Is my ODE a Painlevé equation in disguise?

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1 Is my ODE a Painlevé equation in disguise? arxiv:nlin/ v1 [nlin.si] 8 May 2001 Jarmo HIETARINTA and Valery DRYUMA Department of Physics, University of Turku, FIN Turku, Finland, jarmo.hietarinta@utu.fi Institute of Mathematics AS RM, Academy str. 5, Kishinev, Moldova, valery@gala.moldova.su November 3, 2018 Abstract Painlevéequationsbelongtotheclassy +a 1 y 3 +3a 2 y 2 +3a 3 y +a 4 = 0, wherea i = a i (x,y). This class of equations is invariant under the general point transformation x = Φ(X,Y), y = Ψ(X,Y) and it is therefore very difficult to find out whether two equations in this class are related. We describe R. Liouville s theory of invariants that can be used to construct invariant characteristic expressions (syzygies), and in particular present such a characterization for Painlevé equations I-IV. 1 Introduction Many phenomena in Nature are modeled by ordinary differential equations (ODEs). After such an equation is derived for some physical situation, the natural question is whether that ODE is well known, or at least transformable to a well known equation. For example, onewouldlike to know iftheequation isrelatedtoaknown integrableequation, e.g, inthe case of a second order ODE, to one of the Painlevé equations, or to one of the equations in the Gambier[1]/Ince[2] list. Normally these lists contain only representative equations, e.g., up to some group of transformations. If the equation is of the form y = f(x,y,y ) the usually considered group of transformations is x = φ(x), y = ψ 1(X)Y +ψ 2 (X) ψ 3 (X)Y +ψ 4 (X). However, when we have an equation derived from some physical problem the required transformation may be more complicated. Here we consider the following class of equations y +a 1 (x,y)y 3 +3a 2 (x,y)y 2 +3a 3 (x,y)y +a 4 (x,y) = 0. (1) This class is invariant under a general point transformation x = Φ(X,Y), y = Ψ(X,Y), (2) 1

2 we only need to assume that the transformation is nonsingular, i.e., The point transformation (2) prolongs to := Φ X Ψ Y Φ Y Ψ X 0. (3) y = Ψ X +Ψ Y Y Φ X +Φ Y Y, (4) y = { [Ψ Y Φ X Ψ X Φ Y ]Y +[Φ Y Ψ YY Ψ Y Φ YY ]Y 3 +(Φ X Ψ YY 2Φ XY Ψ Y +2Φ Y Ψ XY Φ YY Ψ X )Y 2 +( Φ XX Ψ Y 2Φ XY Ψ X +2Φ X Ψ XY +Φ Y Ψ XX )Y +Φ X Ψ XX Φ XX Ψ X } /[ΦX +Φ Y Y ] 3, (5) and when these are substituted into (1) the form of the equation (cubic in y ) stays the same, but the coefficients a i change as follows: ã 1 = 1 (Φ YΨ YY Φ YY Ψ Y +Φ 3 YA 4 +3Φ 2 YΨ Y A 3 +3Φ Y Ψ 2 YA 2 +Ψ 3 YA 1 ), (6) ã 2 = 1 [1 3 (Φ XΨ YY 2Φ XY Ψ Y +2Φ Y Ψ XY Φ YY Ψ X )+Φ X Φ 2 Y A 4 +(2Φ X Φ Y Ψ Y +Φ 2 Y Ψ X)A 3 +(Φ X Ψ 2 Y +2Φ YΨ X Ψ Y )A 2 +Ψ X Ψ 2 Y A 1], (7) ã 3 = 1 [1 3 ( Φ XXΨ Y 2Φ XY Ψ X +2Φ X Ψ XY +Φ Y Ψ XX )+Φ 2 XΦ Y A 4 +(Φ 2 X Ψ Y +2Φ X Φ Y Ψ X )A 3 +(2Φ X Ψ X Ψ Y +Φ Y Ψ 2 X )A 2 +Ψ 2 X Ψ YA 1 ], (8) ã 4 = 1 ( Φ XXΨ X +Φ X Ψ XX +Φ 3 X A 4 +3Φ 2 X Ψ XA 3 +3Φ X Ψ 2 X A 2 +Ψ 3 X A 1). (9) Here A i (X,Y) := a i (Φ(X,Y),Ψ(X,Y)). Example: If we apply the transformation x = X + Y, y = XY to the equation y = 0 (in which a i = 0), then we find ã 1 = ã 4 = 0, ã 2 = ã 3 = 2 1, i.e., the equation becomes Y + 2 (Y 2 +Y ) = 0. 3 X Y X Y Since the coefficients of equation (1) transform in such a complicated way it is in general difficult to find a characterization of (1) that is invariant under a general point transformation (2). This classical problem was basically solved more than 100 years ago, the fundamental works being those by Liouville [3] in 1889, Tresse [4] in 1894, with more modern formulations by Cartan [5] in 1924 and Thomsen [6] in Recent wave of interest on this classical problem started with [7]. [For the restricted problem with x = Φ(X) see, e.g., [8].] 2 Relative invariants, absolute invariants and syzygies The invariants we are looking for must be constructed from the coefficients a i and their various derivatives. The transformation rules for a i were given in (6-9) and we are now looking for some combinations that transform in a much simpler way. Let us consider some expression I[x,y] = I(a 1,...,a 4, x a 1,..., x a 1, y a 1,..., y a 4, 2 xa 1,...). Since a i = a i (x,y) this will be a function of x,y. Under the transformation (2) the ingredients a i transform to ã i and the expression constructed from the transformed quantities in exactly the same way is Ĩ[X,Y] = I(ã 1,...,ã 4, X ã 1,..., X ã 1, Y ã 1,..., Y ã 4, 2 Xã1,...). 2

3 This is now a function of X,Y and if it turns out that Ĩ[X,Y] = n I[Φ(X,Y),Ψ(X,Y)], i.e., that I transforms, up to some overall factor, as by substitution then we say that I is a relative invariant of weight n. If furthermore n = 0 or I = 0 we say that I is an absolute invariant. Normally the weight of the relative invariant is indicated by a subscript. Later we will construct several sequences of relative invariants, e.g., i 2n, n = 1,2,3,..., then using them we can construct a sequence of absolute invariants, in this case j 2n := i 2n /i n 2. Although the absolute invariants transform by a simple substitution under (2) they are normally complicated functions of x, y and therefore in practice useless for classification. What we need are relationships between absolute invariants, i.e., syzygies. For example, it turns out later that for one particular equation j 6 6j = 0. This relationship is invariant under the transformation (2) and therefore it is an invariant characterization for that equation. Using this result we can say with certainty that any equation that does not satisfy this relationship cannot be transformed to first equation by any point transformation. The reverse is not true: several equations may satisfy the same syzygy. Note also that the syzygy polynomials can only have numerical coefficients, because parameter values appearing in the equation can be changed with the allowed transformations. 3 Construction of Invariants The construction of the relative and absolute invariants proceeds step by step as follows: First define and then Π 22 = x a 1 y a 2 +2(a 2 2 a 1a 3 ), (10) Π 12 = x a 2 y a 3 +a 2 a 3 a 1 a 4, (11) Π 11 = x a 3 y a 4 +2(a 2 3 a 2 a 4 ), (12) L 2 = x Π 22 y Π 12 a 1 Π 11 +2a 2 Π 12 a 3 Π 22, (13) L 1 = x Π 12 y Π 11 a 2 Π 11 +2a 3 Π 12 a 4 Π 22. (14) Now one finds that the L i transform according to ) ( )( ) ( L1 ΦX Ψ = X L1 [Φ(X,Y),Ψ(X,Y)], (15) L 2 Φ Y Ψ Y L 2 [Φ(X,Y),Ψ(X,Y)] which is already a relatively simple transformation rule. From this we get the first result (R. Liouville): The property L 1 = L 2 = 0 is an absolute invariant and if it holds the equation can be transformed to Y = 0. 3

4 If we define a i and x, y to have weight 1, then L 2 i are of weight 3. Continuing with 2 L i and adding one derivative or a i we find at weight 7 another pair 2 Z 1 = L 1y L 1 3L 1x L 2 +4L 1 L 2x +3a 2 L 2 1 6a 3L 1 L 2 +3a 4 L 2 2, (16) Z 2 = L 2x L 2 +3L 2y L 1 4L 2 L 1y +3a 1 L 2 1 6a 2 L 1 L 2 +3a 3 L 2 2, (17) that transforms similarly: ) ( Z1 Z 2 ( = 3 ΦX Ψ X Φ Y Ψ Y )( ) Z1 [Φ(X,Y),Ψ(X,Y)] Z 2 [Φ(X,Y),Ψ(X,Y)] (18) Using these two we get the first semi-invariant ν 5 = 1 3 [Z 1L 2 Z 2 L 1 ] = L 2 (L 1 x L 2 L 2 x L 1 )+L 1 (L 2 y L 1 L 1 y L 2 ) a 1 L a 2L 2 1 L 2 3a 3 L 1 L 2 2 +a 4L 3 2, (19) which is of weight 5, i.e., transforms as ν 5 = 5 ν 5. Observation: For all Painlevé equations ν 5 = 0. Our ultimate aim is to provide an invariant characterization for the Gambier/Ince list of 50 equations. It contains some equations with L 1 = L 2 = 0, and other equations besides Painlevé s that have ν 5 = 0, but it also contains many with ν 5 0. [We note in passing, that in the standard form all Painlevé equations also have the properties a 1 = 0 and L 2 = 0, but these are not (semi)invariant characterizations.] Let us now continue with ν 5 = 0, i.e. Z 1 L 2 = Z 2 L 1. Let us define R 1 = L 1 x L 2 L 2 x L 1 +a 2 L 2 1 2a 3 L 1 L 2 +a 4 L 2 2, (20) w 1 = [L 3 1 (Π 11L 2 Π 12 L 1 )+R 1 x (L 2 1 ) L2 1 xr 1 +L 1 R 1 (a 3 L 1 a 4 L 2 )]/L 4 1. (21) [If L 1 = 0 there is a similar expression with L 2 as divider.] The expression w 1 is a semi-invariant of weight 1. Observation: For all Painlevé equations w 1 = 0. The Gambier/Ince list contains also equations with ν 5 = 0, w 1 0. We continue further with ν 5 = 0, w 1 = 0. A sequence of semi-invariants can now be constructed starting with with higher members given recursively by i 2 = 2R 1 /L 1 + x L 2 y L 1, (22) i 2(m+1) = L 1 y i 2m L 2 x i 2m +2mi 2m ( x L 2 y L 1 ). (23) If i 2 0 a sequence of absolute invariants is given by j 2m = i 2m i m 2. (24) In the next section we will use the j 2m to characterize Painlevé equations I-IV. 4

5 At this point we can see that the classification using j 2m cannot be sharp. Consider the special case y + a 4 (x,y) = 0. Then Π 11 = y a 4, Π 12 = Π 22 = 0,L 1 = 2 ya 4,L 2 = 0,ν 5 = 0, w 1 = 0, and i 2 = 3 ya 4, i 2(m+1) = L 2m+1 1 y (i 2m /L 2m 1 ), The semi-invariants therefore depend only on 2 y a 4 and its higher y-derivatives and are insensitive to the possible linear in y part in a 4. 4 Invariant characterization of P I P IV The first steps were given above: All Painlevé equations have the properties 1) at least one of L 1,L 2 is nonzero, 2) ν 5 = 0, 3) w 1 = 0. 1 If the candidate equation fails any of these properties it cannot be transformed to a Painlevé equation using the transformation (2). Here we want to go further and derive conditions which differentiate between Painlevé equations. The following results have been derived using the symbolic algebra language REDUCE[9]. It should be noted that we give here only the lowest degree syzygy. 4.1 Painlevé I Any equation of the form y = 6y 2 +f 1 (x)y +f 0 (x), (25) has the property i 2 = 0. In principle i 2 is only a relative invariant, but since its value in this case is 0, this property is an absolute invariant. Equation (25) contains Painlevé I for which f 1 = 0, f 0 = x. For all other Painlevé equations i Painlevé II All equations of form y = 2y 3 +f 1 (x)y +f 0 (x), (26) have the property i 2 = 12,i 4 = 288 and therefore j 4 = 2 is the syzygy for this class of equations. [In fact it is easy to see that j 2(m+1) = 2 m m!] Painlevé II is contained as the special case f 1 = x, f 0 =const. 4.3 Painlevé III The following results are valid for Cases 12,13,13 1 in the Ince/Gambier classification, they all have four parameters, α,β,γ,δ: y y 2 y γy3 αy 2 β δ/y = 0, y y 2 y + y x γy3 1 x (αy2 +β) δ/y = 0, y y 2 y ex (αy 2 +β) e 2x (γy 3 +δ/y) = 0. 1 This was first observed by V. Dryuma in late 1980 s 5

6 1: If any 3 of the parameters α,β,γ,δ are zero, then j 2(m+1) = m! 2a: If γ = δ = 0 or α = β = 0 then j 6 6j 4 +4 = 0. (27) These two cases are connected: x = 1 2 X2,y = Y 2 takes (α,β,0,0) (0,0,α,β). 2b: If α = γ = 0 or β = δ = 0 then 2j 8 13j j 4 21j4 2 = 0. Transformation x = X,y = 1/Y takes (0,β,0,δ) ( β,0, δ,0). 2c: If β = γ = 0 or α = δ = 0 then 2j 8 23j j 4 11j4 2 = 0. Transformation x = X,y = 1/Y takes (α,0,0,δ) (0, α, δ,0). 3: If α 2 δ +β 2 γ = 0 (which contains 2a,2b) then we get a degree 12 relation 4: For the generic case we obtain 4j 12 76j j j j j j 4 j j j 4j 8 115j 2 6 = 0. 4j j j 10 j j j 8 j j 8 j j j j 6 j j 6j j j j j = Painlevé IV Let us again consider a more general form y = 1 2y y 2 +e y3 +4(e 2 x+e 3 )y 2 +2(e 4 x 2 +e 5 x+e 6 )y +e 7 1 y, Painlevé IV corresponds to e 1 = e 2 = e 4 = 1, e 3 = e 5 = 0, e 6 = α, e 7 = β 2 /2. The classification is sensitive only to e 1,e 2,e 3,e 7. 1: If e 1 = e 2 = e 3 = e 7 = 0 then L 1 = L 2 = 0 2a: If e 2 = e 3 = e 7 = 0 then j 2(m+1) = ( 4 3) m m! 2b: If e 2 = e 3 = e 1 = 0 then j 2(m+1) = ( 4 5) m m! 2c: If e 1 = e 7 = 0 then j 2(m+1) = 2 m m! 3a: If e 7 = 0 then j 8 11j j 4 7j 2 4 = 0. 6

7 3b: If e 1 = 0 then 5j 8 95j j 4 +21j 2 4 = 0. 3c: if e 2 = e 3 = 0 then 4: Generic case: 8125j j j 6 j j j j = 0. 3j j j 10 j 4 120j j j j j j 4 j j j 4j j j2 4 j 6 75j 6 j 8 = 0. 5 Conclusions We have presented here the first results of a project aiming to derive syzygies for every equation in the Gambier/Ince list. The biggest problem in finding the characteristic expressions is that we have to solve huge sets of rather complicated nonlinear algebraic equations. For Painlevé V the generic expression is of degree higher than 26 (which is as far as we checked) and for Painlevé VI presumably still much higher. When this classification is eventually finished: We get an algorithmic method for finding out if a given equation of type (1) has any change of being transformed into one of the equations in the Gambier/Ince list. We also get a classification of the equations into essentially different sub-cases (c.f. [10, 11]). The first results that have already been obtained suggest some interesting open questions: Is the type of integrability different in the classes a) ν 5 = w 1 = 0, b) ν 5 = 0, w 1 0, and c) ν 5 0? What does the degree of the minimal syzygy tell us about integrability? Acknowledgment J.H. would like to thank P. Olver for discussions. V.D. would like to thank the Physics Department of University of Turku and INTAS-Program for financial support and kind hospitality. References [1] B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l intégrale générale est a points critiques fixes, Acta Math. 33, 1 55 (1910). [2] E.L. Ince, Ordinary Differential Equations, Dover (1956). 7

8 [3] R. Liouville, Sur les invariants de certaines équations différentielles et sur leurs applications, J. Ecole Polytechnique 59, 7 76 (1889). [4] A. Tresse, Sur les invariants différentiels des groupes continus de transformations, Acta Math. 18, 1 88 (1894). [5] E. Cartan, Sur les variétés a connexion projective, Bull. Soc. Math. France 52, (1924). [6] G.Thomsen, Über die topologischen Invarianten der Differentialgleichung y = f(x,y)y 3 + g(x,y)y 2 +h(x,y)y +k(x,y), Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, 7, , (1930). [7] V. Dryuma, Geometrical properties of nonlinear dynamical systems, In Differentialnye uravneniya i dinamicheskie sistemy, 67 73, Kishinev, Stiintsa (1985). V. Dryuma, On the geometry of differential equation of the second order and its applications, in IVth International workshop solitons and applications, eds. V.G. Makhankov, V.K. Fedyanin, O.K. Pashaev, World Scientific, Singapore (1990). [8] N. Kamran, K. Lamb and F. Shadwick, The local equivalence problem for d 2 y/dx 2 = F(x, y, dy/dx) and the Painlevé transcendents, J. Diff. Geom. 22, (1985). [9] A.C. Hearn, REDUCE User s Manual Version 3.7 (1999). [10] A. Ramani, B. Grammaticos and T. Tamizhmani, Quadratic relations in continuous and discrete Painlevé equations, J. Phys. A:Math. Gen. 33, (2000). [11] C. Cosgrove, unpublished notes and comments on the list of [2]. 8