arXiv subjects

Primitivo B. Acosta-Humanez

Publications and source records attributed to Primitivo B. Acosta-Humanez.

5 recordsLinked to original sources

arXiv2019

Integrability of Stochastic Birth-Death processes via Differential Galois Theory

Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of infinite, coupled ordinary differential equations (the so-called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial differential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard differential Galois theory. We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the (trivial) case in which rates are linear functions of the number of individuals.

math-ph↗

arXiv2012

On Hamiltonian potentials with quartic polynomial normal variational equations

In this paper we prove that there exists only one family of classical Hamiltonian systems of two degrees of freedom with invariant plane $Γ=\{q_2=p_2=0\}$ whose normal variational equation around integral curves in $Γ$ is generically a Hill-Schrödinger equation with quartic polynomial potential. In particular, by means of the Morales-Ramis theory, these Hamiltonian systems are non-integrable through rational first integrals.

math-ph↗

arXiv2012

Nonautonomous Hamiltonian Systems and Morales-Ramis Theory I. The Case $\ddot{x}=f(x,t)$

In this paper we present an approach towards the comprehensive analysis of the non-integrability of differential equations in the form $\ddot x=f(x,t)$ which is analogous to Hamiltonian systems with 1+1/2 degree of freedom. In particular, we analyze the non-integrability of some important families of differential equations such as Painlevé II, Sitnikov and Hill-Schrödinger equation. We emphasize in Painlevé II, showing its non-integrability through three different Hamiltonian systems, and also in Sitnikov in which two different version including numerical results are shown. The main tool to study the non-integrability of these kind of Hamiltonian systems is Morales-Ramis theory. This paper is a very slight improvement of the talk with the almost-same title delivered by the author in SIAM Conference on Applications of Dynamical Systems 2007.

math-ph↗

arXiv2011

On Pasting and Reversing operations over some rings

In this paper we study two operations, Pasting and Reversing, defined from a natural way to be applied over some rings such as the ring of polynomials and the ring of linear differential operators, which is a differential ring. We obtain some properties of these operations over these rings, in particular over the set of natural numbers, in where we rewrite some properties incoming from recreational mathematics.

math.HO↗

arXiv2009

Galoisian Approach to Supersymmetric Quantum Mechanics

This thesis is concerning to the Differential Galois Theory point of view of the Supersymmetric Quantum Mechanics. The main object considered here is the non-relativistic stationary Schrödinger equation, specially the integrable cases in the sense of the Picard-Vessiot theory and the main algorithmic tools used here are the Kovacic algorithm and the \emph{algebrization method} to obtain linear differential equations with rational coefficients. We analyze the Darboux transformations, Crum iterations and supersymmetric quantum mechanics with their \emph{algebrized} versions from a Galoisian approach. Applying the algebrization method and the Kovacic's algorithm we obtain the ground state, the set of eigenvalues, eigenfunctions, the differential Galois groups and eigenrings of some Schrödinger equation with potentials such as exactly solvable and shape invariant potentials. Finally, we introduce one methodology to find exactly solvable potentials: to construct other potentials, we apply the algebrization algorithm in an inverse way since differential equations with orthogonal polynomials and special functions as solutions.

quant-ph↗