Integrating Ordinary Differential Equations

Abstract

AbstractThe problem addressed here is, given an ODE which may admit a single-valued solution, to find it explicitly in closed form. Two kinds of ODEs are considered.The first kind is made of ODEs which pass the Painlevé test, in which case the goal is to find their general solution, and our three main examples will be: the four cases of the Lorenz model isolated in Sect. 2.1.1, the traveling wave reduction of the Korteweg-de Vries equation and of the nonlinear Schrödinger equation. In these three examples, the general solution can indeed be found and is represented either by elliptic functions or by Painlevé functions. The second kind is made of ODEs which fail the Painlevé test, but which do not fail it too badly, leaving open the possibility for particular single-valued solutions. A powerful result of Nevanlinna theory leads us to split this second kind in two classes.The first class is made of ODEs which possess no arbitrary constant in their Laurent series and fulfill another easy to check condition (Eremenko’s theorem); then all their solutions are either elliptic or rational in a single exponential or rational, and they can be and indeed are obtained in closed form. For the second class, made of all other ODEs, there only exist sufficient methods able to yield particular single-valued solutions, mainly those known as “truncation” methods. Several physical examples (Kuramoto-Sivashinsky, complex Ginzburg-Landau, Duffing-van der Pol, Bianchi IX, etc.) are presented in detail for both classes.KeywordsLorenz modelTraveling wave of Korteweg-de Vries equationTraveling wave of nonlinear Schrödinger equationTheorem of EremenkoHermite decompositionElliptic subequation methodAll meromorphic solutions of Kuramoto-Sivashinsky equationAll meromorphic solutions of complex Ginzburg-Landau equationParticular class of polynomials in tanh and sechBianchi IX cosmological model