Painlev e Equations, Elliptic Integrals and Elementary Functions - PowerPoint PPT Presentation

Painlev´ e Equations, Elliptic Integrals and Elementary Functions Henryk ˙ Zo� l¸ adek (University of Warsaw) Jointly with Galina Filipuk (University of Warsaw) Three days on the Painlev´ e equations and their applications Rome, December 18-20, 2014

Plan of the Talk: • The six Painlev´ e equations are written in the Hamiltonian form with time dependent rational Hamilton functions . • By a natural extension of the phase space one gets corresponding au- tonomous Hamiltonian systems with two degrees of freedom . • The B¨ acklund transformations of the Painlev´ e equations are realized as symplectic birational transformations in C 4 . • The cases with classical solutions are interpreted as the cases of partial integrability of the extended Hamiltonian systems . • It is proved that the extended Hamiltonian systems do not have any additional algebraic first integral besides the known special cases of the third and fifth Painlev´ e equations. • It is shown that the original Painlev´ e equations admit the first integrals expressed in terms of the elementary functions only in the special cases mentioned above. In the proofs equations in variations with respect to a parameter and Liouville’s theory of elementary functions are used.

The Painlev´ e Equations x = 6 x 2 + t ¨ ( P I ) x = 2 x 3 + tx + α ¨ ( P II ) � � αx 2 + β + γx 3 + δ x 2 x = ˙ x − ˙ x t + 1 ¨ ( P III ) t x 2 + 4 tx 2 + 2( t 2 − α ) x + β 2 x + 3 x 3 x 2 x = ˙ ¨ ( P IV ) � � x x 2 − ˙ 1 1 x x = ¨ 2 x + ˙ ( P V ) x − 1 t � � + ( x − 1) 2 αx + β + γx t + δx ( x +1) t 2 x x − 1 � � � � x 2 − x = 1 1 1 1 1 1 1 ¨ x + x − 1 + ˙ t + t − 1 + x ˙ ( P V I ) 2 x − t x − t � � + x ( x − 1)( x − t ) ( x − 1) 2 + δ t ( t − 1) α + β t t − 1 x 2 + γ , t 2 ( t − 1) 2 ( x − t ) 2 where α, β, γ, δ are arbitrary complex parameters (and the dot denotes d/dt ).

• The Painlev´ e equations P I − P V I possess the Painlev´ e property . • Solutions of P I − P V I (the Painlev´ e transcendents) are meromorphic functions on the universal cover of CP 1 � { singular points } . • P I − P V I are not integrable in terms of the known functions. • The Painlev´ e equations have numerous applications in mathematics and mathematical physics nowadays. • Equations P I − P V I can be written in the Hamiltonian form dx dt = ∂h dy dt = − ∂h ∂y, ∂x, (1) where h = h ( x, y, t ) is some (time dependent) Hamilton function (papers of K. Okamoto, also J. Malmquist). They have 3/2 degrees of freedom.

Okamoto’s (polynomial) Hamiltonians 1 2 y 2 − 2 x 3 − tx = h I , ˜ h I = � � � � 1 x 2 + 1 α + 1 2 y 2 − ˜ = h II 2 t y − x, 2 � � (1 + 2 θ 0 ) x − 2 η 0 t + 2 η ∞ · tx 2 � � 1 2 x 2 y 2 + ( θ 0 + θ ∞ ) η ∞ · tx − ˜ h III = y , t 2 xy 2 − ( x 2 + 2 tx + 2 κ 0 ) y + θ ∞ x, ˜ = h IV t { x ( x − 1) 2 y 2 − � � 1 κ 0 ( x − 1) 2 + θx ( x − 1) − ηtx ˜ h V = y + κ ( x − 1) } , 1 t (1 − t ) { x ( x − 1)( x − t ) y 2 + κ ( x − t ) ˜ h V I = − [ κ 0 ( x − 1)( x − t ) + κ 1 x ( x − t ) + ( θ − 1) x ( x − 1)] y } , where the parameters above are defined explicitly via the parameters α, β, γ, δ in P J .

Extended Hamiltonian Function dx dt = ∂h dy dt = − ∂h ∂y, ∂x, After renaming the ‘time’ t by a new ‘coordinate’ q , introducing a new ‘mo- mentum’ p and extending the Hamilton function, H ( x, y, q, p ) = h ( x, y, q ) + p, (2) one obtains the autonomous Hamiltonian system x = H ′ y = − H ′ q = H ′ p = − H ′ ˙ ˙ ˙ p = 1 , ˙ (3) y , x , q . Here the dot denotes differentiation with respect to a new time τ. We shall denote the corresponding vector field by X H . E. Horozov and Ts. Stoyanova considered the question of integrability of system (3) in the Liouville–Arnold sense (or of its complete integrability). It means that there should exist a function F ( x, y, q, p ) in involution with H : { H, F } = ˙ F = 0. They applied a version of the Ziglin method, developed by J.-P. Ramis and Morales-Ruiz .

It uses the monodromy group (or the differential Galois group) of the normal variation equation for a particular algebraic solution of the corresponding Hamiltonian system. In the case of complete integrability with meromorphic first integrals the identity component of this differential Galois group should be abelian . Suitable algebraic solutions of the Painlev´ e equations exist for special values of the parameters. By direct computation of the monodromy group (and, for some equations, of Stokes operators) Horozov and Stoyanova show that the identity component of the differential Galois group of the normal variation equation is not abelian . The method works only for special values of the parameters (but not discrete). Our method of proof of the non-integrability is different. By a suitable normal- ization of the variables we arrive at a perturbation of a completely integrable system with two algebraic first integrals. Then we consider the equation in variations with respect to a parameter (denoted by ε ) around a particular solution which is a rather general elliptic curve. Then analysis of few initial terms in powers of ε of a possible first integral of the perturbed system leads to some properties of elliptic integral which cannot be true.

New Hamiltonians The Painlev´ e equations are of the Li´ enard type : x 2 + B ( x, t ) ˙ ¨ x = A ( x, t ) ˙ x + C ( x, t ) (4) with rational coefficients (with possible poles at t = 0 , t = 1 , t = ∞ , x = 0 , x = 1 , x = ∞ and x = t ) . Let y = ˙ x/D ( x, t ) . The divergence of the nonautonomous vector field �� � � � � ∂ V ( x, y, t ) = Dy ∂ y 2 + AD − D ′ B − D ′ ∂x + t /D y + C/D x ∂y equals div V = � � y + � t /D � 2 AD − D ′ B − D ′ = 0 , x which implies D ′ D ′ x /D = 2 A, t /D = B.

Hence, if the condition 2 A ′ t = B ′ (5) x is fulfilled, then Eq. (4) takes the Hamiltonian form in the variables ( x, y ) = ( x, ˙ x/D ) , (6) where �� ( x,t ) � D ( x, t ) = exp 2 A d x + B d t (7) . The corresponding Hamilton function is given by D ( x, t ) y 2 h ( x, y, t ) = 2 + h 0 ( x, t ) , (8) � x C h 0 = − D d x. (9) Moreover, if the 1 − form 2 A d x + B d t has only simple poles with integer residua at them, then the function D ( x, t ) is rational. If, additionally, the 1 − form C D d x has vanishing residua at its poles then the Hamilton function (8) is rational.

List of New Hamiltonians 1 2 y 2 − 2 x 3 − tx, h I = 1 2 y 2 − 1 2 x 4 − 1 2 tx 2 − αx, = h II x 2 t · y 2 2 − αx + β x − γ 2 x 2 t + δ t h III = x 2 , 2 x · y 2 2 − x 3 2 − 2 tx 2 − 2( t 2 − α ) x + β h IV = x, x ( x − 1) 2 · y 2 2 − αx t + β γ tx h V = tx + x − 1 + δ ( x − 1) 2 , t · y 2 x ( x − 1)( x − t ) h V I = t ( t − 1) 2 � � 1 x − γ t − 1 x − 1 − δt ( t − 1) αx − β t − , t ( t − 1) x − t where y = dx/dt.

Symplectic B¨ acklund Transformations B¨ acklund transformations are birational changes of the variables x, t which transform a given equation P J with given parameters to the same P J but with different parameters. In the series of papers of K. Okamoto it is proved that these transformations can be extended to the so-called canonical transformations � � � h ′ � x, y, t, ˜ x ′ , y ′ , t ′ , ˜ h �− → which preserve the canonical form � Ω = d x ∧ d y + d t ∧ d˜ h. h ′ = ˜ The new Hamiltonian ˜ h ′ J is from the same list, but with different param- eters. The corresponding changes of the parameters induce the action on the pa- rameter space . It turns out that the latter action is equivalent (after a proper choice of coordinates) to an action of some group generated by reflections, an affine Weyl group associated with some root system.

The finite Weyl group W ( R ), associated with a root system R ⊂ R n , is → x − 2 ( α,x ) generated by reflections s α : x �− ( α,α ) α, α ∈ R. They are orthogonal reflections with respect to the hyperplanes L α = { ( α, x ) = 0 } . The affine Weyl group W a ( R ), associated with the root system R, is gen- → x − 2 ( α,x ) − k erated by the reflections s α,k : x �− ( α,α ) α, α ∈ R, k ∈ Z ; i.e., by the orthogonal reflections with respect to hyperplanes L α,k = { ( α, x ) = k } . Of course, by rescaling the x ∈ R n we can represent the generators of W a ( R ) as the above reflections, but with k ∈ µ Z for some µ � = 0 . W a ( A 1 ) for P II , W a ( B 2 ) for P III , W a ( A 2 ) for P IV , W a ( A 3 ) for P V , W a ( D 4 ) for P V I .

For new extended Hamilton functions H = H J ( x, y, q, p ) = h J ( x, y, q ) + p, we want to realize the groups above as the groups of symplectic trans- formations in the extended space with coordinates x, y, q, p and with the symplectic form Ω = d x ∧ d y + d q ∧ d p. (10)

Equation P II The new extended Hamitonian function is given by H = H ( α ) = 1 2 y 2 − 1 2 x 4 − 1 2 qx 2 − αx + p. The change � � x ′ , y ′ − ( x ′ ) 2 − 1 2 q ′ , q ′ , p ′ − 1 8( q ′ ) 2 − 1 ( x, y, q, p ) = U ( x ′ , y ′ , q ′ , p ′ ) = 2 x ′ (which is symplectic) transforms the Hamiltonian H ( α ) to the extended Okamoto Hamiltonian � � � � H ( α ) = 1 ( x ′ ) 2 + 1 α + 1 U ∗ H ( α ) = ˜ 2( y ′ ) 2 − y ′ − x ′ + p ′ , 2 q ′ 2 which equals ˜ h II + p ′ .