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Numerical approximation of the fractional Laplacian on R using orthogonal families
(2020-12-01)
In this paper, using well-known complex variable techniques, we compute explicitly, in terms of the F12 Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the complex Higgins functions, the complex ...
Invariant measures for the dnls equation
Lucà, R.
(2020-10-02)
(2020-10-02)
We describe invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schr\"odinger equation (DNLS) constructed in \cite{MR3518561, Genovese2018}. The construction works for small $L^2$ ...
The Schrödinger equation and Uncertainty Principles
(2020-09)
The main task of this thesis is the analysis of the initial data u0 of Schrödinger’s initial value problem in order to determine certain properties of its dynamical evolution.
First we consider the elliptic Schrödinger ...
On the improvement of the Hardy inequality due to singular magnetic fields
Fanelli, L.; Krejčiřík, D.; Laptev, A.; Vega, L. (2020-09-01)
(2020-09-01)
We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type ...
On the unique continuation of solutions to non-local non-linear dispersive equations
Kenig, C. E.; Pilod, D.; Ponce, G.; Vega, L. (2020-08-02)
(2020-08-02)
We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if (Formula presented.) are two suitable solutions of the equation defined in ...
Riemann's non-differentiable function and the binormal curvature flow
Banica, V.; Vega, L. (2020-07-14)
(2020-07-14)
We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object ...
Vortex Filament Equation for a regular polygon in the hyperbolic plane
de la Hoz, F.; Kumar, S.; Vega, L. (2020-07-09)
(2020-07-09)
The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and ...
A geometric and physical study of Riemann's non-differentiable function
(2020-07-08)
Riemann's non-differentiable function is a classic example of a continuous but almost nowhere differentiable function, whose analytic regularity has been widely studied since it was proposed in the second half of the 19th ...
On the energy of critical solutions of the binormal flow
Banica, V.; Vega, L. (2020-07-02)
(2020-07-02)
The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic ...
Pseudospectral Methods for the Fractional Laplacian on R
(2020-07-02)
In this thesis, first, we propose a novel pseudospectral method to approximate accu- rately and efficiently the fractional Laplacian without using truncation. More pre- cisely, given a bounded regular function defined over ...