Equivalence Problem of the Painlevé Equations

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1 Advances in Pure Mahemaics hp://ddoiorg/06/apm00 Pulished Online March 0 (hp://wwwscirporg/journal/apm) Equivalence Prolem of he Painlevé Equaions Sopia Khamrod Deparmen of Mahemaics Facul of Science Naresuan Universi Phisanulok Thailand kunimak@nuach Received Novemer 0; revised Januar 7 0; acceped Januar 6 0 ABSTRACT The manuscrip is devoed o he equivalence prolem of he Painlevé equaions Condiions which are necessar and F o e equivalen o he firs and second sufficien for second-order ordinar differenial equaions Painlevé equaion under a general poin ransformaion are oained A procedure o check hese condiions is found Kewords: Equivalence Prolem; Painlevé Equaions; Poin Transformaion Inroducion Man phsical phenomena are descried differenial equaions Ordinar differenial equaions pla a significan role in he heor of differenial equaions In he 9h cenur an imporan prolem in analsis was he classificaion of ordinar differenial equaions [-] One pe of classificaion prolem is an equivalence prolem: a ssem of equaions is equivalen o anoher ssem of equaions if here eiss an inverile change of he independen and dependen variales (poin ransformaions) which ransforms one ssem ino anoher The si Painlevé equaions (PI-PVI) are nonlinear second-order ordinar differenial equaions which are sudied in man fields of Phsics These equaions and heir soluions he Painlevé ranscenden pla an imporan role in man areas of mahemaics The Painlevé equaions elongs o he class of equaions of he form a a a a 0 () This form is conserved wih respec o an change of he independen and dependen variales u () In fac since under he change of variale () derivaives are changed he formulae PI : 6 PII : 6 PIII : PIV : PV : PVI : () Poin ransformaions are weaker han conac ransformaions S Lie showed ha all second-order equaions are equivalen wih respec o conac ransformaions Coprigh 0 SciRes

2 98 S KHAMROD D u g D g g g u P D Dg Here suscrip means a derivaive for eample 0 Since he Jacoian of he change of variales 0 he equaion ecomes () u 0 a a a u u u u u u u a () (5) (6) Two quaniies pla a major role in he sud of Equaion (5): L u L u u u u Under a poin ransformaion () hese componens are ransformed as follows []: L L L L L L Here ilde means ha a value corresponds o ssem (): he coefficiens i are echanged wih a i he variales and u are echanged wih and respecivel S Lie showed ha an equaion wih L 0 and (7) L 0 is equivalen o he equaion u 0 For he Painlevé equaions L 0 and L 0 R Liouville [] also found oher relaive invarians for eample and v L LL L L L L L LL L 5 u u LL LL L w L L L L R L LR LR L L R LL L L L LL L For he Painlevé equaions v 5 0 and w 0 [5] Up o now he equivalence prolem has een solved in a form more convenien for esing onl for (PI) and (PII) equaions using an eplici poin change of variales was given in [6] The manuscrip is devoed o solving he prolem of descriing all second-order differenial equaions Coprigh 0 SciRes

3 F which are equivalen wih respec o poin ransformaions () o he firs and second Painlevé equaion (PI) and (PII) Eample of he firs Painlevé equaion (PI) is presened Necessar and sufficien condiions for an equaion F o e equivalen o (PI) and (PII) are oained As was noed some of he necessar condiions are [5]: F 0 v 5 0 and w 0 Oher condiions are also epressed in erms of relaions for he coefficiens of Equaion (5) The mehod of he sud is similar o [7-9] I uses analsis of compaiili of an over deermined ssem of parial differenial equaions Equaions Equivalen o he Painlevé Equaions This secion sudies Equaion (5) which are equivalen o he firs and second Painlevé equaion (PI) and (PII) Since an equaion of () elongs o he pe of equaion () he necessar condiion for an equaion F o e equivalen o he firs and second L L Noice ha S KHAMROD 99 Painlevé equaion (PI) and (PII) are ha i has o e of he same pe Since v 5 0 and w 0 are relaive invarians wih respec o () he are also necessar condiion The Firs Painlevé Equaion (PI) For oaining sufficien condiions one has o find condiions for he coefficiens u u u u which guaranee eisence of he funcions ransforming he coefficien of Equaion (6) ino he coefficiens of equaions (PI) Also noe ha he he firs Painlevé equaion has he coefficiens are a 0 a 0 (8) a 0 a 6 Wihou loss of generali i is assumed ha L 0 Since for Equaion (8) he value L 0 and hence he funcions and saisf he equaion L L (9) 0 Susiuing hese coefficiens ino (6) one oains over deermined ssem of parial differenial equaions L L 6LL L L L L L 0 (0) L L L L L L LL L L L L ul LL L L L L L u L LL L L L L L L L L L L L u From Equaions (0)-() one can find he derivaives L L L L L L L 6 LL L L L L 6 L 6 LL L L L L L (5) L L L L L ul L L LL L LL L Taking he mied derivaives 6 6 one oains L 0 (7) Differeniaing his equaion wih respec o and and susiuing found from Equaion (7) one ges 5 L L LL L 7L L L L 6L L L 6 L 6 L 0 u L 6LL L LL L LL L 0 u Finding he derivaives: L u from he equaion v 5 = 0 L from he equaion w = 0 and L from (9) and composing () () () () (6) (8) (9) Coprigh 0 SciRes

4 00 S KHAMROD he equaions one can find he derivaives L L L L 0 u 0 u u L L 8 LL 60 L L 80 L L L 6 L L 90 L L LL 0L L 5L L 0 LL 80 L 00 L L L 5L L L KL u u u u uu L L L LL LL L L LL L L LL 0LL 0LL0 LL0LL 60 L80LLL5L LKL u u u u K L 6LL05L 9LL5L L08 u 0 LL 50 L 60 L 0L L L u L 6LL 0 L 0 L (0) () () Since of () (5) and (8) all second order derivaives of he funcion can e found hen one can compose he equaions 0 and 0 which are reduced o he onl equaion K 600L 0 () The equaion L L u 0 gives u L KuLKLK L LL L () 5 5K L L L L 00L 0 u L 6KL 6KL5KLLK 0 Differeniaing Equaion () wih respec o one oains 7 50 L From his equaion one can find he derivaive 5 6KL 6KL 5KL LK L 50 Noice ha he equaions 0 and 0 Are saisfied and he equaion 0 e comes Q 0 (5) Q 6K 90 KL 780 KLL 850KL 00K LL 70LKL 5 50L u KL 5000L 0KL (6) 00KL 70L KL00KLL 00uKL 800KL 00KL 600KL L80LK 9 K Because of (5) he funcion Q u 0 Differeniaing (5) wih respec o and one ges The derivaive wih respec o is equal o zero Recall ha Equaion (0) is oained from he equaion w = 0 e R5KL 0 (7) KLQ u QL 00QL 0 (8) R KQKL L7L 5LQK KQ (9) Differeniaing Equaion (7) wih respec o and one oains he onl equaion RL R 7L L L 0 (0) Finding he funcion from (7) and susiuing i ino () (6) () one ges Noice ha 5 5 R K 0 L LR R L 5R L L L L u u R L LL L 5 0 () () Thus he necessar and sufficien condiions for equaion F o e equivalen o he firs Painlevé equaion are: he equaion has o e of he form (5) wih he coefficiens i u i saisfing he condiions v 5 = 0 (9)-() () (8) and () he funcions K u Ru and Q u are defined Equaions () (6) (9) The ransformaion is defined (5) and () The Second Painlevé Equaion (PII) Similar o he firs Painlevé equaion one can sud he second Painlevé equaion Painlevé equaion (PII) has he coefficiens are a 0 a 0 () a 0 a Susiuing hese coefficiens ino (6) one oains ov r deermined ssem of parial differenial equaions Coprigh 0 SciRes

5 S KHAMROD 0 L L 6 LL L L L L L 0 () L L L L L L L L L 6 L LL 6 L L L 0 (5) u L L Noice ha L L L L L L 6LL L L L L ul LL L 0 From Equaions ()-(6) one can find he derivaives L u L 6 LL L (6) L (7) L L L L L L L L L L 6 LL L L L L L L L L L L L L L L L L 6 L L L 6 L u Taking he mied derivaives one oains (8) (9) (0) L 0 () Differeniaing his equaion wih respec o and and susiuing Ψ found from Equaion () one ges he funcion K u 5 L L 6L 6L L LL L 7LL L L6L L 0 u is defined he formula () K 0 () K L L 6LL L L L L u L L L Since 0 hen K 0 Hence Equaions () and () define K and he derivaive Thus all second-order derivaives and he derivaive of he funcion are defined Susiuing he epression of ino Equaions (7) and (0) one oains L K L K L u K L L L L K L L L L L 0 u (5) KL 9 L L L 5K K L () (6) Equaions () and (6) define all firs-order derivaives of he funcion Since he second-order derivaives have een found one needs o check he condiions All hese condiions are saisfied ecep he firs one which ecomes L 50K KL 9K Lu L 99KL LL L L 9K L L 0 K 60K L K 5 L L 6 6K L L L L Differeniaing (7) wih respec o and ecluding using (7) one oains KL K L K L K L K L 6K L K L L L L L L LL ul L LL L K L L (8) 6 0 (7) Ecluding he variale α from (7) using (8) Equaion (7) ecomes Q 0 (9) Coprigh 0 SciRes

6 0 S KHAMROD Q L 8L K KL K L K KL K KL 8K L LL 9L L L L ulk L8L ul L (50) Differeniaing (9) wih respec o and one ges respecivel u 5 QL Q L KL KL K L L K 6QK L K L 0 (5) QL QL u QK 0 Since KL 0 he coefficien wih in (5) is no equal o zero Hence Equaions (9) and (5) define he variale and Equaion (8) ecomes 8K L L LL L L L K L 0 L L L L 6 u u L 6L 6K K QL L L L 0K L 6Q KL QL 0K L 8L 0 Remaining equaions are oained differeniaing (5) wih respec o and Ecluding from hem and hese equaions are reduced o he equaion 6Q K L Q KL K L 5 KL 5 KL L K QL Q 6K L L L QL 6K LLL 0KL 86QKLQLQL 0 0K L (5) (5) (5) Thus he necessar and sufficien condiions for an equaion F which can e ransformed o he second Painlevé equaions are: his equaion has o e of he form (5) he coefficiens saisf he equaions v 5 = 0 w = 0 (5) (5)-(5) he funcions K u and Q u are defined Equaions () and (50) The ransformaion of he Equaion (5) ino he second Painlevé equaion (PII) is defined Equaions (9) and (5) Eample of he Resuls Eample The following equaion is equivalen o he firs Painlevé equaion (PI) u u uu This equaion has o e of he form (5) wih he coefficiens u u saisfing he condiions L L K s Q s R s 5 u Equaions (9)-() () (8) and () are saisfied and Equaions (5) and () ecome Q 0 s The changes of vari- 8 8 ale are he following: 0 9 u Conclusion The necessar and sufficien condiions ha an equaion of he form F o e equivalen o he firs and second Painlevé equaion under a general poin ransformaion are oained As was noed some of he necessar condiions are v 5 = 0 and w = 0 Oher condiions are also epressed in erms of relaions for he coefficiens of Equaion () A procedure o check hese condiions is found Since inermediae calculaions in he equivalence prolem are cumersome compuer algera ssem have ecome an imporan compuaional ool 5 Acknowledgemens This research is suppored Commission on Higher Educaion and he Thailand Research Fund under Gran No MRG 9805 Naresuan Universi and Suranaree Universi of Technolog REFERENCES [] S Lie Klassifikaion und Inegraion von Gewonlichen Differenialgleichungen Zwischen Die Eine Gruppe von Transformaionen Gesaen III Archiv for Maemaik og Naurvidenska Vol 8 No 88 pp 7-7 [] R Liouville Sur les Invarians de Ceraines Equaions Differenielles e sur Leurs Applicaions Journal de l École Polechnique Vol pp 7-76 [] A M Tresse Déerminaion des Invarians Poncuels de Coprigh 0 SciRes

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