Related papers: Scattering for the dispersion managed nonlinear Sc…
We consider the focusing cubic nonlinear Schr\"odinger equation \begin{align}\label{CNLSS} i\partial_t U+\Delta U=-|U|^2U\quad\text{on $\mathbb{R}^2\times\mathbb{T}$}.\tag{3NLS} \end{align} Different from the 3D Euclidean case, the…
This article is devoted to a general class of one dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and…
In this paper we prove that the defocusing, quintic nonlinear Schr\"odinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R})$. To do this, we will prove a frequency localized interaction Morawetz…
We study the scattering behavior of global solutions to stochastic nonlinear Schr\"odinger equations with linear multiplicative noise. In the case where the quadratic variation of the noise is globally finite and the nonlinearity is…
We obtain global well-posedness, scattering, and global $L_t^4H_{x}^{1,4}$ spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schr\"odinger equation \[iu_t+\Delta u=u(e^{4\pi |u|^2}-1)\] in two spatial…
We prove several scattering results for dispersion-managed nonlinear Schr\"odinger equations. In particular, we establish small-data scattering for both `intercritical' and `mass-subcritical' powers by suitable modifications of the standard…
We consider the one-dimensional nonlinear Schr\"odinger equation with a nonlinearity of degree $p>1$. We exhibit measures on the space of initial data for which we describe the non trivial evolution by the linear Schr\"odinger flow and we…
We investigate the large time behavior of the solutions to the nonlinear focusing Schr\"odinger equation with a time-dependent damping in the energy sub-critical regime. Under non classical assumptions on the unsteady damping term, we prove…
We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schr\"{o}dinger equations \begin{equation*} \begin{cases} &(i\partial_t+\Delta_\g)u=u|u|^{2\sigma}; &u(0)=\phi, \end{cases} \end{equation*} on the…
Scattering for the mass-critical fractional Schr\"odinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is…
We prove local and global well-posedness results for the Gabitov-Turitsyn or dispersion managed nonlinear Schr\"odinger equation with a large class of nonlinearities and arbitrary average dispersion on $L^2(\mathbb{R})$ and…
In this paper, we consider the Cauchy's problem of global existence and scattering behavior of small, smooth, and localized solutions of cubic fractional Schr\"odinger equations in one dimension, \begin{equation*} \mathrm{i} \partial_t u-…
We consider the nonlinear Schr\"odinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the…
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…
We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…
This article is concerned with one dimensional dispersive flows with cubic nonlinearities on the real line. In a very recent work, the authors have introduced a broad conjecture for such flows, asserting that in the defocusing case, small…
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…
In the present paper, we consider the Cauchy problem of fourth order nonlinear Schr\"odinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schr\"odinger equation with the…
We consider the cubic nonlinear Schr\"odinger equation with an exceptional potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions. We prove the existence of…
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…