Avy Soffer
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View article: The $L^p$-boundedness of wave operators for higher order Schrödinger operator with zero singularities in low odd dimensions
The $L^p$-boundedness of wave operators for higher order Schrödinger operator with zero singularities in low odd dimensions Open
This paper investigates the $L^p$-bounds of wave operators for higher-order Schrödinger operators $H = (-Δ)^m + V$ on $\mathbb{R}^n$, with $m \ge 2$ and real-valued decaying potentials $V$. Our main objective is to establish the sharp $L^p…
View article: Local Decay Estimates
Local Decay Estimates Open
We give a proof of local decay estimates for Schrödinger-type equations, which is based on the knowledge of Asymptotic Completeness. This approach extends to time dependent potential perturbations, as it does not rely on Resolvent Estimate…
View article: The large time asymptotics of nonlinear multichannel Schroedinger equations
The large time asymptotics of nonlinear multichannel Schroedinger equations Open
We consider the Schroedinger equation with a general interaction term, which is localized in space. The interaction may be x, t dependent and non-linear. Purely non-linear parts of the interaction are localized via the radial Sobolev embed…
View article: Decomposition of global solutions for a class of nonlinear wave equations
Decomposition of global solutions for a class of nonlinear wave equations Open
In the present paper we consider global solutions of a class of non-linear wave equations of the form \begin{equation*} \Box u= N(x,t,u)u, \end{equation*} where the nonlinearity~$ N(x,t,u)u$ is assumed to satisfy appropriate boundedness as…
View article: A New Paradigm For Scattering Theory of Linear And Nonlinear Waves: Review And Open Problem
A New Paradigm For Scattering Theory of Linear And Nonlinear Waves: Review And Open Problem Open
I present a review of the recent advancements in scattering theory, which provides a unified approach to studying dispersive and hyperbolic equations with general interaction terms and data. These equations encompass time-dependent potenti…
View article: Scattering and localized states for defocusing nonlinear Schrödinger equations with potential
Scattering and localized states for defocusing nonlinear Schrödinger equations with potential Open
We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schrödinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Mor…
View article: The three-quasi-particle scattering problem: asymptotic completeness for short-range systems
The three-quasi-particle scattering problem: asymptotic completeness for short-range systems Open
We develop an approach to scattering theory for generalized $N$-body systems. In particular we consider a general class of three quasi-particle systems, for which we prove Asymptotic Completeness.
View article: Decomposition of global solutions of bi-laplacian Nonautonomous Schrödinger equations
Decomposition of global solutions of bi-laplacian Nonautonomous Schrödinger equations Open
We study the bi-Laplacian Schrödinger equation with a general interaction term, which may be linear or nonlinear and is allowed to be time-dependent. We show that global solutions to such equations decompose asymptotically into a free wave…
View article: Decay estimates for Beam equations with potentials in dimension three
Decay estimates for Beam equations with potentials in dimension three Open
This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential $$u_{t t}+\big(Δ^2+V\big)u=0, \,\ u(0, x)=f(x),\ u_{t}(0, x)=g(x)$$ in dimension three, where $V$ i…
View article: Soliton resolution for nonlinear Schrödinger type equations in the radial case
Soliton resolution for nonlinear Schrödinger type equations in the radial case Open
We consider the Schrödinger equation with a general interaction term, which is localized in space, for radially symmetric initial data in $n$ dimensions, $n\geq5$. The interaction term may be space-time dependent and nonlinear. Assuming th…
View article: Local Decay Estimates
Local Decay Estimates Open
We give a proof of Local Decay Estimates for Schrödinger type equations, which is based on the knowledge of Asymptotic Completeness (AC). This approach extends to time dependent potential perturbations, as it does not rely on Resolvent Est…
View article: On the Existence of Self-Similar solutions for some Nonlinear Schrödinger equations
On the Existence of Self-Similar solutions for some Nonlinear Schrödinger equations Open
We construct solutions of Schrödinger equations which have asymptotic self similar solutions as time goes to infinity. Also included are situations with two-bubbles. These solutions are global, with constant non-zero $L^2$ norm, and are st…
View article: On The large Time Asymptotics of Klein-Gordon type equations with General Data-I
On The large Time Asymptotics of Klein-Gordon type equations with General Data-I Open
We study the Klein-Gordon equation with general interaction term, which may be linear or nonlinear, and space-time dependent. The initial data is general, large and non-radial. We prove that global solutions are asymptotically given by a f…
View article: On The large Time Asymptotics of Schrödinger type equations with General Data
On The large Time Asymptotics of Schrödinger type equations with General Data Open
For the Schrödinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by a free wave and a weakly loc…
View article: $L^p$ Boundedness of the Scattering Wave Operators of Schrödinger Dynamics -- Part 2
$L^p$ Boundedness of the Scattering Wave Operators of Schrödinger Dynamics -- Part 2 Open
We give another proof of the $L^p$ boundedness of scattering wave operators, at the low frequency part of the data. The proof also allows the control of the commutator of multiplication by $|x|$ with the wave operator in $L^p$. The method …
View article: On Modified Scattering for 1D Quadratic Klein–Gordon Equations With Non-Generic Potentials
On Modified Scattering for 1D Quadratic Klein–Gordon Equations With Non-Generic Potentials Open
We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein–Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to …
View article: Beyond Bogoliubov dynamics
Beyond Bogoliubov dynamics Open
We consider a system of N interacting bosons in the mean-field scaling regime and construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to arbitrary precision. The N-independent corrections are …
View article: Almost sure scattering for the nonradial energy-critical NLS with arbitrary regularity in 3D and 4D cases
Almost sure scattering for the nonradial energy-critical NLS with arbitrary regularity in 3D and 4D cases Open
In this paper, we study the defocusing energy-critical nonlinear Schrödinger equations $$ i\partial_t u + Δu = |u|^{\frac{4}{d-2}} u. $$ When $d=3,4$, we prove the almost sure scattering for the equations with non-radial data in $H_x^s$ fo…
View article: Almost sure well-posedness and scattering of the 3D cubic nonlinear Schrödinger equation
Almost sure well-posedness and scattering of the 3D cubic nonlinear Schrödinger equation Open
We study the random data problem for 3D, defocusing, cubic nonlinear Schrödinger equation in $H_x^s(\mathbb{R}^3)$ with $s<\frac 12$. First, we prove that the almost sure local well-posedness holds when $\frac{1}{6}\leqslant s<\frac 12$ in…
View article: Decay estimates for fourth-order Schrödinger operators in dimension two
Decay estimates for fourth-order Schrödinger operators in dimension two Open
In this paper we study the decay estimates of the fourth order Schrödinger operator $H=Δ^{2}+V(x)$ on $\mathbb{R}^2$ with a bounded decaying potential $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ near the zero thre…
View article: On the wave turbulence theory: ergodicity for the elastic beam wave equation
On the wave turbulence theory: ergodicity for the elastic beam wave equation Open
We analyse a 3-wave kinetic equation, derived from the elastic beam wave equation on the lattice. The ergodicity condition states that two distinct wavevectors are supposed to be connected by a finite number of collisions. In this work, we…
View article: On modified scattering for 1D quadratic Klein-Gordon equations with\n non-generic potentials
On modified scattering for 1D quadratic Klein-Gordon equations with\n non-generic potentials Open
We consider the asymptotic behavior of small global-in-time solutions to a 1D\nKlein-Gordon equation with a spatially localized, variable coefficient\nquadratic nonlinearity and a non-generic linear potential. The purpose of this\nwork is …
View article: Remark On The Notion Of Adapted Conformal And Other Estimates
Remark On The Notion Of Adapted Conformal And Other Estimates Open
I describe a way to modify the multipliers of a-priori estimates, so as to include potential perturbations of the Laplacian.
View article: Global well-posedness for the cubic nonlinear Schr{ö}dinger equation with initial lying in $L^{p}$-based Sobolev spaces
Global well-posedness for the cubic nonlinear Schr{ö}dinger equation with initial lying in $L^{p}$-based Sobolev spaces Open
In this paper we continue our study [DSS20] of the nonlinear Schrödinger 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 global…