A Systematic Analysis of the Properties of the Generalised Painlevé--Ince Equation
aa r X i v : . [ n li n . S I] A ug A Systematic Analysis of the Properties of the GeneralisedPainlev´e–Ince Equation
Andronikos Paliathanasis ∗ Institute of Systems Science, Durban University of TechnologyPO Box 1334, Durban 4000, Republic of South Africa
P.G.L. Leach
Institute of Systems Science, Durban University of TechnologyPO Box 1334, Durban 4000, Republic of South AfricaSchool of Mathematical Sciences, University of KwaZulu-NatalDurban, Republic of South Africa
August 14, 2019
Abstract
We consider the generalized Painlev´e–Ince equation,¨ x + αx ˙ x + βx = 0and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When thefree parameters are related as β = α / x = 1 /y. We concludethat the Painlev´e–Ince equation is integrable is terms of Lie symmetries and of the Painlev´e test.
The Painlev´e–Ince Equation, ¨ x + 3 x ˙ x + x = 0 , (1)is well-known to have a number of interesting features. Firstly it has eight Lie point symmetries [16] with thealgebra sl (3 , R ) in the Mubarakzyanov Classification Scheme [17–20] and so is linearisable by means of a point ∗ Email: [email protected] et al [15] showed thatthe singularity was a simple pole and that the resonances for a = 1 were a perfectly normal 1 and the generic − a = 2 they were − −
2. This result was not acceptable to some workersin the field, but a subsequent programme initiated by Mac Feix [14] gave substance to the result and a fullerexposition is to be found in the paper of Andriopoulos et al [5] in which the whole question of positive andnegative resonances was answered in terms of regions in the complex plane centred on the singularity. Thirdlythe Painlev´e–Ince Equation is a member – the second – of the Riccati Hierarchy [10–13] based on the recursionoperator, D + x with D = ddt , applied to ˙ x + x as initial member .The generalised Painlev´e–Ince Equation is defined as¨ x + αx ˙ x + βx = 0 (2)in which α and β are constants, by means of some ingenious manipulations. In this paper we approach thequestion of integrability of (2) by means of the systemic methods of analysis, namely the search for Lie pointsymmetries and the determination of the existence of a Laurent series about a singularity. The Lie point symmetries of (2) are easily calculated using the Mathematica add-on, SYM [4, 7–9]. For generalvalues of α and β there are two symmetries, namely,Γ = ∂ t and (3)Γ = t∂ t − x∂ x (4)except when the parameters are related according to β = α / . (5)Then there are the eight symmetries of the Painlev´e–Ince Equation up to the effect of the rescaling by α .The invariants for the symmetry, Γ are u = x and (6) v = ˙ x. (7)In the new variables (2) becomes vv ′ = 0 . (8)In the new variables Γ becomes u∂ u + v∂ v when the superfluous term in t is ignored. The invariants for theonce extended form of this symmetry are w = vu and (9) z = v ′ u . (10)With the invariants (9) and (10 the first-order equation (8) becomes the algebraic equation, wz + αw + q = 0 . (11) One could simply start at x . Singularity Analysis
We now turn to the analysis of (2) in terms of the Painlev´e-Test as summarized in the ARS algorithm. Thefirst step is to determine whether a singularity exists and, if so, to calculate its coefficient. To this end we makethe substitution x = aτ r (12)into (2), where τ = t − t and t is the location of the putative singularity. We find that the terms balance inexponent when r = − ie the singularity is a simple pole. Moreover all terms are dominant. When we replace r with −
1, the equation for the leading-order coefficient is2 aτ − αa τ + βa τ = 0 (13)which has the solutions a = 0 , α − sqrtα − β β and α + sqrtα − β β . (14)The first solution must be rejected as being irrelevant. The other two can take any value as we are working inthe complex time plane.The second step of the ARS algorithm is the location of the resonances, that is the exponents at which theremaining constants of integration enter the Laurent expansion. We make the substitution x = aτ − + mτ − s (15)into (2) (recall that all terms in (2) are dominant) and collect the terms linear in m for these are the terms atwhich a new constant enters into the expansion. For that constant to be arbitrary the coefficient of m must bezero. The coefficient is a polynomial in s and the values of s which render it zero are the resonances. When wedo this, we obtain s = − , α p α − β − α + 8 β β (16)in the former case and s = − , − α p α − β − α + 8 β β (17)in the latter case. The value − et al could be regarded as commonplace. Naturally there are also a multitude of values forwhich the second (nongeneric) resonance is irrational and/or complex. We have seen that the generalised Painlev´e–Ince Equation possesses two Lie point symmetries and is reducibleto an algebraic equation for all values of the parameters. From singularity analysis we further see that thesingularity is always a simple pole. Various possibilities exist for the nature of the Laurent expansion about thissimple pole. It can be either a Right – expansion over a disc centered on the pole – or a Left – expansion about3he pole over the complex plane without a disc – Painlev´e Series depending upon the values of the parameters.Alternatively it can be a mess, thereby indicating nonintergability.An interesting feature occurs if one inverts the dependent variable by setting x ( t ) = 1 y ( t ) . (18)Then equation (2) takes the following form y ¨ y − y + α ˙ y − β = 0 (19)The two symmetries look the same under the coordinate transformation, except that the plus sign becomes aminus sign in Γ . Naturally we are not considering the particular case in which α = 9 β. In terms of the singularity analysis for equation (19) only the first two terms are dominant, the singularityis a simple pole and the coefficient of the leading-order term is unspecified. (A zeroth-order possibility alsoexists, but that is not a singularity.) The resonances are at − α and β which give rise to equations of maximal symmetry, that is equations of Painlev´e–Ince form. Hence, we can infer that the generalized Painlev´e–Ince equation (2) passes the Painlev´e test underthe coordinate transformation (18) if and only if α = 9 β .That is not the first example of a differential equation which pass the singularity test under a coordinatetransformation. A recent discussion on that property can be found in [21]. Acknowledgements
PGLL thanks the Durban University of Technology, the University of KwaZulu-Natal and the National ResearchFoundation of South Africa for support. This work was undertaken while we enjoyed the gracious hospitalityof Surananee University o Technology, Thailand.
References [1] Ablowitz M J, Ramani A & Segur H (1978) Nonlinear Evolution Equations and Ordinary DifferentialEquations of Painlev´e Type
Lett Nuovo Cimento J Math Phys J Math Phys Physics Letters A
Applied Mathematics andInformation Sciences (3) 484-499[7] Dimas S & Tsoubelis D (2004, October). SYM: A new symmetry-finding package for Mathematica. InProceedings of the 10th International Conference in Modern Group Analysis (Unmiversity of Cyprus, pp.64-70).[8] Dimas S & Tsoubelis D (2006, June). A new Mathematica-based program for solving overdeterminedsystems of PDEs. In 8th International Mathematica Symposium (Avignon).[9] Dimas S “Partial Differential Equations, Algebraic Computing and Nonlinear Systems.” PhD thesis, Uni-versity of Patras, Greece (2008).[10] Euler M, Euler N and Leach PGL (2007) The Riccati and Ermakov-Pinney hierarchies Journal of NonlinearMathematical Physics Theoretical and Mathematical Physics
Journal of Nonlinear Mathematical Physics
Lobachevskii Journal of Mathematics Journal of PhysicsA: Mathematical and General y ′′ + yy ′ = 0 JPhys A: Math Gen QuæstionesMathematicæ Izvestia Vysshikh Uchebn Za-vendeni˘ı Matematika , 161-171[18] Mubarakzyanov GM (1963) On solvable Lie algebras Izvestia Vysshikh Uchebn Zavendeni˘ı Matematika Izvestia VysshikhUchebn Zavendeni˘ı Matematika Izvestia Vysshikh Uchebn Zavendeni˘ı Matematika International Journal of Geometric Methods in Modern Physics13