Related papers: On scattering for two-dimensional quintic Schr\"od…
The nonlinear Schr\"odinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation which was first derived by means of…
We consider the cubic nonlinear Schr\"odinger equation with harmonic trapping on $\mathbb{R}^D$ ($1\leq D\leq 5$). In the case when all but one directions are trapped (a.k.a "cigar-shaped" trap), following the approach of…
The aim of this text is to develop on the asymptotics of some 1-D nonlinear Schrodinger equations from both the theoretical and the numerical perspectives, when a caustic is formed. We review rigorous results in the field and give some…
We investigate the scattering theory for the nonlinear Schr\"{o}dinger equation $i \partial_{t}u+ \Delta u+\lambda|u|^\alpha u=0$ in $\Sigma=H^{1}(\mathbb{R}^{d})\cap L^{2}(|x|^{2};dx)$. We show that scattering states $u^{\pm}$ exist in…
In this paper, we consider the quadratic nonlinear Schr\"odinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved…
We present an iterative algorithm to compute numerical approximations of the potential for the Schr\"odinger operator from scattering data. Four different types of scattering data are used as follows: fixed energy, fixed incident angle,…
We consider the cubic nonlinear Schr\"odinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^d$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\leq…
We investigate the following inhomogeneous nonlinear Schr\"odinger equation in the radial regime, featuring a focusing energy-critical nonlinearity and a defocusing perturbation: $$ i\partial_t u +\Delta u =|x|^{-a} |u|^{p-2} u - |x|^{-b}…
We consider the problem of identifying sharp criteria under which radial $H^1$ (finite energy) solutions to the focusing 3d cubic nonlinear Schr\"odinger equation (NLS) $i\partial_t u + \Delta u + |u|^2u=0$ scatter, i.e. approach the…
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…
In this paper we consider the nonlinear Schr\"odinger system (NLS) with quadratic interaction in five dimensions. We determine the global behavior of the solutions to the system with data below the ground state. Our proof of the scattering…
We consider the problem of large data scattering for the defocusing cubic nonlinear Schr\"odinger equation on $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$. This equation is critical both at the level of energy and mass. The key ingredients…
In this paper, we explore the integrable fractional derivative nonlinear Schr\"odinger (fDNLS) equation by using the inverse scattering transform. Firstly, we start from the recursion operator and obtain a formal fDNLS equation. Then the…
We prove a scattering result near certain steady states for a Hartree equation for a random field. This equation describes the evolution of a system of infinitely many particles. It is an analogous formulation of the usual Hartree equation…
The aim of this paper is to develop the inverse scattering transform (IST) for multi-component generalisations of nonlocal reductions of the nonlinear Schrodinger (NLS) equation with PT-symmetry related to symmetric spaces. This includes:…
In this paper, we study the $2$D cubic nonlinear Schr\"odinger equation (NLS) with the partial harmonic potential. First, we prove the local well-posedness in Bourgain spaces by establishing a key bilinear estimate associated with the…
In this paper we consider the Cauchy initial value problem for the defocusing quintic nonlinear Schr\"odinger equation in $\mathbb{R}^2$ with general data in the critical space $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$. We show that if a…
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…
For the first time, a nonlinear Schr\"odinger equation of the general form is considered, depending on time and two spatial variables, the potential and dispersion of which are specified by two arbitrary functions. This equation naturally…
The semiclassical limit of the focusing Nonlinear (cubic) Schr\" odinger Equation (NLS) corresponds to the singularly perturbed Zakharov Shabat (ZS) system that defines the direct and inverse scattering transforms (IST). In this paper, we…