相关论文: On the two-power nonlinear Schr\"odinger equation …

We consider a two-dimensional nonlinear Schr\"odinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well…

We derive new results about existence and uniqueness of local and global solutions for nonlinear Schrodinger equation, including self-similar global solutions. Our analysis is performed in the framework of Marcinkiewicz spaces.

We prove the well-posed results in sub-critical and critical cases for the pure power-type nonlinear fractional Schr\"odinger equations on $\mathbb{R}^d$. These results extend the previous ones in \cite{HongSire} for $\sigma\geq 2$. This…

We consider stationary $p$-Schr\"odinger equations on the whole space with integrable data and potentials that are confining in measure. We introduce asymptotic energy solutions in an asymptotic $L^p$ framework and establish existence and…

We consider the stochastic nonlinear Schr\"odinger equations (SNLS) posed on $d$-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness…

We study the well posedness of the nonlinear Schr\"odinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so…

In this paper we prove local well-posedness of a space-time fractional generalization of the nonlinear Schr\"odinger equation with a power-type nonlinearity. The linear equation coincides with a model proposed by Naber, and displays a…

We consider a dispersive equation of Schr{\"o}dinger type with a non-linearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with LHY…

We study the Cauchy problem for the non-linear Schr\"odinger equation with singular potentials. For point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-L^{p} spaces. Specific…

We study the nonlinear Schr\"odinger equation with initial data in $\mathcal{Z}^s_p(\mathbb{R}^d)=\dot{H}^s(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, where $0<s<\min\{d/2,1\}$ and $2<p<2d/(d-2s)$. After showing that the linear Schr\"odinger…

We establish the existence of an entire solution for a class of stationary Schr\"{o}dinger equations with subcritical discontinuous nonlinearity and lower bounded potential that blows-up at infinity. The abstract framework is related to…

We present numerical simulations of the defocusing nonlinear Schrodinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig-Merle and Kilip-Visan who considered some energy…

We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schr\''{o}dinger operator under the presence of a singular nonlinear term. Among other new facts, with respect…

Local and global existence of localized solutions of a discrete nonlinear Schrodinger (DNLS) equation, with arbitrary on-site nonlinearity, is proved. In particular, it is shown that an initially localized excitation persists localized…

Schr\"odinger equations with nonlinearities concentrated in some regions of space are good models of various physical situations and have interesting mathematical properties. We show that in the semiclassical limit it is possible to…

In this work, we consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^n$ \begin{align} i\partial_t u + \Delta u + \gamma |x|^{-b}|u|^{\alpha} u = 0, \end{align} where $\gamma=\pm 1$, and $\alpha$ and $b$ are…

We study the Cauchy problem for the $1$-d periodic fractional Schr\"odinger equation with cubic nonlinearity. In particular we prove local well-posedness in Sobolev spaces, for solutions evolving from rough initial data. In addition we show…

We consider a class of nonlinear Schr\"odinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$)…

We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schr\"odinger type and have recently been obtained in \cite{DLS}…

We investigate the global well-posedness and modified scattering for the one-dimensional Schr\"odinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we…