Related papers: Global well-posedness for the nonlinear Schr\"{o}d…

As a continuation of the previous work \cite{Wu}, we consider the global well-posedness for the derivative nonlinear Schr\"odinger equation. We prove that it is globally well-posed in energy space, provided that the initial data $u_0\in…

We prove that the 1D Schr\"odinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>2/3$ for small $L^{2}$ data. The result follows from an application of the ``I-method''. This method allows to…

We consider the supercritical inhomogeneous nonlinear Schr\"odinger equation (INLS) $$i\partial_t u+\Delta u+|x|^{-b}|u|^{2\sigma}u=0,$$ where $(2-b)/N<\sigma<(2-b)/(N-2)$ and $0<b<\min\{2,N\}$. We prove a Gagliardo-Nirenberg type estimate…

We study the Derivative Nonlinear Schr\"odinger (DNLS). equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities corresponding to algebraic solitons). We show…

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…

In this paper, we consider the well-posedness of the weakly damped stochastic nonlinear Schr\"odinger(NLS) equation driven by multiplicative noise. First, we show the global existence of the unique solution for the damped stochastic NLS…

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 consider the Schr\"odinger equation in dimension two with a fixed, pointwise, focusing nonlinearity and show the occurrence of a blow-up phenomenon with two peculiar features: first, the energy threshold under which all solutions blow up…

We show that the derivative nonlinear Schr\"odinger (DNLS) equation is globally well-posed in the weighted Sobolev space $H^{2,2}(\mathbb{R})$. Our result exploits the complete integrability of DNLS and removes certain spectral conditions…

We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schr\"odinger equation (NLS), $i \partial_t u+\Delta u+ |u|^{p-1}u=0, $ with initial data $u_0\in H^1,\; x \in R^n$. The nonlinearity power…

We prove that the derivative nonlinear Schr\"{o}dinger equation is globally well-posed in $H^{\frac 12} (\mathbb{R})$ when the mass of initial data is strictly less than $4\pi$.

We prove, for the energy critcal, focusing NLW, that for Cauchy data (u_0, u_1) whose energy is smaller than that of (W,0), where W is the well-known radial positive solution to the corresponding ellipyic equation, the following dichotomy…

We study the well-posedness of the generalized derivative nonlinear Schr\"odinger equation (gDNLS) $$iu_t+u_{xx}=i|u|^{2\sigma}u_x,$$ for small powers $\sigma$. We analyze this equation at both low and high regularity, and are able to…

In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schr\"odinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some…

We investigate the local and global well-posedness of the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on $\mathbb{R}$, described by \[ i\partial_t u + \partial_x^2 u = i\alpha \partial_x (|u|^2 u) + i\beta \partial_x…

In this paper we continue our study [DSS20] of the nonlinear Schr\"odinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on $\mathbb{R}$ was proved for real analytic data. Here we prove…

We prove that the derivative nonlinear Schr\"odinger equation in one space dimension is globally well-posed on the line in $L^2(\mathbb{R})$, which is the scaling-critical space for this equation.

This paper is dedicated to the study of the derivative nonlinear Schr\"odinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global…

We study the defocusing energy-critical inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_tu+\Delta u=|x|^{-b}|u|^{\frac{4-2b}{d-2}}u, \qquad (t,x)\in\R\times\R^d, \] with initial data $u_0\in\dot H_x^1(\R^d)$, where $d\ge 3$ and…

We review recent results on global wellposedness and long-time behavior of smooth solutions to the derivative nonlinear Schr\"{o}dinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and…