Related papers: The cubic nonlinear Schr\"odinger equation in two …
In this paper, we consider the following inhomogeneous nonlinear Schr\"odinger equation (INLS) \[ i\partial_t u + \Delta u + \mu |x|^{-b} |u|^\alpha u = 0, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d \] with $b, \alpha>0$. First, we…
We consider the focusing inhomogeneous biharmonic nonlinear Schr\"odinger equation in $H^2(\mathbb{R}^N)$, \begin{equation} iu_t + \Delta^2 u - |x|^{-b}|u|^{\alpha}u=0 \end{equation} when $b > 0$ and $N \geq 5$. We first obtain a small data…
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…
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…
In this paper, we first prove that the cubic, defocusing nonlinear Schr\"odinger equation on the two dimensional hyperbolic space with radial initial data in $H^s(\mathbb{H}^2)$ is globally well-posed and scatters when $s > \frac{3}{4}$.…
We consider the 3-dimensional nonlinear Schr\"{o}dinger equation (NLS) with average nonlinearity. This is a limiting model of NLS with strong magnetic confinement and a generalized model of the resonant system of NLS with a partial harmonic…
The focusing cubic nonlinear Schr\"odinger equation in two dimensions admits vortex solitons, standing wave solutions with spatial structure, Qm(r,theta) = e^{i m theta} Rm(r). In the case of spin m = 1, we prove there exists a class of…
We study the following nonlinear Schr\"odinger equation with a forth order dispersion term \[ \Delta^2u-\beta\Delta u=g(u) \quad \text{in } \mathbb{R}^N \] in the positive and zero mass regimes: in the former, $N\geq 2$ and $\beta >…
We prove a general, non-perturbative result about finite-time blowup solutions for the $L^2$-critical boson star equation $i\partial_t u = \sqrt{-\Delta+m^2} \, u - (|x|^{-1} \ast |u|^2) u$ in 3 space dimensions. Under the sole assumption…
In this paper, we consider the following equation: \[ i\frac{\partial u}{\partial t}+\Delta u+g(x)|u|^{\frac{4}{N}}u-Wu=0. \] We construct a critical-mass solution that blows up at a finite time and describe the behaviour of the solution in…
In this paper, we study the scattering theory for the cubic inhomogeneous Schr\"odinger equations with inverse square potential $iu_t+\Delta u-\frac{a}{|x|^2}u=\lambda |x|^{-b}|u|^2u$ with $a>-\frac14$ and $0<b<1$ in dimension three. In 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…
In this article, we will show the global wellposedness and scattering of the cubic defocusing nonlinear Schr\"odinger equation on waveguide $\mathbb{R}^2\times \mathbb{T}$ in $H^1$. We first establish the linear profile decomposition in…
In this paper we consider the long time behavior of solutions to the cubic nonlinear Schr\"odinger equation posed on the spatial domain $\mathbb{R}\times\mathbb{T}^{d}$, $1\leq d\leq4$. For sufficiently small, smooth, decaying data we prove…
In this paper, we study the probabilistic local well-posedness of the cubic Schr\"odinger equation (cubic NLS): \[ (i\partial_{t} + \Delta) u = \pm |u|^{2} u \text{ on } [0,T) \times \mathbb{R}^{d}, \] with initial data being a Wiener…
In this paper, we introduce two minimization problems on non-scattering solutions to nonlinear Schr\"odinger equation. One gives us a sharp scattering criterion, the other is concerned with minimal size of blowup profiles. We first…
We consider the following nonlinear Schr\"{o}dinger equation with double power nonlinearities and an inverse power potential: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u+C_1|u|^{p-1}u+\frac{C_2}{|x|^{2\sigma}}u=0 \] in…
We make two observations concerning the generalised Korteweg de Vries equation $u_t + u_{xxx} = \mu (|u|^{p-1} u)_x$. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for…
We prove global existence and stability of solution to the mass-critical stochastic nonlinear Schr\"odinger equation in $d=1$ at $L^2$ regularity. Our construction starts with the existence of solution to the truncated subcritical problem.…
We consider large time behavior of solutions to the nonlinear Schr\"odinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We treat the case in which the nonlinearity contains…