Related papers: Global well-posedness for the derivative nonlinear…

The Cauchy problem for the 1-dimensional Zakharov system is shown to be globally well-posed for large data which not necessarily have finite energy. The proof combines the local well-posedness result of Ginibre, Tsutsumi, Velo and a general…

We consider the periodic fractional nonlinear Schr\"{o}dinger equation $$ iu_t -(-\Delta)^{\frac{s}{2}} u + \mathcal{N}(|u|)u=0, \quad x\in \mathbb{T}^N,\, \, t \in \mathbb R, \, \, s>0, $$ where the nonlinearity term is expressed in two…

We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first…

We prove global well-posedness for the cubic nonlinear Schr\"odinger equation for periodic initial data in the mass-critical dimension $d=2$ for initial data of arbitrary size in the defocusing case and data below the ground state threshold…

In this paper, we prove that the cubic nonlinear Schr\"odinger equation with the fractional Laplacian on the unit disk is globally well-posed for certain radial initial data below the energy space. The result is proved by extending the…

We prove that the Cauchy problem for the Schr\"odinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sovolev spaces $L^2(\R)\times H^{-{3/4}}(\R)$. The new ingredient is that we use the $\bar{F}^s$…

In this paper we study the Cauchy problem associated to the Maxwell-Schr\"odinger system with a defocusing pure-power non-linearity. This system has many applications in physics, for instance in the description of a charged non-relativistic…

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…

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 study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr\"{o}dinger (IBNLS) equation \[iu_{t} +\Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $d\in \mathbb N$,…

In this paper we establish an almost optimal well-posedness and regularity theory for the Klein-Gordon-Schr\"odinger system on the half line. In particular we prove local-in-time well-posedness for rough initial data in Sobolev spaces of…

This article is devoted to the study of a nonlinear and nonlocal parabolic equation introduced by Stefan Steinerberger to study the roots of polynomials under differentiation; it also appeared in a work by Dimitri Shlyakhtenko and Terence…

We consider the nonlinear Schr\"odinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced…

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].

We consider the Cauchy problem for the nonlinear Schr\"odinger equation on $\mathbb{R}^2$, $iu_t + u_{xx} + u_{yy} + \lambda|u|^\sigma u =0$, $\lambda\in \mathbb{R}$, $\sigma>0$. We introduce new functional spaces over which the initial…

The local and global well-posedness for the one dimensional fourth-order nonlinear Schr\"odinger equation are established in the modulation space $M^{s}_{2,q}$ for $s\geq \frac12$ and $2\leq q <\infty$. The local result is based on the…

We consider the Cauchy problem for the fourth order cubic nonlinear Schr\"odinger equation (4NLS). The main goal of this paper is to prove low regularity well-posedness and mild ill-posedness for (4NLS). We prove three results. First, we…

We consider the Cauchy problem for the nonlinear Schr\"odinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $\mathbb{R}^d$, $d=5,6$, and energy-critical NLS…

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…

The Cauchy problem for a higher order modification of the nonlinear Shcrodinger equation (MNLS) on the line is shown to be well-posed in Sobolev spaces with exponent $\ge 0$. This result is achieved by demonstrating that the associated…