Related papers: Cubic-quintic NLS: scattering beyond the virial th…
We consider the Cauchy problem of the cubic nonlinear Schr\"odinger equation (NLS) on $\mathbb R^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling critical regularity $s_\text{crit} =…
We consider the nonlinear Schr{\"o}dinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a com-plete scattering theory is available, showing that both the potential and…
The Fourier transforms of the products of two respectively three solutions of the free Schroedinger equation in one space dimension are estimated in mixed and, in the first case weighted, L^p - norms. Inserted into an appropriate variant of…
We present a comprehensive study of stationary states in a coherent medium with a quadratic or Kerr nonlinearity in the presence of localized potentials in one dimension (1D) for both positive and negative signs of the nonlinear term, as…
We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schroedinger equations on the disc of the plane $\R^2$. We also prove an estimate giving some intuition to what may happen in 3…
We prove asymptotic completeness in the energy space for the nonlinear Schrodinger equation posed on hyperbolic space in the radial case, in space dimension at least 4, and for any energy-subcritical, defocusing, power nonlinearity. The…
We consider a dispersive equation of Schr{\"o}dinger type with a non-linearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with LHY…
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 (NLS) on $\mathbb{R}^3$ with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By…
In this article we study some aspects of dispersive and concentration phenomena for the Schr\"odinger equation posed on hyperbolic space $\mathbb{H}^n$, in order to see if the negative curvature of the manifold gets the dynamics more stable…
In this short note we present a new proof of the global well-posedness and scattering result for the defocusing energy-critical NLS in four space dimensions obtained previously by Ryckman and Visan. The argument is inspired by the recent…
This paper deals with the cubic-quintic nonlinear Schr\"{o}dinger equation on R^3. Two monotonicity conjectures for solitons posed by Killip, Oh, Pocovnicu and Visan are completely resolved: one concerning frequency monotonicity, and the…
This paper investigates the Cauchy problem for the nonlinear Schr\"odinger equation (NLS) in the mass-supercritical and energy-subcritical regime within three spatial dimensions. For initial data in the critical homogeneous Sobolev space…
We formulate scattering in one dimension due to the coupled Schr\"{o}dinger equation in terms of the $S$ matrix, the unitarity of which leads to constraints on the scattering amplitudes. Levinson's theorem is seen to have the form $\eta(0)…
We consider the one-dimensional nonlinear Schr\"odinger equation $$ iu_t + u_{xx} + \mathcal{N}(u)u=0, \quad x,t \in \mathbb R, $$ with the nonlinearity term that is expressed as a sum of powers, possibly infinite: $$ \mathcal{N}(u) = \sum…
We present a new proof of global existence and long range scattering, from small initial data, for the one-dimensional cubic gauge invariant nonlinear Schr\"odinger equation, and for Hartree equations in dimension $n \geq 2$. The proof…
We construct solutions with prescribed scattering state to the cubic-quintic NLS $$ (i\partial_t+\Delta)\psi=\alpha_1 \psi-\alpha_{3}\vert \psi\vert^2 \psi+\alpha_5\vert \psi\vert^4 \psi $$ in three spatial dimensions in the class of…
We prove sharp $L^\infty$ decay and modified scattering for a one-dimensional dispersion-managed cubic nonlinear Schr\"odinger equation with small initial data chosen from a weighted Sobolev space. Specifically, we work with an averaged…
We prove symplectic non-squeezing for the cubic nonlinear Schr\"odinger equation on the line via finite-dimensional approximation.
We adapt the arguments in the recent work of Duyckaerts, Landoulsi, and Roudenko to establish a scattering result at the sharp threshold for the $3d$ focusing cubic NLS with a repulsive potential. We treat both the case of short-range…