Related papers: On scattering asymptotics for the 2D cubic resonan…
We consider the derivation of the defocusing cubic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{R}^{3}$ from quantum $N$-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering…
In this paper, we investigate a resonant system comprising $N$ closely packed spherical resonators ($N>2$). We analyze how the spatial arrangement of these resonators influences the distribution of resonant frequencies, focusing on…
We transpose work by K.Yajima and by T.Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schr\"odinger equation (NLS) in 2D. As an application we extend to…
We study two resonant Hamiltonian systems on the phase space $L^2(\mathbb{R} \rightarrow \mathbb{C})$: the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified…
We consider nonlinear Schr\"{o}dinger equation with strong magnetic fields in 3D. This model was derived by R L. Frank, F. M\'{e}hats, C. Sparber in 2017. We prove modified scattering for small initial data and the existence of modified…
In this paper, we consider the resonance problem for the cubic nonlinear Helmholtz equation in the subwavelength regime. We derive a discrete model for approximating the subwavelength resonances of finite systems of high-contrast resonators…
This work establishes a rigorous mathematical framework for the analysis of nonlinear dielectric resonances in wave scattering by high-index resonators with Kerr-type nonlinearities. We consider both two- and three-dimensional settings and…
We consider the cubic defocusing nonlinear Schr\"odinger equation in one dimension with the nonlinearity concentrated at a single point. We prove global well-posedness in the scaling-critical space $L^2(\mathbb{R})$ and scattering for all…
Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schr\"odinger equation below a mass-energy threshold $M[u]E[u] < M[Q]E[Q]$ and satisfying an initial mass-gradient bound $\|u_0\|_{L^2} \|\nabla u_0 \|_{L^2} <…
The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schr\"odinger equation $$ i u_t +\Delta u+|x|^{-b}|u|^2 u = 0, $$ where $0<b<1/2$. Let $Q$ be the ground state solution of $-Q+\Delta Q+ |x|^{-b}|Q|^{2}Q=0$ and…
An S matrix approach is developed to describe elastic scattering resonances of systems where the scattered particle is asymptotically confined and the scattering potential lacks continuous symmetry. Examples are conductance resonances in…
We consider the magnetic nonlinear inhomogeneous Schr\"odinger equation $$i\partial_t u -\left(-i\nabla+\frac{\alpha}{|x|^2}(-x_2,x_1)\right)^2 u =\pm|x|^{-\varrho}|u|^{p-1}u,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^2,$$ where…
In this paper we generalize a weak sequential result of \cite{fan20182} to a non-scattering solutions in dimension $d \geq 2$. No symmetry assumptions are required for the initial data. We build on a previous result of \cite{dodson20202}…
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…
Consider the focusing inhomogeneous nonlinear Schr\"odinger equation in $H^1(\mathbb{R}^N)$, $$iu_t + \Delta u + |x|^{-b}|u|^{p-1}u=0,$$ when $b > 0$ and $N \geq 3$ in the intercritical case $0 < s_c <1$. In previous works, the second…
We revisit the scattering result of Holmer and Roudenko on the radial focusing cubic NLS in three space dimensions. Using the radial Sobolev embedding and a virial/Morawetz estimate, we give a simple proof of scattering below the ground…
We consider the dispersion managed nonlinear Schr\"dinger equations with quintic and cubic nonlinearities in one and two dimensions, respectively. We prove the global well-posedness and scattering in $L_x^2$ for small initial data employing…
We review some recent results on the theory of scattering and more precisely on the local Cauchy problem at infinity in time for some long range nonlinear systems including some form of the Schr"odinger equation. We consider in particular…
We observe and analyze d-wave resonant scattering of bosons in tightly confining harmonic waveguides. It is shown that the d-wave resonance emerges in the quasi-1D regime as an imprint of a 3D d-wave shape resonance. A scaling relation for…
We show that if a solution of the defocusing cubic NLS in 3d remains bounded in the homogeneous Sobolev norm of order 1/2 in its maximal interval of existence, then the interval is infinite and the solution scatters. No radial assumption is…