Topological Construction of Transseries and Introduction to Generalized Borel Summability

Power and Exponential-power Series Solutions of Evolution Equations

1.1. Brief overview. The theory of partial differential equations when one, or more variables, is in the complex domain, and approaches a characteristic variety has only recently started to develop. In their paper [11], generalized in [12], O. Costin and S. Tanveer proved existence and uniqueness of solutions with given initial conditions, for quasilinear systems of evolution equations in a lar...

Transseries and Écalle-Borel summability

On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE’s

In this paper we study analytic (linear or) nonlinear systems of ordinary differential equations, at an irregular singularity of rank one, under nonresonance conditions. It is shown that the formal asymptotic exponential series solutions (transseries solutions: countable linear combinations of formal power series multiplied by small exponentials) are Borel summable in a generalized sense along ...

Large Order Behaviour of 2 D Gravity Coupled to d < 1 Matter

We discuss the large order behaviour and Borel summability of the topological expansion of models of 2D gravity coupled to general (p, q) conformal matter. In a previous work it was proven that at large order k the string susceptibility had a generic a k Γ(2k − 1 2) behaviour. Moreover the constant a, relevant for the problem of Borel summability, was determined for all one-matrix models. We he...

Rodica D. Costin Research Statement

My research is focused on differential equations, with emphasis on the study of singularities and asymptotic behavior of solutions, regularity, normal forms, the study of non-integrable systems and connection to chaos. I am applying new methods—analysis of equations in Borel space (after inverse Laplace transform) and a theory of generalized Borel summation of general formal solutions (transser...