arXiv subjects

Luciano Medina

Publications and source records attributed to Luciano Medina.

6 recordsLinked to original sources

arXiv2026

Existence of Ground State and Excited Spinning $Q$-Vortex Solitons on Finite Domains

We establish the existence of spinning $Q$-vortex solitons in a complex scalar field theory with a sextic potential on a finite domain. By reducing the governing equation to a nonlinear boundary value problem, we use variational methods to prove the existence of at least two distinct types of solutions: a ground state solution obtained via constrained minimization and an excited state of the saddle-point type obtained via the Mountain Pass Theorem. We derive bounds for the angular frequency $ω$, the wave amplitude, and the domain size $P$, and provide explicit estimates for the exponential decay of the solutions. Furthermore, we implement a spectral-Galerkin formulation to numerically compute the profiles of fundamental $Q$-vortices, illustrating the saturation behavior of the soliton's amplitude and the asymptotic dependence of the frequency on a prescribed reduced norm and vortex winding number, as well as verifying the theoretical results and visualizing the topological phase structure of the solutions.

math-ph↗

arXiv2025

Existence of Coupled Optical Vortex Solitons Propagating in a Quadratic Nonlinear Medium

We consider the coupled propagation of an optical field and its second harmonic in a quadratic nonlinear medium governed by a coupled system of Schrodinger equations. We prove the existence of ring-profiled optical vortex solitons appearing as solutions to a constrained minimization problem and as solutions to a min-max problem. In the case of the constrained minimization problem solutions are shown to be positive with undetermined wave propagation constants, but in the min-max approach the wave propagation constants can be prescribed. The quadratic nonlinearity introduces some interesting properties not commonly observed in other coupled systems in the context of nonlinear optics, such as the system not accepting any semi-trivial solutions, meaning, that optical solitons cannot be observed when, say, one of the beams are off. Additionally, the second harmonic always remains positive.

math-ph↗

arXiv2023

Existence of Optical Vortex Solitons in Photorefractive Media

Optical propagation and vortices in nonlinear media have been intensively studied in modern optical physics. In this paper, we establish constraints regarding the propagation constant and provide an existence theory and numerical computations for positive exponentially decaying solutions for a class of ring-profiled solitons in a type of nonlinear media known as a photorefractive nonlinearity. Our methods include constrained minimization and finite element formalism, and we study the vortex profile and its propagation by fixing the energy flux.

math.AP↗

arXiv2017

On the existence of optical vortex solitons propagating in saturable nonlinear media

In this paper, an existence theory is established for ring-profiled optical vortex solitons. We consider such solitons in the context of an electromagnetic light wave propagating in a self-focusing nonlinear media and governed by a nonlinear Schrödinger type equation. A variational principle and constrained minimization approach is used to prove the existence of positive solutions for an undetermined wave propagation constant. We provide a series of explicit estimates related to the wave propagation constant, a prescribed energy flux, and vortex winding number. Further, on a Nehari manifold, the existence of positive solutions for a wide range of parameter values is proved. We also provide numerical analysis to illustrate the behavior of the soliton's amplitude and wave propagation constant with respect to a prescribed energy flux and vortex winding number.

math-ph↗

arXiv2016

Localized Optical Vortex Solitons in Pair Plasmas

The dynamics of short intense electromagnetic pulses propagating in a relativistic pair plasma with small temperature asymmetry is of current interest. Such pulses were shown to be governed by a nonlinear Schrodinger equation with a new type of focusing-defocusing nonlinearity. We provide an existence theory and numerical analysis for a special class of solutions, known as "ring-profiled" optical vortex solitons. Our methods include both a direct and constrained minimization. We give a necessary condition for the existence of nontrivial solutions and a series of interesting estimates in terms of the relevant parameters describing the electromagnetic pulse. Additionally, we use the constrained minimization approach and a finite element formalism to numerically study the behavior of the profiles of soliton solutions.

math.AP↗

arXiv2015

Vortex equations governing the fractional quantum Hall effect

An existence theory is established for a coupled non-linear elliptic system, known as "vortex equations", describing the fractional quantum Hall effect in 2-dimensional double-layered electron systems. Via variational methods, we prove the existence and uniqueness of multiple vortices over a doubly periodic domain and the full plane. In the doubly periodic situation, explicit sufficient and necessary conditions are obtained that relate the size of the domain and the vortex numbers. For the full plane case, existence is established for all finite-energy solutions and exponential decay estimates are proved. Quantization phenomena of the magnetic flux are found in both cases.

math-ph↗