Non-integrability Of Painlevé Equations

The problem of solving differential equations is one of the most important problems in the theory of differential equations.From the date of birth of the differential equations,this problem has been valued by mathematicians and physicists.The classic Galois theory is one of the most important works in the algebra, Picard and Vessiot discovered differential Galois theory in the late nineteenth-century. The modern rigorous form of differential Galois theory is due to E.Kolchin, S.L.Ziglin, J.J.Morales-Ruiz and J.P.Ramis.Painleve and his colleagues discovered six Painleve equations as follows in an investi-gation of nonlinear second-order differential equations about a hundred years ago. whereα,β,γandδare arbitrary constants. Although first discovered from strictly mathe-matical considerations, the Painleve equations have arisen in a variety of important physical applications including statistical mechanics, nonlinear waves and quantum gravity.We introduce the fundamental definition and fundamental theorem of differential Galois theory in brief.The main result of this paper is two theoremsTheorem 1 Forα=1, PainleveⅡequation is not integrable by means of rational first integrals.Theorem 2 Forα∈Z, PainleveⅡequation is not integrable by means of rational first integrals.We also introduce the new result of Morales-Ramis theory about the non-integrability of PainleveⅥequation, and non-linear differential Galois theory about the non-integrability of Painleve equations.T.Stoyanova proved several theorems as follows in 2007 and 2009 by using Morales-Ramis theoryTheorem 3 Forα=β=γ=δ=0, PainleveⅥequation is non integrable by means of meromorphic first integrals.Theorem 4 Assumeθ4=θ1+θ2+θ3, at least oneθj∈Z and at least oneθ(?)kQ.Then PainleveⅥequation is not integrable.Theorem 5 Assumeθ4=θ1+θ2+θ3 and at least twoθj are integers. Then Painleve VI equation is not integrable.We can prove PainleveⅠequation is not integrable by means of rational functions, and for all values of the parameters Painleve VI equation is not integrable by mean of rational functions.Because of the importance of the Painleve equations, many mathematicians and physi-cists are interested in this problem.But integrability of Painleve equations has not been solved, some mathematicians guess they are not integrable for all values of the parame-ters.Using the non linear version of differential Galois theory, we can avoid the choice of a particular solution, which is good. But unfortunately there is a price to pay:the proof and computations are more difficult.