Related papers: The cubic fourth-order Schrodinger equation

In this paper, we consider the defocusing cubic nonlinear wave equation $u_{tt}-\Delta u+|u|^2u=0$ in the energy-supercritical regime, in dimensions $d\geq 6$, with no radial assumption on the initial data. We prove that if a solution…

In this paper, we study the local well-posedness of the cubic Schr\"odinger equation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I \times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ being an…

We prove global well-posedness for the cubic, defocusing, nonlinear Schr{\"o}dinger equation on $\mathbf{R}^{2}$ with data $u_{0} \in H^{s}(\mathbf{R}^{2})$, $s > 1/4$. We accomplish this by improving the almost Morawetz estimates in [9].

In this paper, we consider a class of nonlinear fourth-order Schr\"odinger equation, namely \[ \left\{ \begin{array}{rcl} i\partial_t u +\Delta^2 u &=&-|u|^{\nu-1} u, \quad 1+ \frac{8}{d}<\nu <1+\frac{8}{d-4},\\ u(0)&=&u_0 \in…

We study the global well-posedness of the two-dimensional defocusing fourth-order Schr\"odinger initial value problem with power type nonlinearities $\vert u\vert^{2k}u$ where $k$ is a positive integer. By using the $I$-method, we prove…

We consider the fourth-order Schr\"odinger equation $$ i\partial_tu+\Delta^2 u+\mu\Delta u+\lambda|u|^\alpha u=0, $$ where $\alpha>0,\mu=\pm1$ or $0$ and $\lambda\in\mathbb{C}$. Firstly, we prove local well-posedness in…

In this paper, we study the local well-posedness of the cubic Schr\"odinger equation $$(i\partial_t + \mathcal{L}) u = \pm |u|^2 u \qquad \textrm{on} \quad \ I\times \mathbb{R}^d ,$$ with initial data being a Wiener randomization at unit…

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…

We establish that the quadratic non-linear Schr\"odinger equation $$ iu_t + u_{xx} = u^2$$ where $u: \R \times \R \to \C$, is locally well-posed in $H^s(\R)$ when $s \geq -1$ and ill-posed when $s < -1$. Previous work of Kenig, Ponce and…

We investigate the initial value problem for some defocusing coupled nonlinear fourth-order Schrodinger equations. Global well-posedness and scattering in the energy space are obtained.

In this paper, we first prove global well-posedness for the defocusing cubic nonlinear Schr\"odinger equation (NLS) on 4-dimensional tori - either rational or irrational - and with initial data in $H^1$. Furthermore, we prove that if a…

We consider the cubic defocusing nonlinear Schr\"odinger equation in one dimension with the nonlinearity concentrated at a single point. We prove global well-posedness in the scaling-critical space $L^2(\mathbb{R})$ and scattering for all…

We establish that the initial value problem for the quadratic non-linear Schr\"odinger equation $$ iu_t - \Delta u = u^2$$ where $u: \R^2 \times \R \to \C$, is locally well-posed in $H^s(\R^2)$ when $s > -1$. The critical exponent for this…

We investigate the defocusing inhomogeneous nonlinear Schr\"odinger equation $$ i \partial_tu + \Delta u = |x|^{-b} \left({\rm e}^{\alpha|u|^2} - 1- \alpha |u|^2 \right) u, \quad u(0)=u_0, \quad x \in \mathbb{R}^2, $$ with $0<b<1$ and…

In this paper, we prove the global well-posedness of defocusing 3D quadratic nonlinear Schr\"odinger equation \begin{align*} i\partial_t u + \frac12\Delta u = |u| u, \end{align*} in its sharp critical weighted space $\mathcal F \dot…

In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schr\"odinger equation. Recently, Dodson [arXiv:2004.09618] studied the global well-posedness in a critical Sobolev space $\dot{W}^{11/7,7/6}$. In this…

We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - \Delta u + u \pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in the…

We consider the defocusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation (INLS) $iu_t + \Delta u = |x|^{-b}|u|^{k}u$ in $\mathbb{R} \times \mathbb{R}^{n}$ where $n \geq 3$, $0<b<\min(2, n/2)$, and $k=(4-2b)/(n-2)$. We…

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 focusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation \[ iu_t + \Delta u = -|x|^{-b}|u|^{\alpha}u \] where $n \geq 3$, $0<b<\min(2, n/2)$, and $\alpha=(4-2b)/(n-2)$. We prove the global well-posedness and…