Related papers: Uniqueness of the modified Schroedinger map in H^{…

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H \subseteq V^*$: \begin{align*} \left\{ \begin{aligned} dX(t) & = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in…

We modify the Schr\"{o}dinger equation in a way that preserves its main properties but makes use of higher order derivative terms. Although the modification represents an analogy to the Doebner-Goldin modification, it can differ from it…

We use the dispersive properties of the linear Schr\"{o}dinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $\mathbb{R}^d$ for $d\geq 2$. The proofs…

We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schr\"odinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space…

We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in Sugimoto (2011), of the intersection $M^s_{p,q}(\mathbb{R}^d) \cap…

The solution of the Chern-Simons-Higgs model in Lorenz gauge with data for the potential in $H^{s-1/2}$ and for the Higgs field in $H^s \times H^{s-1}$ is shown to be unique in the natural space $C([0,T];H^{s-1/2} \times H^s \times…

We give necessary conditions for a set E to be removable for Holder continuous quasiregular mappings in the plane. We also obtain some removability results for Holder continuous mappings of finite distortion.

The aim of the paper is twofold. We establish refined Strichartz estimates for the Schr\"odinger equation on tori within the framework of partial regularity. As a result, we reveal that the solution of the free Schr\"odinger equation has…

In this paper, we prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schr\"odinger equation. We show that a unique solution exists for $u_{0} \in H^{s}(\mathbf{R})$, $s > {8/29}$. This…

We consider equivariant solutions for the Schr\"odinger map problem from $\mathbb{R}^{2+1}$ to $\mathbb{S}^2$ with energy less than $4\pi$ and show that they are global in time and scatter.

In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a…

We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of…

We study the random data problem for 3D, defocusing, cubic nonlinear Schr\"odinger equation in $H_x^s(\mathbb{R}^3)$ with $s<\frac 12$. First, we prove that the almost sure local well-posedness holds when $\frac{1}{6}\leqslant s<\frac 12$…

We study the presence of exact localized solutions in a quadratic-cubic nonlinear Schr\"odinger equation with inhomogeneous nonlinearities. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one,…

We prove that the modified KdV equation is unconditionally well-posed in H s (T) for s $\ge$ 1/3.

We prove sharp $L^\infty$ decay and modified scattering for the Schr\"odinger-Bopp-Podolsky equation in $2$ and $3$ spatial dimensions with small initial data chosen from a weighted Sobolev space.

In this paper we prove that the 1D Schr\"odinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>\frac12$ for data small in $L^{2}$. To understand the strength of this result one should recall that…

We consider a randomization of a function on $\mathbb R^d$ that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to…

This paper constructs solutions to linear and nonlinear Schr\"odinger-type equations in two and three spatial dimensions that exhibit prescribed, extraordinary gradient amplification and localization. For any finite time interval $[0,T]$,…

Using the theory developed by Kenig, Ponce, and Vega, we prove that the Hirota-Satsuma system is locally well-posed in Sobolev spaces $H^s(\mathbb{R}) \times H^{s}(\mathbb{R})$ for $3/4<s\le1$. We introduce some Bourgain-type spaces…