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

Non-integrability of some few body problems in two degrees of freedom

The basic theory of Differential Galois and in particular Morales--Ramis theory is reviewed with focus in analyzing the non--integrability of various problems of few bodies in Celestial Mechanics. The main theoretical tools are: Morales--Ramis theorem, the algebrization method of Acosta--Bl\'azquez and Kovacic's algorithm. Morales--Ramis states that if Hamiltonian system has an additional meromorphic integral in involution in a neighborhood of a specific solution, then the differential Galois group of the normal variational equations is abelian. The algebrization method permits under general conditions to recast the variational equation in a form suitable for its analysis by means of Kovacic's algorithm. We apply these tools to various examples of few body problems in Celestial Mechanics: (a) the elliptic restricted three body in the plane with collision of the primaries; (b) a general Hamiltonian system of two degrees of freedom with homogeneous potential of degree -1; here we perform McGehee's blow up and obtain the normal variational equation in the form of an hypergeometric equation. We recover Yoshida's criterion for non--integrability. Then we contrast two methods to compute the Galois group: the well known, based in the Schwartz--Kimura table, and the lesser based in Kovacic's algorithm. We apply these methodology to three problems: the rectangular four body problem, the anisotropic Kepler problem and two uncoupled Kepler problems in the line; the last two depend on a mass parameter, but while in the anisotropic problem it is integrable for only two values of the parameter, the two uncoupled Kepler problems is completely integrable for all values of the masses.Comment: 33 page

Darboux Transformations for orthogonal differential systems and differential Galois Theory

Darboux developed an algebraic mechanism to construct an infinite chain of "integrable" second order differential equations as well as their solutions. After a surprisingly long time, Darboux's results had important features in the analytic context, for instance in quantum mechanics where it provides a convenient framework for Supersymmetric Quantum Mechanics. Today, there are a lot of papers regarding the use of Darboux transformations in various contexts, not only in mathematical physics. In this paper, we develop a generalization of the Darboux transformations for tensor product constructions on linear differential equations or systems. Moreover, we provide explicit Darboux transformations for \sym^2 (\mathrm{SL}(2,\mathbb{C})) systems and, as a consequence, also for $\mathfrak{so}(3, C_K)$ systems, to construct an infinite chain of integrable (in Galois sense) linear differential systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials.Comment: 22 page

'Drop-in' de Lausanne : étude d'une année d'activité de consultation et d'information

We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrability with some higher transcendent functions, like the error functionPeer ReviewedPostprint (author's final draft

Differential galois theory and non-integrability of planar polynomial vector fields

We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrability with some higher transcendent functions, like the error functionPeer Reviewe