Aspects of proper differential sequences of ordinary differential equations

Asp ects of prop er differ e n tial sequences of ordinary differe ntial equations N EULER and PGL LEACH 1 Dep art ment of Mathematics, Lule ˚ a University of T e chnolo g y SE-971 87 Lule ˚ a, Swe den Email: Norb ert.Eu ler@ sm.luth.se Abstract : W e define a prop er d ifferen tial sequence of ordinary differenti al equations and in tro du ce a method to deriv e an alternativ e sequence of in tegrals for suc h a sequence. W e describ e some general prop erties which are illustrated b y sev eral examples. 1 In tro duction In a recen t pap er M Euler and the present authors rep orted a symmetry analysis and P ainlev ´ e analysis of t wo sequences of ord inary differenti al equ ations, namely a Riccati sequence and a E rmak o v-Pinney sequen ce [5]. An driop oulos and Leac h [1] used the sin- gularit y analysis and explicit solution of the Riccati Sequence as a v ehicle to demonstrate some sp ecific results whic h can arise during the course of the singularit y analysis. Sub- sequen tly Andr iop oulos et al [2] made a detailed study of the s ymmetry and singu larity prop erties of the Riccati Sequence. The aim of the presen t pap er is to define the p r op er differ ential se quenc e and discuss its in tegrabilit y . W e also int ro duce an alternative se quenc e w here the equations of the higher order mem b ers in the sequence d o not increase in ord er, but are fi xed b y the fi rst equation in the sequence. This approac h can help to integrate the sequence and pr o vides in some cases a direct route to the firs t integ rals of the equations in the differen tial sequence. The pap er is organised as follo ws: In Section 2 w e giv e defin itions regarding prop er differen tial sequences, th eir Lie symmetrey algebra and their integ rability . In S ection 3 w e in tro d uce a metho d to deriv e an alternativ e sequence and pro vide sev eral examples to illustrate the concept of c omp atible and c ompletely c omp atible se quenc es . I n an App endix w e pro vide details of the Lie p oin t s y m metry analysis of some of the sequences d iscussed in Section 3. 1 p ermanent address: School of Mathematical Sciences, U niversit y of KwaZulu-Natal, Priv ate Bag X54001, Durb an 40 00, Repub lic of South Africa Email: leac hp@uk zn.ac.za; leac h@math.aegean.gr 2 N Euler and PGL Leac h 2 General description Consider th e v ariables x an d u , wh ere u = u ( x ) w ith u x = du/dx, u xx = d 2 u/dx 2 and the additional notation u nx = d n u/dx n . W e now consider a differentia l sequen ce of m equations, { E 1 , E 2 , . . . , E m } , (2.1) in the follo wing form: E 1 := F ( u, u x , u xx , . . . , u nx ) = 0 E 2 := R [ k ] [ u ] F ( u, u x , u xx , . . . , u nx ) = 0 E 3 := ( R [ k ] [ u ]) 2 F ( u, u x , u xx , . . . , u nx ) = 0 (2.2) . . . E m := ( R [ k ] [ u ]) m − 1 F ( u, u x , u xx , . . . , u nx ) = 0 , where R [ k ] [ u ] is a k th-order integrodifferential op erator of the form R [ k ] [ u ] = G k D k x + G k − 1 D k − 1 x + · · · + G 0 + Q D − 1 x ◦ J. (2.3) The adjoin t of R [ k ] [ u ] has the form ( R [ k ] ) ∗ [ u ] = k X i =0 ( − 1) i D i x ◦ G i − J D − 1 x ◦ Q. (2.4) W e term E 1 the se e d e quation of the different ial s equence (2.2). Note that th e second equation, E 2 := R [ k ] [ u ] F ( u, u x , u xx , . . . , u nx ) = 0 , (2.5) is of order n + k , the third equation, E 3 := ( R [ k ] ) 2 [ u ] F ( u, u x , u xx , . . . , u nx ) = 0 , (2.6) is of order n + 2 k and the m th equation E m is of order n + ( m − 1) k . Let L E i [ u ] denote the linear op erator L E i [ u ] = ∂ E i ∂ u + ∂ E i ∂ u x D x + ∂ E i ∂ u 2 x D 2 x + . . . ∂ E i ∂ u q x D q x (2.7) and L ∗ E i [ u ] the adjoin t to L E i [ u ], namely L ∗ E i [ u ] = ∂ E i ∂ u − D x ◦ ∂ E i ∂ u x + D 2 x ◦ ∂ E i ∂ u 2 x + . . . + ( − 1) q D q x ◦ ∂ E i ∂ u q x . (2.8) Asp ects of prop er differentia l sequences 3 W e den ote by Z i ( E i ) the v ertical symmetry generator of the equ ation E i in th e sequence (2.2), namely Z i ( E i ) = Q ( x, u, u x , u xx , u 3 x , . . . , u j x ) ∂ u (2.9) where the necessary and sufficient inv ariance condition for equation E i is L E i Q     E i =0 = 0 . (2.10) Note that (2.9) includes the p oint s ymmetry generators Γ i = ξ ( x, t, u ) ∂ x + η ( x, t, u ) ∂ u (2.11) with sym metry charac teristic Q ( x, u, u x ) = ξ ( x, u ) u x − η ( x, u ) and equiv alent v ertical form Z i = [ ξ ( x, u ) u x − η ( x, u )] ∂ u . (2.12) Definition 1: The se qu enc e (2.2) admits a p - dimensional Lie symmetry algebr a, L , sp anne d by the line arly indep endent symmetry gener ators { Z i 1 ( E i ) , Z i 2 ( E i ) , . . . , Z i p ( E i ) } (2.13) if e ach e qu ation in the se quenc e (2.2 ), { E 1 , E 2 , . . . , E m } , admits a p -dimensional Lie symmetry algebr a, L ′ , isomorphic to L . Definition 2: J = J ( x, u, u x , u xx , . . . ) is an inte g r ating factor f or the differ ential se quenc e (2.2) if J is an inte gr ating factor f or e ach e quation in the se qu enc e. Definition 3: The op er ator R [ k ] [ u ] of the form (2.3) is define d as a k th-or der r e cursion op er ator of the differ ential se quenc e (2.2) under the fol lowing c onditions: h L E i [ u ] , R [ k ] [ u ] i = 0 , i = 1 , 2 , . . . , m, (2.14a ) ( R [ k ] ) ∗ [ u ] J k = αJ l ∀ k , l = 1 , 2 , . . . , p, (2.14b) wher e α is a nonzer o c onstant, i = 1 , 2 , . . . m and p is the total numb er of inte gr ating factors, J l , valid for al l memb ers of the se quenc e. F or some values of l , J l may b e zer o. Definition 4: A pr op er differ ential se quenc e of or dinary differ ential e q u ations is a differ- ential se qu enc e which admits at le ast one r e cursion op er ator of the f orm (2.3). Definition 5: An inte gr able differ ential se quenc e is define d as a pr op er differ ential se- quenc e of or dinary differ ential e quations for which e ach e quation in the se quenc e is inte- gr able. 4 N Euler and PGL Leac h Remark: By an inte gr able ordin ary differential equation of n th ord er, w e mean an equa- tion wh ic h admits a solution, u = φ ( x ; c 1 , . . . , c n ), wh er e c j , j = 1 , n are in dep endent ar- bitrary constan ts. In less strict sense w e requir e th at the n th-order equation admits n − 1 functionally ind ep endent first integ rals su c h that the general solution can b e expressed as a quadrature. In tegrabilit y of a nonlinear ordinary differential equation can also b e expressed in terms of its singularit y stru cture in the complex domain. This is kno wn as the Painlev´ e P r op e rty an d requires that the solutions p ossess only mov able p oles as sigu- larities (see for example [4] for some recent reviews on the Pa inlev´ e Prop erty) . It can b e of in terest to study the P ainlev ´ e Prop ert y of p rop er d ifferen tial sequences, bu t this falls outside the scop e of the curr ent pap er. W e refer the to the pap ers by Andriop ou los et al [1, 2] in w hic h singu larity analysis is used to stud y the in tegrabilit y of a Riccati sequence. Let E i := u q x − f i ( x, u, u x , u xx , . . . , u ( q − 1) x ) = 0 , (2.15) where q = n + ( m − 1) k . W e int ro d uce the f ollo wing total deriv ativ e op erator D E i = D x     E i =0 = ∂ ∂ x + q − 1 X j =0 u j x ∂ ∂ u ( j − 1) x + f i  x, u, u x , . . . , u ( q − 1) x  ∂ ∂ u ( q − 1) x . (2.16) Prop osition 1: J s is an inte gr ating factor for the se qu enc e (2.2) if and only if the f ol- lowing c onditions ar e satisfie d: L ∗ E i [ u ] J s ( x, u, u x , . . . )     E i =0 = 0 , i = 1 , 2 , . . . , m, (2.17a ) ∂ J s ∂ u ( q − 2 r ) x + 2 r − 1 X j =1 ( − 1) j − 1 ∂ ∂ u ( q − 1) x  D j − 1 E i  ∂ f i ∂ u ( j + q − 2 r ) x J s  + ∂ ∂ u ( q − 1) x  D 2 r − 1 E i J s  = 0 , s = 1 , 2 , . . . , p, r = 1 , 2 , . . . , h q 2 i . (2.17b) Her e  q 2  is the lar gest natur al numb er less than or e qual to the numb er q 2 , i = 1 , 2 , . . . m , and p is the total numb er of inte gr ating factors, J s , valid for al l memb ers of the se quenc e, i.e. s = 1 , 2 , . . . , p. Remark: The d eriv ation of th e n ecessary and s u fficien t cond itions f or in tegrating factors of single ordinary differen tial equations of ord er n are deriv ed in the b o ok of Bluman and Anco [3 ]. Prop osition 1 is a natural extension of this r esult to prop er d ifferen tial sequences of ordin ary d ifferen tial equations. Note that condition (2.17a) ensu res that eac h J s is an Asp ects of prop er differentia l sequences 5 adjoin t symm etry for eac h equ ation in th e sequ ence and (2.17b) ensu res that this adjoint symmetry is an inte grating factor for eac h memb er of th e sequence. Example 1: W e consider the seed equation u xx + u 2 x = 0 . (2.18) with the differentia l op erator R [ u ] = D x + u x . (2.19) This giv es a pr op er differen tial sequen ce R j [ u ]  u xx + u 2 x  = 0 (2.20) with zeroth-order int egrating f actors J 1 ( x, u ) = e u , J 2 ( x, u ) = xe u . (2.21) Here R ∗ [ u ] e u = 0 , R ∗ [ u ]( xe u ) = − e u . (2.22) In the next section we sho w that the sequ en ce (2.20) is an integ rable sequence of ordinary differenti al equations and discuss its pr op erties. 3 An alternativ e description It is of in terest to constru ct an alternative se quenc e , { ˜ E 1 , ˜ E 2 , . . . , ˜ E m } , (3.1) to (2.2 ), ie { E 1 , E 2 , . . . , E m } , n amely one in whic h the order of the differentia l equations in th e s equ ence (3.1) do es not increase bu t is fi x ed by the s eed equation ˜ E 1 . F or the same seed equation, E 1 = ˜ E 1 , the t w o sequences (2.1) and (3.1) should then b e c omp atible or c ompletely c omp atible . Definition 6: Two e quations, E j and ˜ E j fr om the se quenc es (2.2) and (3.1) r esp e c tively, ar e c al le d c omp atible if the e q uations admit at le ast one c ommo n solution. The two e qu a- tions ar e c al le d c ompletely c omp atible if the gene r al solution of ˜ E j gives the gener al solution for E j . Two se quenc es of m e quations, { E 1 , E 2 , . . . , E m } and { ˜ E 1 , ˜ E 2 , . . . , ˜ E m } with the same se e d e quation E 1 = ˜ E 1 , is c al le d c omp atible if e ach e quation in the se que nc e admits at le ast one c ommon solution b etwe e n the c orr esp onding memb ers in the two se quenc es. The se quenc es ar e c al le d c ompletely c omp atible if the gener al solution of ˜ E j pr ovides the gener al solution f or E j for al l memb ers of the se qu enc e, ie for al l j = 1 , 2 , . . . , m . The se qu enc e (3.1) is terme d an alternative se quenc e to (2.2) if the two se quenc es ar e at le ast c omp atible. 6 N Euler and PGL Leac h Since the order of the equations in an alternativ e sequence (3.1) do es not increase, the equations that mak e up an alternativ e s equence should d efine integ rals for the equations in the prop er differen tial sequence (2.1) to gu arantee compatibilit y of its solutions. W e in tro du ce the follo wing Prop osition 2: Consider a pr op er differ ential se quenc e { E 1 , E 2 , . . . , E m } with r e cursion op er ator R [ k ] [ u ] . An alternative se quenc e, { ˜ E 1 , ˜ E 2 , . . . , ˜ E m } , of the form ˜ E 1 := F ( u, u x u xx , . . . , u nx ) = 0 (3.2a) ˜ E j +1 := F ( u, u x u xx , . . . , u nx ) = Q j ( x, u, u x , . . . , ω 1 , ω 2 , . . . , ω ℓ ; c 1 , c 2 , . . . , c s ) (3.2b) j = 1 , 2 , . . . , m − 1 , is c omp atible with the pr op er differ e ntial se quenc e { E 1 , E 2 , . . . , E m } with E 1 = ˜ E 1 if R [ k ] Q 1 = 0 (3.3a) R [ k ] Q i = Q i − 1 , i = 2 , 3 , . . . , m. (3.3b) Her e ω 1 , ω 2 . . . , ω ℓ ar e nonlo c al c o or dinates define d by dω 1 dx = g 1 ( u ) , (3.4a) dω 2 dx = g 2 ( ω 1 ) , dω 3 dx = g 3 ( ω 2 ) , . . . , dω ℓ dx = g ℓ ( ω ℓ − 1 ) (3.4b) for some differ entiable f unctions g k . Belo w we discu s s several examples of pr op er differentia l sequences which are compatible or completely compatible with their alternativ e sequen ces. Th e examples are sufficient ly simple to demonstrate th e metho d of construction and to in vesti gate the prop erties of the sequences. A classification is n ot at all attempted here, but will b e addressed in future w orks. F or our examples we consider the follo wing four seed equations which are part of the list of second-order linearisable evo lution equ ations in (1 + 1) d im en sions rep orted in [6]: u xx + u 2 x = 0 u xx + h ( u ) u 2 x = 0 u xx + λu x − h ′ ( u ) h ( u ) u 2 x + h ( u ) = 0 u xx + uu x = 0 for arbitrary d ifferentiable functions h , where pr ime denotes the d eriv ativ e with resp ect to u . Asp ects of prop er differentia l sequences 7 Example 2: Firstly w e consid er the prop er differenti al sequence, (2.20), already intro- duced in Example 1, where R [ u ] = D x + u x . The prop er differentia l sequence is E 1 := F ( u, u x u xx ) = u xx + u 2 x = 0 (3.5a) E 2 := R [ u ] F ( u, u x u xx ) = u 3 x + 3 u x u xx + u 3 x = 0 (3.5b) E 3 := R 2 [ u ] F ( u, u x u xx ) = u 4 x + 4 u x u 3 x + 3 u 2 xx + 6 u 2 x u xx + u 4 x = 0 (3.5c) . . . E m := R m − 1 [ u ] F ( u, u x u xx ) = u ( m +1) x + · · · = 0 . ( 3.5d) W e apply Prop osition 2 to calculate the alternativ e sequence with seed equation (3.5a ). The second mem b er in th e alternativ e sequence is u xx + u 2 x = Q 1 ( x, u, u x , . . . ) (3.6) under the condition R [ u ] Q 1 ( x, u, u x , . . . ) = 0 . (3.7) Condition (3.7) is of the form D x ( Q 1 ) = − u x Q 1 (3.8) with general solution Q 1 ( u, c 1 ) = c 1 e − u , (3.9) where c 1 is an arb itrary constant of inte gration. Thus th e second memb er in the alternativ e sequence is u xx + u 2 x = c 1 e − u . (3.10) The third mem b er, (3.5c), b ecomes u xx + u 2 x = Q 2 ( x, u, u x , . . . ) (3.11) in the alternativ e sequen ce under the condition R [ u ] Q 2 ( x, u ; c 1 , c 2 ) = Q 1 ( u ; c 1 ) , (3.12) whic h admits the general solution Q 2 ( x, u ; c 1 , c 2 ) = c 1 xe − u + c 2 e − u (3.13) 8 N Euler and PGL Leac h with c 2 another constant of in tegration. Thus the third member in th e alternativ e sequence is u xx + u 2 x = e − u ( c 1 x + c 2 ) (3.14) whic h can b e presen ted in the form u xx + u 2 x = e − u D − 1 x c 1 . (3.15) The next mem b er in th e alternativ e sequence is u xx + u 2 x = e − u  1 2 c 1 x 2 + c 2 x + c 3  ≡ e − u D − 2 x c 1 . (3.16) The functions Q k in E k +1 are Q k = e − u q ( x ) with q ( k ) ( x ) = 0 ⇔ q ( x ) = k X j =1 c j ( k − j )! x k − j . (3.17) W e th us conclud e th at the alternativ e sequ ence to the differen tial sequ en ce (3.5a) - (3.5d) has the form ˜ E 1 := u xx + u 2 x = 0 (3.18a ) ˜ E j := u xx + u 2 x = e − u D − ( j − 2) x c 1 , j = 2 , 3 , . . . , m, (3.18b) where D − q x are q ∈ N comp ositions of the int egral op erator D − 1 x and D 0 x := 1. In explicit form the alternativ e sequ ence is ˜ E 1 := u xx + u 2 x = 0 (3.19a ) ˜ E 2 := u xx + u 2 x = Q 1 with Q 1 = e − u c 1 (3.19b) ˜ E 3 := u xx + u 2 x = Q 2 with Q 2 = e − u ( c 1 x + c 2 ) (3.19c ) ˜ E 4 := u xx + u 2 x = Q 3 , with Q 3 = e − u  1 2 c 1 x 2 + c 2 x + c 3  (3.19d) . . . (3.19e ) ˜ E m := u xx + u 2 x = Q m − 1 with Q m − 1 = e − u   m − 1 X j =1 c j ( m − j − 1)! x m − j − 1   . (3.19f ) It is easy to establish that th e prop er d ifferential sequence (3.5a) – (3.5d) is an inte grable differen tial sequence since eac h memb er of the sequence is linearisable by the c hange f or v ariables w ( X ) = u x e u , X = x. (3.20) Asp ects of prop er differentia l sequences 9 Moreo v er the alternative sequence (3.19a) – (3.19f) is linearisable by the c hange of v ariables w ( X ) = e u , X = x. (3.21) T o establish the compatibilit y or complete compatibilit y of the t wo sequences (3.5a) – (3.5d) and (3.19a) – (3.19f) we take a closer lo ok at the corresp ondin g member s . • Compare the memb ers E 2 and ˜ E 2 : A first in tegral for E 2 is giv en by ˜ E 2 , namely c 1 = e u  u xx + u 2 x  . (3.22) Therefore the general solution of ˜ E 2 giv es the general solution of E 2 with c 1 as one of the constants of int egration for E 2 . Hence the t wo equations, E 2 and ˜ E 2 , are completely compatible. • Compare the memb ers E 3 and ˜ E 3 : A second in tegral f or E 3 is giv en by ˜ E 3 , namely c 1 x + c 2 = e u  u xx + u 2 x  . (3 .23) Therefore the general solution of ˜ E 3 giv es the general solution of E 3 (with c 1 and c 2 as t wo of the constants of in tegration for E 3 ) and the t wo equ ations E 3 and ˜ E 3 are completely compatible. A similar arguman t follo ws for all equations in the prop er differen tial s equ ence (3.5a) – (3.5d). We c onclude that the two se quenc es (3.5a ) – (3.5d) and (3.19a) – (3.19f) ar e c ompletely c omp atible. Another in teresting pr op ert y of the sequences (3.5a) – (3.5d) and (3.19a ) – (3.19f) is that the symmetry charac teristics, η j , of the solution symmetries, Γ s j = η j ( x, u ) ∂ u , (3.24) for the equations in (3.5a) – (3.5d) are give n b y the fu nctions Q 1 , Q 2 , . . . in the alternativ e sequence (3.19a) – (3.19f). In p articular The symmetry char acteristic, η j , for the solution symmetry of E j in (3.5a) – (3.5d) is given by Q j +1 of the e quation ˜ E j +2 in (3.19a) – (3.19f) for al l j = 1 , 2 , . . . , m . F or example E 1 := u xx + u 2 x = 0 admits the solution symmetry Γ s 1 = Q 2 ∂ u , (3.25) where Q 2 = e − u ( c 1 x + c 2 ) corr esp onds to the right -hand exp ression in ˜ E 3 . Note th at the complete set of all p oint symmetries for the p rop er differen tial sequence (3.5a) – (3.5d) is the follo wing: 10 N Euler and PGL Leac h F or E 1 the complete set of Lie p oin t symmetries are  e − u ∂ u , x e − u ∂ u , ∂ u , ∂ x , x∂ + 1 2 ∂ u , x 2 ∂ x + x∂ u , e u ∂ x , x e u ∂ x + e u ∂ u  (3.26) F or E k , k = 2 , 3 , . . . , m , the complete set of Lie p oint sym metries are  q ( x )e − u ∂ u , ∂ u , ∂ x , x∂ x + 1 2 ( n − 1) ∂ u , x 2 ∂ x + ( n − 1) x∂ u  , (3.27) where n is the order of the differential equation, E k , and q ( n ) ( x ) = 0 . (3.28) The Lie p oin t s y m metry prop erties of the alternativ e sequence (3.19 a ) – (3.19 f ) are dis- cussed in the App end ix. Example 3: Consider the seed equation u xx + h ( u ) u 2 x = 0 (3.29) with recursion op erator R [ u ] = D x + h ( u ) u x . (3.30) This defines the prop er d ifferential sequence of the form E 1 := u xx + h ( u ) u 2 x = 0 (3.31a ) E j +1 := R j [ u ]  u xx + h ( u ) u 2 x  = 0 , j = 1 , 2 , . . . , m − 1 . (3.31b) W e app ly Prop osition 2 and calculate the fun ctions Q j in th e same m anner as in Example 2. This leads to the follo wing alternativ e sequence ˜ E 1 := u xx + u 2 x = 0 (3.32a ) ˜ E j +1 : = u xx + u 2 x = exp  − Z h ( u )u .  D − ( j − 1) x c 1 , j = 1 , 2 , . . . , m − 1 . (3.32b) It is easy to sho w that the sequence (3.31 a ) – (3.31b) and its alternativ e sequence (3.32 a ) – (3.32b) are completely compatible and that (3.31a) – (3.31b) is an int egrable s equence. The linearisation and Lie p oin t symmetries of (3.32a ) – (3.32b) are discussed in the App end ix. Example 4: Consider the seed equation u xx + λu x − h ′ ( u ) h ( u ) u 2 x + h ( u ) = 0 (3.33) with the recursion op er ator R [ u ] = D x − h ′ ( u ) h ( u ) u x . (3.34) Asp ects of prop er differentia l sequences 11 This giv es the p rop er differen tial s equ ence E 1 := u xx + λu x − h ′ ( u ) h ( u ) u 2 x + h ( u ) = 0 (3.35a ) E j +1 := R j [ u ]  u xx + λu x − h ′ ( u ) h ( u ) u 2 x + h ( u )  = 0 , j = 1 , 2 , . . . , m − 1 , (3.35b) with its alternativ e sequence ˜ E 1 := u xx + λu x − h ′ ( u ) h ( u ) u 2 x + h ( u ) = 0 (3.36a ) ˜ E j +1 := u xx + λu x − h ′ ( u ) h ( u ) u 2 x + h ( u ) = h ( u ) D ( j − 1) x c 1 , j = 1 , 2 , . . . , m − 1 , (3. 36b) where h is an arb itrary differentia ble f unction. Ju s t lik e th e sequences in E xample 2 and Example 3 the sequences (3.35a) – (3.35b) and (3.36 a ) – (3.36b) are completely compatible and (3.35a) – (3.35b) is an in tegrable sequence. Details are giv en in the Ap p endix. Example 5: T o the Burgers equation u xx + uu x = u t (3.37) one can (with standard symm etry redu ction follo wing the t -translation in v ariance) asso- ciate u xx + uu x = 0 (3.38) whic h shares the same integrodifferential recursion op erator, R [ u ] = D x + 1 2 u + 1 2 u x D − 1 x ◦ 1 . (3.39) The prop er differentia l sequence, which we n ame the Bu r gers Se quenc e , is E 1 := u xx + uu x = 0 (3.40a ) E j +1 := R j [ u ] ( u xx + uu x ) = 0 , j = 1 , 2 , . . . , m. (3.40b) W e no w construct an alternativ e Burger’s Sequance follo wing Prop osition 2. The solution of R [ u ] Q 1 = 0 is Q 1 =  − 2 A exp  − 1 2 Z u x .  + 2 B exp  − 1 2 Z u x .  Z exp  1 2 Z u x .  x .  x , (3.41) where A and B are constants of in tegration. Equation (3.41) is an in tegro differenti al equation. It can b e rend ered as an ordinary different ial equation b y defining w = Z exp  1 2 Z u x .  x . (3.42) 12 N Euler and PGL Leac h so that u xx + uu x =  − 2 A exp  − 1 2 Z u x .  + 2 B exp  − 1 2 Z u x .  Z exp  1 2 Z u x .  x .  x (3.43) b ecomes w 4 x w x − w xx w 3 x w x 2 = Aw xx w x 2 + B  1 − ww xx w x 2  . (3.44) In a similar fashion the equation R [ u ] Q 2 = Q 1 has the solution Q 2 =  2 C exp  − 1 2 Z u x .  Z exp  1 2 Z u x .  x . − 2 Ax exp  − 1 2 Z u x .  +2 B exp  − 1 2 Z u x .  Z  Z exp  1 2 Z u x .  x .  x .  x , (3.45) where C is also a constan t of integ ration, and the integ ro d ifferen tial equation is u xx + uu x =  2 C exp  − 1 2 Z u x .  Z exp  1 2 Z u x .  x . − 2 Ax exp  − 1 2 Z u x .  +2 B exp  − 1 2 Z u x .  Z  Z exp  1 2 Z u x .  x .  x .  x . (3.46) The corresp ondin g higher-order ordin ary differen tial equation is w 5 x w xx − w 3 x w 4 x w xx 2 = C  1 − w x w 3 x w xx 2  + A  xw 3 x w xx 2 − 1 w xx  + B  w x w xx − ww 3 x w xx 2  , (3.47) where no w w = Z  Z exp  1 2 Z u x .  x .  x . (3.48) or equiv alent ly u = 2 w 3 x w xx . (3.49) Eviden tly one could con tinue in like fashion. It contrast to Examples 2, 3 and 4 in whic h the recur sion op erator did not con tain an inv erse deriv ativ e to obtain an ordin ary differen tial equation one m us t redefine the d ep endent v ariable as in (3.42) and (3.47) (and the ob vious extension f or higher elemen ts of the sequence). W e note that the terms on the left sides of (3.44) and (3.46) h a v e the same form apart fr om the increase in the ord er of eac h d er iv ativ e. Consequent ly , if w e wish to hav e a d ifferential sequence b ased u p on the differen tial equ ation (3.38) and its recursion op erator, (3.39), of m elemen ts in terms of differen tial equations, all differential equations b elonging to the sequence m ust b e written in terms of a d ifferen tial equ ation of order m + 2. Th e alternativ e is an integrodifferential equation of increasing nonlo calit y . Asp ects of prop er differentia l sequences 13 W e th us conclude that the fir s t three terms in the alternativ e sequence tak e the follo wing forms ˜ E 1 ( w ) := w 5 x w xx − w 3 x w 4 x w xx 2 = 0 (3.50a ) ⇔  w 4 x w xx  x = 0 (3.5 0b) ⇔ w 4 x = k 1 w 2 x (3.50c ) ˜ E 2 ( w ) := w 5 x w xx − w 3 x w 4 x w xx 2 = Aw 3 x w xx 2 + B  1 − w x w 3 x w xx 2  (3.50d) ⇔  w 4 x w xx  x = −  A w xx  x + B  w x w xx  x (3.50e ) ⇔ w 4 x = a 1 w xx + B w x − A (3.50f ) ˜ E 3 ( w ) := w 5 x w xx − w 3 x w 4 x w xx 2 = C  1 − w x w 3 x w xx 2  + A  xw 3 x w xx 2 − 1 w xx  + B  w x w xx − ww 3 x w xx 2  (3.50g) ⇔  w 4 x w xx  x = C  w x w xx  x − A  x w xx  x + B  w w xx  x (3.50h) ⇔ w 4 x = a 2 w xx + C w x + B w − Ax. (3.50i) Ho w ev er, suc h a differen tial sequence should n ot b e confused with th e normal type of differen tial sequence sin ce, as the v alue of m increases, b oth the left side of th e equ ations and the recursion op erator must b e r edefined. In order to establish compatibilit y of the fi rst three members of the t wo s equ ences, (3.40a) – (3.40b) and (3.50a) – (3.50g), n eed to b e written in the same v ariable w , that is, w e need to apply the transform ation (3.49) and transform the fir st thr ee mem b ers of the differentrial sequen ce (3.40a) – (3.40b) in order to write the equations in terms of th e v ariable w . W e obtain the follo wing: E 1 ( w ) := w 5 x w xx − w 3 x w 4 x w xx 2 = 0 ⇔  w 4 x w xx  x = 0 ⇔ w 3 x = k 1 w x + k 11 (3.51a ) E 2 ( w ) := w 6 x w xx − w 3 x w 5 x w xx 2 = 0 ⇔  w 5 x w xx  x = 0 ⇔ w 4 x = k 2 w x + k 21 (3.51b) E 3 ( w ) := w 7 x w xx − w 3 x w 6 x w xx 2 = 0 ⇔  w 6 x w xx  x = 0 ⇔ w 5 x = k 3 w x + k 31 (3.51c ) 14 N Euler and PGL Leac h All k s are constan ts of in tegration. Comparing (3.50f) with (3.51b) and (3.50i) with (3.51c) it is clear that these tw o sequences are compatible but not completely compatible. In particular the expr ession w 4 x = B w x − A, (3.52) whic h is (3.50f) w ith a 1 = 0, is a second integ ral for E 2 , namely equ ation (3.51b ). Ho w- ev er, the same second integral for (3.51c) follo ws from (3.50i), wh ere the tw o additional parameters a 2 and C ha ve to b e zero for compatibilit y . Th erefore the higher memb er s of the alternativ e sequence do not provide additional parameters for the in tegration of the differentia l sequence (3.40a) – (3.40b) and w e conclud e that the t wo sequences only share sp ecial s olutions. Th e prop er differentia l sequen ce, (3.40a) – (3.40b), is, how ever, in tegrable sin ce ev ery mem b er of the sequence can b e lin earised. The s ame is of course true for the alternate sequence (3.50a), (3.50d ) and (3.50g). Unlik e the Examples 2,3 and 4 there is no preserv ation of Lie p oint symmetries in th e alternate sequence. In the cases of (3.50 a ), (3.50d) and (3.50g ) we obtain Γ 1 = ∂ x , Γ 2 = ∂ w , Γ 3 = x∂ x , Γ 4 = x∂ w , Γ 5 = w∂ w (3.53a ) Γ 1 = ∂ x , Γ 2 = ∂ w , Γ 6 = ( Ax − B w ) ∂ w (3.53b) Γ 7 = B ∂ x + A∂ w , Γ 8 = ( AB x − B 2 w − AC ) ∂ w , (3.53c ) resp ectiv ely . Ab o ve w e only lo oked at the first thr ee mem b ers of th e sequence. W e end this example with the follo wing statemen ts ab out the fu ll sequence (3.40a) – (3.40b). The n th element of the alternative Bur gers Differ ential Se quenc e written in the inte gr o- differ ential form u xx + uu x = exp  − 1 2 Z u x .  n − 1 X i =1 B i D − i x exp  1 2 Z u x .  ! (3.54) is line arise d to W ( n +1) x = B n − 2 + B n − 1 W , (3.55) wher e W = D − ( n − 1) x exp  1 2 R u x .  . Remark: This pro cedure for the linearisation of (3.5 4 ) is a n atur al generalisation of the Cole-Hopf transformation whic h can also b e d eriv ed via the x -generalised ho d ograph transformation for ev olution equations [6]. The n th element of the Bur gers D iffer ential Se quenc e (3.40a) – (3.40b), ie R n − 1 [ u ] ( u xx + uu x ) = 0 , (3.56) Asp ects of prop er differentia l sequences 15 wher e R [ u ] = D x + 1 2 u + 1 2 u x D − 1 x , is line arise d to v ( n +1) = Ω n +1 n v , (3 .57) wher e u = 2 v x /v and Ω ar e arbitr ary c onstants . Remark: Here we mak e u s e of the relationship b et ween th e elements of the Burgers Differen tial S equence (3.40 a ) – (3.40b) and th e sequence in Example 2. 4 Discussion Although n o general Th eorem has b een p ro vided to in ve stigate th e integrabilit y of a prop er differen tial s equ ence, the pap er give s defin itions of these ob jects and s u ggests some ro ots for the inv estigatio ns illustrated by sev eral examples. In particular, the foregoing examples strongly suggest that the alternativ e sequence an d Prop osition 2, wh ich addresses the compatibilit y/complete compatibilit y of sequences, provides a usefu l ro ot to the integ rals of a prop er differen tial sequ ence. In this sense, the cur ren t p ap er should b een appreciated as a s tarting p oint for the in vestig ations of prop er differential sequen ces r ather than a concluding pap er on this sub j ect. W e d elib erately concentrate on simple examples, n amely pr op er differentia l sequ en ces for w h ic h the general solution can b e derived via linearisations of the equations in th e sequences, in order to gain an un derstanding of the prop erties. W e recall th at equations of the sequence in the last example, the Burger’ S equence of Example 5, are linearisable by a Cole-Ho pf t yp e transformation, whereas all other sequen ces are examples of equ ations linearisable b y p oint transformations. The p oin t-linearisable sequences ha ve b eautiful prop ereties in view of the Lie symmetry structure of the sequence and the usefulness of the alternativ e sequ ence f or the construction of the complete s et of first in tegrals of the prop er d ifferen tial sequences. F or the Burgers’ Sequence w e hav e to introd uce nonlo cal v ariables f or the general solution of the op erator equation R [ u ] Q 1 = 0 (4.1) whic h then results in a h igher order alternativ e sequen ce in terms of local v ariables. This example clearly su ggests that fu rther inv estiga tions are necessary in order to h andle suc h cases, n amely when n onlo cal v ariables come into pla y . W e susp ect that non lo cal symme- tries and nonlo cal integ rating factors will pla y an imp ortant role for this inv estiga tion. A App endix The sequences discuss ed in Examples 2, 3 and 4 are integrable sequences and their alter- nativ e sequen ces p reserv e the maximal Lie symmetry algebra of p oint sym metries as given b y th e corresp ond ing seed equ ations. A d etailed Lie p oin t sy m metry analysis is the aim of this App en dix. W e determine the Lie p oin t symmetries of the general equations in the systems u xx + u 2 x = 0 (A.1a) u xx + u 2 x = e − u D − ( j − 1) x c 1 , j = 1 , m, (A .1b) 16 N Euler and PGL Leac h u xx + u 2 x = 0 (A.2a) u xx + u 2 x = exp  − Z h ( u ) du  D − ( j − 1) x c 1 , j = 1 , m, (A.2b) and u xx + λu x − h ′ ( u ) h ( u ) u 2 x + h ( u ) = 0 (A.3a) u xx + λu x − h ′ ( u ) h ( u ) u 2 x + h ( u ) = h ( u ) D ( j − 1) x c 1 . (A.3b) It is eviden t that (A.1a ) – (A.1b) is subsu med int o (A.2a) – (A.2b). Equation (A.2b) b elongs to the class of equ ations u xx + u 2 x = exp  − Z h ( u )u .  f ( x ) (A.4) and (A.3b) to the class u xx + λu x − h ′ ( u ) h ( u ) u 2 x + h ( u ) = h ( u ) f ( x ) , (A.5) where f ( x ) is at least C 1 . This is more than adequately co v ers the p olynomials of the original equations. If in (A.4) we mak e th e c hange of d ep endent v ariable, w = Z exp  Z h ( u )u .  u . , (A.6) (A.4) b ecomes w xx = f ( x ) . (A.7) In lik e mann er the change of d ep endent v ariable, w = Z u . h ( u ) , (A.8) con v erts (A.5) to w xx + λw x = f ( x ) − 1 . (A.9) Since the transformation in the dep endent v ariable is a p oint transformation in b oth cases, the Lie p oin t symmetries of (A.7) and (A.9), wh ich are reasonably easy to calculate, lead directly to the Lie p oin t symmetries of (A.4) and (A.5). Since (A.7) and (A.9) are linear Asp ects of prop er differentia l sequences 17 second-order ordinary differentia l equations, they eac h p ossess eight Lie p oin t symmetries with the algebra sl (3 , R ). The co efficien t fu nctions for a Lie p oint symmetry of (A.7), Γ = ξ ( x, w ) ∂ x + η ( x, w ) ∂ w , (A.10) ha v e the forms ξ = a ( x ) + b ( x ) w (A.11a) η = b x ( x ) w 2 + c ( x ) w + d ( x ) , (A.11b) in whic h the f unctions of x satisfy b xx = 0 (A.12a) c xx = bf x (A.12b) a xx = b + 2 c x (A.12c) d xx = af x + (2 a x − c ) f . (A.12d) W e solv e the equ ations in (A.12a) – (A.12d) in turn to obtain b = B 0 + B 1 x c = C 0 + C 1 x + Z Z ( bf x ) dx dx a = A 0 + A 1 x + Z Z ( b + 2 c x ) dx dx d = D 0 + D 1 x + Z Z [ af x + (2 a x − c ) f ] dx dx. W e did n ot giv e the explicit formulæ for the integ rals for general f ( x ) as they are not informativ e. The imp ortant thing to n ote is th at th ere are eigh t Lie p oin t symmetries. The Lie p oin t symmetries of (A.9) ha v e the same dep enden ce up on w as giv en in (A.11a). No w th e equations to b e satisfied by the fu nctions of x are b xx − λb x = 0 c xx + λc x = bf x + 2 λb ( f − 1) a xx − λa x = 2 c x − 3 b ( f − 1) d xx + λd x = af x + (2 a x − c )( f − 1) whic h can easily b e solve d and again provides eigh t arbitrary constan ts and hence eigh t Lie p oint symmetries. The exp licit Lie p oin t symm etries of th e original equations, (A.2b) and (A.3b), require b oth the in version of th e transformations (A.6) and (A.8) an d the s p ecification of the function f ( x ). T o main tain a mo dicum of simplicit y we tak e (A.1b), for wh ic h u = log w , 18 N Euler and PGL Leac h and f ( x ) = c 3 + c 2 x + 1 2 c 1 x 2 , ie we list th e Lie p oint symmetries of the equation whic h is completely compatible with the fourth element of th e differen tial sequence (2.20). Γ 1 = e − u ∂ u Γ 2 = x e − u ∂ u Γ 3 = 6 ∂ x +  3 c 2 x 2 + c 1 x 3  e − u ∂ u Γ 4 =  24 −  12 c 3 x 2 + 4 c 2 x 3 + c 1 x 4  e − u  ∂ u Γ 5 = 6 x∂ x +  6 c 3 x 2 + 3 c 2 x 3 + c 1 x 4  e − u ∂ u Γ 6 = 24 x 2 ∂ x +  24 x +  12 c 3 x 3 + 8 c 2 x 4 + 3 c 1 x 5  e − u  ∂ u Γ 7 =  144e u − 216 c 3 x 2 − 24 c 2 x 3 − 6 c 1 x 4  ∂ x +  72 c 2 x 2 + 24 c 1 x 3 −  144 c 2 3 x 3 + 108 c 3 c 2 x 4 + 12 c 2 2 x 5 + 36 c 3 c 1 x 5 + 7 c 2 c 1 x 6 + c 2 1 x 7  e − u  ∂ u Γ 8 =  576e u x − 288 c 3 x 3 − 96 c 2 x 4 − 24 c 1 x 5  ∂ x +  576 e u + 96 c 2 x 3 +48 c 1 2 x 4 −  144 c 2 3 x 4 + 144 c 3 c 2 x 5 + 32 c 2 2 x 6 + 48 c 3 c 1 x 6 + 20 c 2 c 1 x 7 +3 c 2 1 x 8  e − u  ∂ u . Ac kno wledgemen ts The work rep orted in this pap er is part of a pr o ject fun ded un der the Swedish-South African Agreement, Gran t Numb er 60935, sp onsored by sida/VR and the National Re- searc h F oundation of the Repub lic of South Africa. P GLL thanks th e Department of Mathematics, Lu le ˚ a Unive rs ity of T ec hnology , for the p ro vision of facilitie s wh ile the bulk of this work w as undertake n. P GLL thanks th e Univ ersity of Kwa Zu lu-Natal for its con- tin ued sup p ort. References [1] And riop oulos K & Leac h PGL (2006) An in terpretation of th e p resence of b oth p osi- tiv e and negativ e nongeneric resonances in the singularit y analysis Physics L etters A 359 199-2 03. [2] And riop oulos K, Leac h PGL & Mahara j A (2007) On differen tial sequences (p reprint: Sc ho ol of Mathematica l S ciences, Univ ersit y of KwaZulu-Natal, Priv ate Bag X54001 Durban 4000, Republic of South Africa) arXiv:0704.32 43 [3] Bluman GW and Anco SC Symmetriy and Inte gr ation Metho ds for Differ ential Eq ua- tions Springer, New Y ork, 2002 Asp ects of prop er differentia l sequences 19 [4] Conte R (ed) The P ainlev ´ e Pr op erty: One Century L ater Spr inger, New Y ork, 1999. [5] Eu ler M, Euler N & Leac h PGL (2007) The Riccati and Ermak o v-Pinney hierarchies Journal of N online ar Mathematic al Physics 14 290-310 [6] Eu ler M, Euler N & Pe tersson N (2003) L inearizable hierarc hies of ev olution equations in (1 + 1) d imensions Studies in Applie d Mathematics 111 315-337