite{bule}, we prove stochastic versions of Strichartz estimates for Schr\"odinger with harmonic potential. As a consequence, we show that the nonlinear Schr\"odinger equation with quadratic…

We consider the nonlinear Schr\"odinger equation on ${\mathbb R}^N $, $N\ge 1$, \begin{equation*} \partial _t u = i \Delta u + \lambda | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \end{equation*} with $\lambda \in {\mathbb…

We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\partial_t u - \Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3,…

This paper is concerned with the existence of normalized solutions of the nonlinear Schr\"odinger equation \[ -\Delta u+V(x)u+\lambda u = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in the mass supercritical and Sobolev subcritical case…

We study the scattering theory for the Schr\"odinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of the paper concerns with an inverse-square potential in both of subcritical and…