Painlevé equations, elliptic integrals and elementary functions

Abstract

The six Painlevé equations can be written in the Hamiltonian form, with time dependent Hamilton functions. We present a rather new approach to this result, leading to rational Hamilton functions. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems with two degrees of freedom. We realize the Bäcklund transformations of the Painlevé equations as symplectic birational transformations in C 4 and we interpret the cases with classical solutions as the cases of partial integrability of the extended Hamiltonian systems. We prove that the extended Hamiltonian systems do not have any additional algebraic first integral besides the known special cases of the third and fifth Painlevé equations. We also show that the original Painlevé equations admit the first integrals expressed in terms of the elementary functions only in the special cases mentioned above. In the proofs we use equations in variations with respect to a parameter and Liouville's theory of elementary functions.

Nonintegrability of the Painlevé IV equation in the Liouville–Arnold sense and Stokes phenomena

“…Since the Bäcklund transformations are birational canonical transformations, 25,35,36 we can extend the result of Theorem 1 to the main theorem of this paper.…”

“…From the paper of Murata, 22 it follows that all the rational solutions of the system () can be obtained by Bäcklund transformations from the following two particular solutions: $$$$\begin{eqnarray*} (\textrm {I})\quad & & v_1=v_2=v_3=0 \quad (\textrm {respectively,} \quad a=1, b=0 ), \quad y=p=0,\\ (\textrm {II})\quad & & v_1=-v_3=\frac{1}{3}, v_2=0 \quad {\left(\textrm {respectively,} \quad a=0, b=-\frac{2}{9} \right)}, \quad y=-p=-\frac{2}{3}\,t\,. \end{eqnarray*}$$$$Since the Bäcklund transformations are birational canonical transformations, 25,35,36 we can extend the result of Theorem 1 to the main theorem of this paper. Theorem Assume that $$$$\begin{equation*} \, a=m,\quad b=-\frac{2}{9}\,(1+3 n)^2\,, \, \end{equation*}$$$$where $$m, n\in {\mathbb {Z}}$$ such that …”

“…Proof of Theorem The proof follows from the fact that the auto‐Bäcklund transformations are birational canonical transformations 25,35,36 ▪$\blacksquare$…”

“…Noumi and Okamoto 23 prove that except for some special particular solutions (rational, algebraic, Riccati solutions) any solution of the $$P_{IV}$$ equation is nonclassical in the sense of Umemura. Zoladek and Filipuk 36 prove nonintegrability of the fourth Painlevé equation as a Hamiltonian system by algebraic first integrals. Their approach is based on Liouville's theory of elementary functions and some properties of elliptic integrals.…”

Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville

“…For the recent development in the study of the non-integrability of the other Painlevé equations we refer to [28]. In a very recent paper [32]Żo ladek and Filipuk have proved that the classical Painlevé equations do not admit a first integral that can be expressed in terms of elementary functions, except for some known cases of P III and P V .…”