Journal of Mahemaics and Saisics 7 (: 49-54, ISSN 549-644 Science Publicaions A New Perurbaive Approach in Nonlinear Singulariy Analysis Ta-Leung Yee Deparmen of Mahemaics and Informaion Technology, The Hong Kong Insiue of Educaion, Tai Po, New Terriories, Hong Kong Absrac: Problem saemen: The sudy is devoed o he mirror mehod which enables one o sudy he inegrabiliy of nonlinear differenial equaions. Approach: A perurbaive exension of he mirror mehod is inroduced. Resuls: The mirror sysem and is firs perurbaion are hen uilized o gain insighs ino cerain nonlinear equaions possessing negaive Fuchs indices, which were poorly undersood in he lieraures. Conclusion/Recommendaions: In paricular, for a nonprincipal bu maximal Painleve family he firs-order perurbed series soluion is already a local represenaion of he general soluion, whose convergence can also be proved. Key words: Mirror ransformaion, painleve es, singulariy analysis, Ordinary Differenial Equaions (ODE, singulariy analysis, mirror sysem, maximal family, perurbaion expansion, negaive Fuchs indices INTRODUCTION poin of view. Secondly, Hu e al. ( showed ha he mirror ransformaions are canonical for The relevan lieraure sudy daes back o one finie-dimensional Hamilonian sysems. Moreover, cenury ago when Painleve made an in-deph sudy of Yee ( demonsraed ha he linearizaion of singulariies and iniiaed he (now named Painleve mirror sysems near movable poles gives he analysis of inegrabiliy. Painleve se up he problem of possibiliy o consruc he associaed Backlund ransformaions of some parial differenial equaions deermining all differenial equaions whose general and he Schlesinger ransformaions of some ordinary soluion are single-valued. Following he pioneering differenial equaions. work of Painleve 9, he mehods of Gambier In he curren work our primary goal is o 99 Bureau (964; Ablowiz e al. (98 and inroduce an improvemen of he mirror mehod so Weiss e al. (98 have been evolved and hey were ha negaive indices ( resonances can be reaed. successful o apply in many cases. However, he The srucure of he sudy can now be explained. The main drawback of he mehods is ha none of hem perurbaive Painleve mehod is firs inroduced. We can build necessary condiions a all ineger values of demand single-valuedness no only for any pole-like resonances. To be specific, negaive Fuchs indices expansion as in he Painleve es, bu also for every canno be handled by hese mehods. The reason why soluion close o i, represened as a perurbaion he mehods canno handle negaive indices lies in series in a small parameer ε. The usage of he idea he fac ha heir Lauren series is assumed o be of he perurbaive mehod proves o be bounded from below. remendously beneficial for he mirror mehod as a The mirror mehod uses he new ool in new improvemen. Order-zero is he usual mirror singulariy analysis: mirror ransformaions and sysem. Order-one reduces o a linearizaion of regular mirror sysems, which was firs inroduced by mirror sysem near a regular singulariy and allows Hu and Yan (999;. By his mehod hey were he inroducion of all missing arbirary coefficiens. successful in he following several aspecs. Firsly, Higher orders lead o he analysis of a linear, he success of consrucing mirror ransformaions Fuchsian ype inhomogeneous sysem. In paricular, enables us o rea each principal balance in he negaive indices give rise o doubly infinie Lauren Painleve es, singulariy srucures and symplecic series. An illusraive example of Bureau s equaion srucures of Hamilonian sysems from a common is also presened and finally he conclusion follows. 49
MATERIALS AND METHODS Perurbaive Painleve analysis: Now we firs presen he perurbaive mehod originally developed by Cone e al. (99. The mehod allows us o exrac he informaion conained in he negaive indices, hus building infiniely many necessary condiions for he absence of movable criical singulariies of he logarihmic ype. Le us consider a nonlinear ordinary differenial equaion: E K (u, x = ( which is polynomial in u and is derivaives, analyic in x. The sandard Painleve expansion akes he form (X is he expansion variable, X x =: ( ( j+ p j j= u = u u X ( ( j+ q j j= E = E E X = J. Mah. & Sa., 7 (: 49-54, ( in which he negaive inegers p and q are he respecive singulariy order of u and E. We seek a Lauren expansion for any soluion which is near o he soluion obained by he sandard Painleve mehod. We do his by considering a perurbaion expansion. For a non-principal bu maximal Painleve family he perurbaion exends he paricular soluion ino a represenaion of he general soluion. Le us define he Painleve expansion (u (, E ( as he soluion of unperurbed problem, and look for a nearby soluion formally represened by an infinie perurbaion series in powers of small parameer ε: n (n u = Taylor( ε ε u n = n (n K(u, x E = ε E = n = ( Le us denoe, for he equaion E =, R k = he se of indices for k h family = {, -, }, wih he following assumpions: ( all indices are disinc inegers, and s o be he smalles index (s -; ( all k families are maximal (families wih a number of indices equal o he order of he equaion; ( a leas one of he k families is principal (any maximal family wih, apar from -, all ineger indices non-negaive. Now, he condiion ha he perurbaion expansion sill be a soluion generaes an infinie sequence of successive differenial equaions: E ( = K(u (, x =, E ( = K (u ( u ( =,, where, K is he Freche operaor acing on u (n. A each level of perurbaion, we consruc a pole expansion, bu he order of he pole increases wih he order of he perurbaion. The resuling infinie perurbaion expansion is a doubly infinie Lauren expansion: u = ε u X = u X n (n j+ p j+ p j j n = j= ns j= n (n j+ q j+ q E = ε E j X = E jx = n = j= ns j= In general, perurbaion heory pracically always yields divergen series. However, by considering he perurbaion series soluion u ( +εu ( and expanding in X, wih coefficiens dependen of ε, we can prove he convergence. This can be done by inroducing a new ransformaion for he mirror sysem. Evenually we succeed o deduce a regular exended mirror sysem wih regular iniial daa. The Cauchy-Kowalevski heorem is hen applied and convergence follows accordingly. The imporance of he perurbed soluion u ( +εu ( is ha: for a non-principal bu maximal Painleve family i is already a local represenaion of he general soluion. In he following we invesigae in deail each order of ε: Wih n =, E ( (u ( K(u (, x = where, u ( is a (eiher paricular or general soluion of he original nonlinear equaion, which is deermined by sandard Painleve analysis: = X p (A + B X +, A (4 Wih n =, E ( (u (, u ( K (u ( u ( = where, u ( is (he general soluion of homo equaion + (a paricular soluion of inhomo equaion: = X p (A X s + B X s- + + (5 where, A, B, are arbirary coefficiens inroduced a level one. The Painleve series u (, a Lauren series which is bounded below, is subsiued ino he linearized equaion K (u ( u ( =, he resuling equaion for u ( is of Fuchsian ype, he movable 5
J. Mah. & Sa., 7 (: 49-54, singulariy X = of he original ODE is a regular singulariy for he linearized equaion and is Fuchs indices are i+p, where i runs over he Painleve resonances. A his firs order, an arbirary coefficien is inroduced a each index. No all of hese are new since we already have a coefficien in u (, corresponding o each posiive ineger index. The coefficiens inroduced ino u ( a he corresponding indices (i+p, i a posiive ineger jus perurb he already arbirary coefficiens, so add nohing new and i is no harmful o se hem zero a his level. However, all oher indices give rise o new arbirary coefficiens. Therefore he expression u ( +εu ( already conains as many arbirary coefficiens as here are indices in he family: Wih n, E (n (u (,, u (n K (u ( u (n - R (n (u (,, u (n = Where: u (n = X p (A n X s + B n X s- + +X p (C n X sn + D n X sn- + (6 where, A n, B n, are arbirary (independen of A i, B i, i n- n-h level coefficiens ha can be absorbed by u (. Wihou he loss of generaliy, we se A n = B n = for each n. Therefore, for n, we only concern abou a paricular soluion of each inhomogeneous equaion. The coefficiens C n, D n, are dependen of he previous useful coefficiens which belong o a subse of {A i, B i, i =, }. A hese n-h orders (n each funcion u (n saisfies an inhomogeneous, linear differenial equaion. The indicial equaion is he same for all n bu for n he leading behaviour of u (n is deermined by he singulariy order of he rhs funcion R (n, no by K (u (. RESULTS K (u, u +uu +(u +(u -c u -c u-d, (7 where, c and d are funcions of. By he sandard Painleve es we obain wo families of soluions wih singulariy orders and Fuchs indices in he following: (F p = -, u =, {-,, }, (F p = -, u =, {-,-, } The Painleve series of (F-(F are respecively: ( c ( u = T + r + ( r + T + rt +..., (8 And: ( c ( u = T + T + st 6 + + + 6 c ( d ( c ' T... (9 where, T := - and, r, r, s are arbirary. Now we are applying he perurbaive Painleve analysis o he second family (F and he resul reads: ( u = T + c ( T + st +..., 6 ( u = T AT + T + A ( c s T 6 + A ( c ( d ( + c ' T + T +... 4 ] ( Thus, he resuling infinie perurbaion expansion for (F is a doubly infinie Lauren expansion: ( ( ( n (n j j j= u u u u... u... T u T = + ε + ε + + ε + = ( The local represenaion of he general soluion is given by: We begin o illusrae, hrough a simple bu insrucive example, ha he analysis on mirror sysems migh be performed in a perurbaive approach such ha negaive and posiive indices can be reaed a he same ime. We illusrae he algorihm of performing he new perurbaive approach on he mirror sysem hrough he following ODE example of hird-order kind, namely he Bureau s equaion. We also aim a showing he proof of convergence of he no principal balance of mirror sysem. The Bureau s hird-order ODE is E = K(u, =, where: 5 u = u + ε u = ( ε A T + T * ( ( +ε + 6 A ( c s T... ( To demonsrae explicily he exension of he mirror mehod we inroduce he corresponding mirror ransformaion for he original equaion given by (7: u = θ, v = θ + η θ, w = θ η θ + c θ + η (
J. Mah. & Sa., 7 (: 49-54, The regular mirror sysem is given by: θ ' = ηθ η ' = c η + η θ η ' = d (4 The above mirror sysem can be expressed as K(Θ =, where Θ = ( θ, η, η. Based on he dominan balance we obain he wo families of soluions of he mirror sysem wih he following singulariy orders and Fuchs indices: (F p = (,-,-, Θ = (,,, {-,, }, (F p = (,-,-, Θ = (,,, {-,-, } The Painleve series of (F-(F are respecively: θ = T r T + (r c ( T + ( r r + rc ( T +..., ( ( η = T + rt + ( 4r + c ( T + (4r + 4r rc ( T +... ( η = T + T + T + ( 8r + 8r + rc ( c T +... and: ( θ = T + T c ( T st +... 4 4 ( 5 η = T + T + c ( T + st +... ( ' η = T + T + T + (s c ( T +... ] (5 (6 The perurbaive expansion for he mirror sysem is: n (n (n (n ( (, (,, (7 n = Θ = ε Θ Θ = θ η η K( Θ =, K(Θ ( =. The firs few erms are deermined by: ( ( ( Θ : K '( Θ Θ = Θ ( Θ ( Θ ( ( Θ ( Θ ( = : K '( R (, Θ ( Θ ( Θ ( ( Θ ( Θ ( Θ ( = : K '( R (,, ( ( + η θ ( ( ( ( K '( Θ η + η θ ( ( θ η ( ( ( ( R ( η + θ η ( ( ( ( θ η θ η ( ( ( ( ( ( ( R η η + θ η + θ η Finally he successive linearizaions of mirror sysem can be deermined now. We consider he nonprincipal balance (F only. Wih n = : ( ( ( θ = η θ ( η = c ( η + η θ ( ( η = d ( ( ( ( (8 ( ( ( ( which deermines Θ = ( θ, η, η as given by (6. Wih n = : ( ( ( ( ( θ + η θ + θ η = ( ( ( ( ( ( ( ( η η θ + η η θ η = ( ( η = which gives: ( θ = T α T + T α c ( T +... ( η = T T + T + T + (α s α c T +... ( η = T T + T + T + T +... (9 ( where s and α are independen arbirary consans inroduced a he zero and he firs level, which correspond o indices - and -, respecively. So, α is he new (imporan parameer ha we are looking for. A his level, we se anoher wo arbirary consans o zero wihou any loss of generaliy since he arbirary consans (a indices - and are already represened ino Θ (. Since he family (F is maximal hen he perurbed soluion Θ ( +εθ ( is a local represenaion of he general soluion. One indeed can coninue o look for higher level perurbaion in order o obain a doubly infinie expansion. We jus lis he second level linearizaion of mirror sysem below and he informaion up o n = is good enough for our purpose. Where: Wih n = : 5
J. Mah. & Sa., 7 (: 49-54, ( ( ( ( ( ( ( θ + η θ + θ η = θ η ( ( ( ( ( ( ( ( η η θ + η η θ η = ( ( ( ( η + θ η ( ( η = DISCUSSION ( The new ransformaion for he mirror sysem can be deermined based on he above resuls. In he following, le us also prove he convergence of he perurbaion series soluion Θ ( +εθ ( of (F. Again, we shall use a new ransformaion o conver he original mirror sysem ino a new regular sysem of firs-order differenial equaions wih regular iniial daa. Wih he family (F, we deduce he following Lauren series based on (6 and (: The exended mirror sysem becomes: ' δ = ξξδ cδ ' ξ = ξ ξ + ( c ξ δ ' ξ = d The Lauren series for (δ, ξ, ξ are: δ = T + αε(c s T +..., ξ = α ε + α ε (c s T +..., ξ = s c +..., which gives he iniial daa: (6 (7 ( δ, ξ, ξ ( = (, αε,s c (8 αε T + ( αεc ( T 6 5 6 s c +... ( ( Θ Θ + εθ = +αε( s + c T +... T + αε(s c +... ( The convergence of he general soluion can now be discussed. For a non-principal bu maximal family (F he firs-order perurbed series soluion ( is already a local represenaion of he general soluion. In order o show he convergence of (, we need he ransformaions: where, α and s are he Painleve resonances a Fuchs indices - and, respecively. We easily see ha he soluion blows up when T, or. We observe ha fac ha η is he only resonance variable blowing up in he order of So we firs inroduce he new O(. variable δ by η and formally inver ( - ino a series of δ. In his example, i is: = δ + α ε(s c δ 4 + ( c ( + αε (c s δ + O( δ ( Nex, we formally expand θ and η ino series of δ: θ = αεδ + αε (c s + ( α εc ( δ +..., η = s c +... (4 (u,u',u'' ( θ, η, η ( δ, ξ, ξ By he Cauchy-Kowalevski heorem, he exended mirror sysem (6 wih he iniial daa (8 has a unique analyic soluion (δ(, ξ (, ξ ( near =. Then (θ, η, η = (ξ δ -, δ -, ξ is a soluion of he original mirror sysem (4 near =. Moreover, from he ordinary power series mehod, we can find he expansions for (δ, ξ, ξ. Then an easy calculaion reveals ha he Lauren series of (θ, η, η = (ξ δ -, δ -, ξ are exacly (. The convergen power series soluions of he exended mirror sysem lead o convergen Lauren series soluions of he original mirror sysem, because of he equivalence beween he sysems. This proves he convergence of Θ ( +εθ ( in (. In paricular, he series of θ ( + εθ ( is convergen. From: By runcaing he δ-series for θ a he locaion of he firs resonance α o inroduce a new variable ξ, and similarly, runcaing he δ-series for η a he locaion of s o inroduce ξ we hen deduce he new ransformaion (θ, η, η (δ, ξ, ξ : θ = ξ δ, η = δ, η = ξ (5 5 ( u ( + ε u ( +... = θ ( + εθ ( +... Or: ( ( ( ( ( ( ( u θ + ε( u θ + u θ ( ( +ε ( u θ +... +... =
J. Mah. & Sa., 7 (: 49-54, we can find he expansions for (u (, u ( and hey are exacly (, up o he order where all he resonances appear. This proves he convergence of Lauren series soluion u ( +εu (, which is locally represening he general soluion for (F. CONCLUSION In his sudy we are rying o inroduce a pach o he mirror mehod so ha he negaive Fuchs indices can be reaed. This consideraion exends he use of mirror ransformaions o a larger class of differenial equaions. Based on he examples under consideraion, we are successful in reaing he negaive Fuchs indices. Order-zero perurbaion gives he ordinary mirror sysem. Order-one reduces o a linearizaion of mirror sysem near a regular singulariy and allows he inroducion of all missing arbirary coefficiens. The mehod reveals ha u ( +εu ( is already a represenaion of he general soluion, whose convergence can also be proved. REFERENCES Ablowiz, M.J., A. Ramani and H. Segur, 98. A connecion beween nonlinear evoluion equaions and ordinary differenial equaions of P-ype. J. Mah. Phys. :75-7. DOI:.6/.5449 Bureau, F.J., 964. Differenial Equaions wih fixed criical poins, Annal. Ma. Pura Appli., 64: 9-64. DOI:.7/BF454 Cone, R., A. P. Fordy and A. Pickering, 99. A perurbaive Painleve approach o nonlinear differenial equaions. Physica D, 69: -58. DOI:.6/67-789(9979-5 Hu, J. and M. Yan, 999. Singulariy analysis for inegrable sysems by heir mirrors. Nonlineariy, : 5-54. DOI:.88/95-775//6/6 Hu, J. and M. Yan,. The mirror sysems of inegrable equaions. Sud. Appl. Mah., 4: 67-9. DOI:./467-959. Hu, J., M. Yan and T. L. Yee,. Mirror ransformaions of Hamilonian sysems. Phys. D: Nonlinear Phenomena, 5: -. DOI:.6/S67-789(64-6 Weiss, J. M. Tabor and G. Carnevale, 98. The Painleve propery for parial differenial equaions, J. Mah. Phys., 4: 5-56. DOI:.6/.557 Yee, T.L.,. Linearizaion of mirror sysems. J. Nonlinea. Mah. Phys., 9: 4-4. hp://cieseerx.is.psu.edu/viewdoc/download?doi =...64.759&rep=rep&ype=pdf 54