Related papers: Finite-dimensional approximation and non-squeezing…
We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time…
We prove approximate controllability of the bilinear Schr\"odinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and…
We consider an 1D partial integro-differential equation (PIDE) comprising of an 1D parabolic partial differential equation (PDE) and a nonlocal integral term. The control input is applied on one of the boundaries of the PIDE. Partitioning…
This project investigates the approximate controllability of a class of stochastic integrodifferential equations in Hilbert space with non-local beginning conditions. In a departure from the conventional concerns expressed in the…
In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schr\"odinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a…
From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
We propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. It is based on the original exploitation of a result related to…
This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations…
The appearance of a fundamental long-time asymptotic regime in the two space one time dimensional hyperbolic nonlinear Schr\"odinger (HNLS) equation is discussed. Based on analytical and extensive numerical simulations an approximate…
In this paper, we study the local well-posedness of the cubic Schr\"odinger equation $$(i\partial_t + \mathcal{L}) u = \pm |u|^2 u \qquad \textrm{on} \quad \ I\times \mathbb{R}^d ,$$ with initial data being a Wiener randomization at unit…
We prove global well-posedness for the cubic nonlinear Schr\"odinger equation for periodic initial data in the mass-critical dimension $d=2$ for initial data of arbitrary size in the defocusing case and data below the ground state threshold…
This paper addresses the focusing cubic-quintic nonlinear Schrodinger equation in three space dimensions. Especially, we study the global dynamics of solutions whose energy and mass equal to those of the ground state in the sprits of…
A simple formula is derived for the maximum squeezing rate which occurs at the initial stages of the squeezing process: the rate only depends on the second partial derivatives of a classical Hamiltonian. Rules for optimum rotation of the…
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…
We consider equations of nonlinear Schrodinger type augmented by nonlinear damping terms. We show that nonlinear damping prevents finite time blow-up in several situations, which we describe. We also prove that the presence of a quadratic…
We introduce a second-order stochastic model to explore the variability in growth of biological shapes with applications to medical imaging. Our model is a perturbation with a random force of the Hamiltonian formulation of the geodesics.…
An exactly solvable position-dependent mass Schr\"odinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems…
We explore how the analysis of the Carleman linearization can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. We demonstrate the well-posedness and convergence of the truncated Carleman…
We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence)…
We obtain almost-sure scattering for the cubic defocusing Schr{\"o}dinger equation in the Euclidean space {$\mathbb{R}^3$}, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness…