B"acklund Transformations for First and Second Painleve Hierarchies

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 5 (2009), 024, 11 pages B¨ ac klund T ransformations for First and Second P ainlev ´ e Hie rarc hies Ayman Hashem SAKKA Dep artment of Mathematics, Islamic University of Gaza, P.O. Box 108 , R imal, Gaza, Palestine E-mail: asakka@iugaza.e du.ps Received Nov em ber 25, 2 0 08, in f inal form F ebruar y 2 4, 2 009; Published online March 02, 2009 doi:10.38 42/SIGMA.20 09.024 Abstract. W e give B¨ acklund transformations for f irst and seco nd Painlev´ e hier a rchies. These B¨ ac klund transformations are generalization of k nown B¨ acklund t ransfor mations of the f irst and second Painlev ´ e equations and they relate the considered hierar chies to new hierarchies of Painlev´ e-type equations . Key wor ds: Painlev´ e hierarchies; B¨ acklund transformations 2000 Mathematics S ubje ct Classific ation: 3 4M55; 33E17 1 In tro duction One ce n tury ago Pai nlev ´ e and Gam b ier ha v e disco v ered the six P ainlev ´ e equations, PI–PVI. These equations are the only second-order ordinary dif feren tial equations whose general solutions can not be expressed in te rms of elemen tary and cla ssical sp ecial functions; thus they d ef ine new transcendent al fu nctions. P ainlev ´ e transcendental functions app ear in m an y areas of moder n mathematics and physics and they paly the same role in nonlinear pr oblems as the cl assical sp ecial functions pla y in linear problems. In rec en t y ears there is a considerable in terest in studying hierarc hies of P ainlev ´ e equations. This in terest is du e to the connection b et w een these hierarc hies of Pa inlev ´ e equatio ns and co m- pletely in tegrable partial d if ferent ial equations. A P ainlev´ e hierarc h y is an inf inite s equence of nonlinear ordinary dif ferential equations whose f irst member is a P ainlev ´ e equation. Airault [1] w as th e f irs t to deriv e a Pa inlev ´ e hierarc h y , namely a second P ainlev ´ e hierarc h y , as the similari- t y reduction of the mo d if ied Kortewe g–de V r ies (mKdV) hierarc h y . A f irst Painlev ´ e h ierarc h y w as giv en by Kudryasho v [2]. Later on several hierarc hies of Painlev ´ e equations w ere introd u - ced [3 , 4, 5, 6, 7, 8, 9, 10, 11]. As it is well kno wn, Pa inlev ´ e equatio ns possess B¨ ac klun d t ransformations; that is, mappings b et w een solutions of the same Painlev ´ e equation or b et w een solutions of a particular Painlev ´ e equation and other second-order P ainlev´ e- t yp e equations. V arious metho d s to derive these B¨ ac klund tran s formations can b e found for example in [12, 13, 14, 15]. B ¨ ac klund transf ormations are now adays considered t o b e one of t he main prop erties of int egrable n onlinear ordinary dif ferentia l equations, and there is m uc h int erest in their deriv ation. In the presen t a rticle, we generali ze kno wn B¨ ac klund transformations of the f ir st and se- cond P ainlev ´ e equations to the f irst and second P ainlev ´ e hierarc hies giv en in [6, 11]. W e giv e a B¨ acklund tran s formation b et w een the considered f irst Painlev ´ e hierarc hy and a n ew hierarch y of P ainlev ´ e-t yp e equations. In addition, we giv e t w o new hierarc hies of P ainlev´ e-t yp e equations related, via B¨ ac klund transformations, to the considered s econd Painlev ´ e hierarc h y . Then w e deriv e auto-B¨ ac klu n d transf orm ations for this second Pa inlev ´ e hierarc h y . B¨ ac klund transforma- tions o f th e sec ond P ainlev´ e hierarc h y ha ve been s tudied in [6, 16]. 2 A.H. S akk a 2 B¨ ac klund transformations for PI hierarc h y In this section, w e w ill derive a B ¨ ac klund transf ormation for the f ir st Painlev ´ e hierarc h y (PI hierarc h y) [6 ] n +1 X j =2 γ j L j [ u ] = γ x, (2.1) where t he oper ator L j [ u ] satisf ies the Lenard recursion rela tion D x L j +1 [ u ] =  D 3 x − 4 uD x − 2 u x  L j [ u ] , L 1 [ u ] = u. (2.2) The sp ecial case γ j = 0, 2 ≤ j ≤ n , of this h ierarc h y is a similarit y redu ction of the Sc h w arz– Kortew eg–de V ries hierarch y [2, 4]. Moreo v er its memb er s ma y def ine new transcenden tal func- tions. The P I h ierarc h y (2.1 ) can b e w ritten in the follo wing form [11] R n I u + n X j =2 κ j R n − j I u = x, (2.3) where R I is the recursion o p erator R I = D 2 x − 8 u + 4 D − 1 x u x . In [1 7, 18] , it is sho wn that t he B¨ ac klun d transformation u = − y x , y = 1 2  u 2 x − 4 u 3 − 2 xu  , (2.4) def ines a one-to-one corr esp ondence b etw een the f irst P ainlev ´ e equation u xx = 6 u 2 + x. (2.5) and t he S D-I.e equation of C osgro v e and Scouf is [17] y 2 xx = − 4 y 3 x − 2( xy x − y ) . (2.6) W e will sho w that there is a generalization of this B¨ ac klund transformation to all members of the PI hierarch y (2.3). Let y = − xu + D − 1 x u x " R n I u + n X j =2 κ j R n − j I u # . (2.7) Dif feren tiating (2.7) and using (2.3), w e f ind u = − y x . (2.8) Substituting u = − y x in to (2.7), w e obtain the follo wing hierarc hy of dif ferential equation for y D − 1 x y xx " S n I y x + n X j =2 κ j S n − j I y x # + ( xy x − y ) = 0 , (2.9) where S I is the recursion o p erator S I = D 2 x + 8 y x − 4 D − 1 x y xx . B¨ ac klund T ransformations f or First and Second Painlev ´ e Hierarchies 3 The f irst m em b er of the hierarc hy (2.9) is the SD-I.e equation (2.6). Th us we shall call this hierarc h y SD-I.e hierarc h y . Therefor we h a v e derive d the B¨ ac klund tr ansformation (2.7)–(2.8) b et w een solutions u of th e f irst P ainlev´ e hierarc h y (2. 3) and solutions y of the SD-I.e h ierarc h y (2.9 ). When n = 1, the B¨ ac klund transformation (2.7)–(2.8) giv es the B¨ ackl und trans f ormation (2.4) b et w een the f irst Pa inlev ´ e equation (2.5) and the SD-I.e equation (2.6). Next w e w ill consider the ca ses n = 2 and n = 3. Example 1 ( n = 2) . The second m em b er of the PI h ierarc h y (2.3 ) is the fourth -ord er equation u xxxx = 20 uu xx + 10 u 2 x − 40 u 3 − κ 2 u + x. (2.10) In t his ca se, the B¨ ac klund transformation (2.7) reads y = 1 2  2 u x u xxx − u 2 xx − 20 uu 2 x + 20 u 4 + κ 2 u 2 − 2 xu  . (2.11) Equations (2.11) and (2.8) giv e one-to-one corresp ondence b et w een (2.10) and th e follo wing equation 2 y xx y xxxx − y 2 xxx + 20 y x y 2 xx + 20 y 4 x + κ 2 y 2 x + 2( xy x − y ) = 0 . (2.12) Equation (2 .12) a nd the B¨ ac klund transformation (2.8 ) and (2.11) were giv en b efore [19]. Example 2 ( n = 3) . Th e third mem b er of the PI hierarc h y (2 .3) reads u xxxxxx = 28 uu xxxx + 56 u x u xxx + 42 u 2 xx − 280 u 2 u xx − 280 uu 2 x + 280 u 4 − κ 2  u xx − 6 u 2  − κ 3 u + x. (2.13) In t his ca se, the B¨ ac klund transformation (2.7) has the form y = 1 2  2 u x u xxxxx − 2 u xx u xxxx + u 2 xxx − 56 uu x u xxx + 28 uu 2 xx − 56 u 2 x u xx + 280 u 2 u 2 x − 112 u 5 + κ 2  u 2 x − 4 u 3  + κ 3 u 2 − 2 xu  . (2.14) Equations (2.8 ) a nd (2.14 ) give one-to-one corresp ondence b et w een solutio ns u of ( 2.13) a nd solutions y of the follo wing equation 2 y xx y xxxxxx − 2 y xxx y xxxxx + y 2 xxxx + 56 y x y xx y xxxx − 28 y x y 2 xxx + 56 y 2 xx y xxx + 280 y 2 x y 2 xxx + 112 y 5 x + κ 2  y 2 xx + 4 y 3 x  + κ 3 y 2 x + 2( xy x − y ) = 0 . (2.15) Equation (2 .15) is a new sixth-order P ainlev ´ e-t yp e equation. 3 B¨ ac klund transformations for second P ainlev ´ e hierarc h y In th e p resen t section, we will study B¨ ac klund transformations of the second P ainlev ´ e hierarc h y (PI I hierarc hy) [6] ( D x − 2 u ) n X j =1 γ j L j  u x + u 2  + 2 γ xu − γ − 4 δ = 0 , where the oper ator L j [ u ] is def in ed by (2. 2). The sp ecial case γ j = 0 , 1 ≤ j ≤ n − 1 , of this h ierarc h y is a simila rit y reduction of the mo dif ied Korteweg –de V ries hierarc hy [2 , 4 ]. The mem b ers of this hierarc h y ma y d ef ine new transcendental functions. This hierarc hy can b e wr itten in the follo wing alternativ e form [11] R n II u + n − 1 X j =1 κ j R j II u − ( xu + α ) = 0 , (3.1) where R II is the recursion o p erator R II = D 2 x − 4 u 2 + 4 uD − 1 x u x . 4 A.H. S akk a 3.1 A hierarc h y of SD-I.d equation As a f irst B¨ ac klund transformation for the PI I hierarch y (3.1), w e will g eneralize the B¨ ac klund transformation b et w een the sec ond P ainlev´ e equation and the SD-I.d equation of Cosgro v e and Scouf is [1 7, 18]. Let y = D − 1 x " u x R n II u + n − 1 X j =1 κ j R j II u !# − 1 2 xu 2 − 1 2 (2 α − ǫ ) u, (3.2) where ǫ = ± 1. Dif ferentia ting (3.2) a nd using (3.1), w e f ind u x = ǫ  u 2 + 2 y x  . (3.3) No w we will sho w that D − 1 x  u x R j II u  = 1 2  u 2 H j [ y x ] + D − 1 x y x H j x [ y x ]  , (3.4) where t he oper ator H j [ p ] satisf ies the Len ard r ecursion rela tion D x H j +1 [ p ] =  D 3 x + 8 pD x + 4 p x  H j [ p ] , H 1 [ p ] = 4 p. (3.5) Firstly , w e w ill use indu ction to sho w that for any j = 1 , 2 , . . . , R j II u = 1 2 ( ǫD x + 2 u ) H j [ y x ] . (3.6) F or j = 1, R II u = u xx − 2 u 3 . Using (3.3), w e f ind that u xx = 2 u 3 + 4 y x u + 2 ǫy xx . (3.7) Th us R II u = 4 uy x + 2 ǫy xx = 1 2 ( ǫD x + 2 u ) H 1 [ y x ] . Assume t hat it is tr ue for j = k . Then 2 R k +1 II u = R II ( ǫD x + 2 u ) H k [ y x ] = ǫH k xxx [ y x ] + 2 uH k xx [ y x ] + 4 u x H k x [ y x ] + 2 u xx H k [ y x ] − 4 u 2  ǫH k x [ y x ] + 2 uH k [ y x ]  + 4 uD − 1 x  ǫu x H k x [ y x ] + 2 uu x H k [ y x ]  . (3.8) In tegration b y parts giv es D − 1 x  ǫu x H k x [ y x ] + 2 uu x H k [ y x ]  = u 2 H k [ y x ] + D − 1 x  ǫu x − u 2  H k x [ y x ]  . Hence (3.8 ) can b e w ritten as 2 R k +1 II u = ǫH k xxx [ y x ] + 2 uH k xx [ y x ] + 4 u x H k x [ y x ] + 2 u xx H k [ y x ] − 4 u 2  ǫH k x [ y x ] + 2 uH k [ y x ]  + 4 u  u 2 H k [ y x ] + D − 1 x  ǫu x − u 2  H k x [ y x ]  . (3.9) Using (3 .3) to substitute u x and ( 3.7) to substitute u xx , (3 .9) b ecomes 2 R k +1 II u = ǫ  H k xxx [ y x ] + 8 y x H k x [ y x ] + 4 y xx H k [ y x ]  + 2 u  H k xx [ y x ] + 4 y x H k [ y x ] + 4 D − 1 x y x H k x [ y x ]  = ( ǫD x + 2 u )  H k xx [ y x ] + 4 y x H k [ y x ] + 4 D − 1 x y x H k x [ y x ]  . B¨ ac klund T ransformations f or First and Second Painlev ´ e Hierarchies 5 Since D x  H k xx [ y x ] + 4 y x H k [ y x ] + 4 D − 1 x y x H k x [ y x ]  = H k xxx [ y x ] + 8 y x H k x [ y x ] + 4 y xx H k [ y x ] , w e ha v e H k xx [ y x ] + 4 y x H k [ y x ] + 4 D − 1 x y x H k x [ y x ] = H k +1 [ y x ], see (3.5), and hen ce the pro of by induction is f inish ed . No w u sing (3.6 ) we f ind 2 u x R k II ( u ) =  ǫu x − u 2  H k x [ y x ] + D x  u 2 H k [ y x ]  . (3.10) Using (3 .3) to substitute u x in to (3 .10) and then in tegrating, we obtain (3 .4). Therefore (3.2 ) ca n be used to obtain the follo wing quadratic equatio n for u − x + H n [ y x ] + n − 1 X j =1 κ j H j [ y x ] ! u 2 − (2 α − ǫ ) u + 2 D − 1 x y x H n x [ y x ] + n − 1 X j =1 κ j H j x [ y x ] ! − 2 y = 0 . (3.11) Eliminating u b et w een (3.3) and (3.11) give s a one-to-o ne corresp ondence b et w een the second P ainlev ´ e hierarch y (3.1) a nd the follo wing hierarc hy of second-degree equ ations H n x [ y x ] + n − 1 X j =1 κ j H j x [ y x ] − 1 ! 2 + 8 H n [ y x ] + n − 1 X j =1 κ j H j [ y x ] − x ! × D − 1 x y x H n x [ y x ] + n − 1 X j =1 κ j D − 1 x y x H j x [ y x ] − y ! = (2 α − ǫ ) 2 . (3.12) Therefore w e h a v e deriv ed the B¨ ac k lu nd transform ation (3.2) and (3.11) b etw een the PI I hierarc h y ( 3.1) and the new hierarc h y (3.12 ). Next we will giv e the explicit forms of the ab o v e r esults wh en n = 1 , 2 , 3. Example 3 ( n = 1) . T h e f irst mem b er of th e second P ainlev ´ e hierarc h y (3.1) is the sec ond P ainlev ´ e equation u xx = 2 u 3 + xu + α. In t his ca se, (3.2) and (3.1 1) read y = 1 2  u 2 x − u 4 − xu 2 − (2 α − ǫ ) u  and (4 y x − x ) u 2 − (2 α − ǫ ) u + 4 y 2 x − 2 y = 0 , resp ectiv ely . T he second-degree equation for y is (4 y xx − 1) 2 + 8(4 y x − x )  2 y 2 x − y  = (2 α − ǫ ) 2 . (3.13) The c hange of v ariables w = y − 1 8 x 2 transforms (3.13) into th e SD-I.d equation of Cosgro v e and Scouf is [1 7] w 2 xx + 4 w 3 x + 2 w x ( xw x − w ) = 1 16 (2 α − ǫ ) 2 . Th us when n = 1, the B¨ ac klu nd tr ansformation (3.2) and (3.11) is the kno wn B¨ ac klund transformation b et w een the second P ainlev ´ e equation and the S D-I.d equation (3.12). Since the f irst mem b er o f the hierarc h y (3 .12) is the SD-I.d equation, w e shall call it SD-I.d hierarch y . 6 A.H. S akk a Example 4 ( n = 2) . Th e second mem b er of the sec ond P ainlev´ e hierarc h y (3. 1) reads u xxxx = 10 u 2 u xx + 10 uu 2 x − 6 u 5 − κ 1  u xx − 2 u 3  + xu + α. (3.14) Equation (3 .14) is lab elled in [20, 21] as F-XVII. In t his ca se, (3.2) and (3.1 1) read y = 1 2  2 u x u xxx − u 2 xx − 10 u 2 u 2 x + 2 u 6 + κ 1  u 2 x − u 4  − xu 2 − (2 α − ǫ ) u  (3.15) and  4 y xxx + 24 y 2 x + 4 κ 1 y x − x  u 2 − (2 α − ǫ ) u + 8 y x y xxx − 4 y 2 xx + 32 y 3 x + 4 κ 1 y 2 x − 2 y = 0 , (3.16) resp ectiv ely . Equations (3. 15) and (3.16) g iv e on e-to-one corresp ondence betw een (3.14) and the foll o wing fourth-order second-degree equatio n [4 y xxxx + 48 y x y xx + 4 κ 1 y xx − 1] 2 (3.17) + 8  4 y xxx + 24 y 2 x + 4 κ 1 y x − x  4 y x y xxx − 2 y 2 xx + 16 y 3 x + 2 κ 1 y 2 x − y  = (2 α − ǫ ) 2 . Equation (3 .17) is a f irs t in tegral of the follo wing f ifth -order equ ation y xxxxx = − 20 y x y xxx − 10 y 2 xx − 40 y 3 x − κ 1 y xxx − 6 κ 1 y 2 x + xy x + y . (3.18) The tr ansformation y = − ( w + 1 2 γ z + 5 γ 3 ), z = x + 30 γ 2 transforms (3 .18) in to the equati on w z z z z z = 20 w z w z z z + 10 w 2 z z − 40 w 3 z + z w z + w + γ z . (3.19) The B¨ acklund transformatio n [2 2] v = w z , w = v z z z z − 20 v v z z − 10 v 2 z + 40 v 3 − z v − γ z , (3.20) giv es a one-to-one corresp ondence betw een (3.19) and Cosgro v e’ s Fif-I I I equation [20] v z z z z z = 20 v v z z z + 40 v z v z z − 120 v 2 v z + z v z + 2 v + γ . (3 .21) Therefore we hav e rederived the kno w n rela tion v = − 1 2  ǫu x − u 2 + γ  , u = − ǫ [ v z z z − 12 v v z + 4 γ v z + ǫ 2 α ] 2[ v z z − 6 v 2 + 4 γ v + 1 4 z − 4 γ 2 ] . b et w een Cosgro ve’ s equations Fif-I I I (3.2 1) and F-XVI I (3.14) [20]. Example 5 ( n = 3) . Th e third mem b er of the sec ond P ainlev´ e hierarc h y (3. 1) reads u xxxxxx = 14 u 2 u xxxx + 56 uu x u xxx + 42 uu 2 xx + 70 u 2 x u xx − 70 u 4 u xx − 140 u 3 u 2 x + 20 u 7 − κ 2 ( u xxxx − 10 u 2 u xx − 10 uu 2 x + 6 u 5 ) − κ 1 ( u xx − 2 u 3 ) + xu + α. (3.22) In t his ca se, (3.2) and (3.1 1) ha v e the fol lo wing forms resp ectiv ely 2 y = 2 u x u xxxxx − 2 u xx u xxxx + u 2 xxx − 28 u 2 u x u xxx + 14 u 2 u 2 xx − 56 uu 2 x u xx − 21 u 4 x + 70 u 4 u 2 x − 5 u 8 + κ 2 (2 u x u xxx − u 2 xx − 10 u 2 u 2 x + 2 u 6 ) + κ 1 ( u 2 x − u 4 ) − xu 2 − (2 α − ǫ ) u (3.23) and 4  y xxxxx + 20 y x y xx + 10 y 2 xx + 40 y 3 x + κ 2  y xxx + 6 y 2 x  + κ 1 y x − 1 4 x  u 2 B¨ ac klund T ransformations f or First and Second Painlev ´ e Hierarchies 7 − (2 α − ǫ ) u + 4  2 y x y xxxxx − 2 y xx y xxxx + y 2 xxx + 40 y 2 x y xxx + 60 y 4 x  + 4 κ 2  2 y x y xxx − y 2 xx + 8 y 3 x  + 4 κ 1 y 2 x − 2 y = 0 . (3.24) Equations (3.23) and (3 .24) gi v e one-to-one co rresp ondence b et w een (3.22) a nd the fo llo wing six-order s econd-d egree equati on  y xxxxxx + 20 y x y xxxx + 40 y xx y xxx + 120 y 2 x y xx + κ 2 ( y xxxx + 12 y x y xx ) + κ 1 y xx − 1 4  2 + 2  y xxxxx + 20 y x y xx + 10 y 2 xx + 40 y 3 x + κ 2  y xxx + 6 y 2 x  + κ 1 y x − 1 4 x  ×  4 y x y xxxxx − 4 y xx y xxxx + 2 y 2 xxx + 80 y 2 x y xxx + 120 y 4 x + 2 κ 2  2 y x y xxx − y 2 xx + 8 y 3 x  + 2 κ 1 y 2 x − y  = 1 16 (2 α − ǫ ) 2 . (3.25) The B¨ acklund transformatio n (3 .23), (3.2 4) and th e equation ( 3.25) are not g iv en before. 3.2 A hierarc h y of a second-order fourth-degree equ ation In this su bsection, we will generalize the B¨ acklund transformation gi v en in [23] b et w een t he second P ainlev´ e equation and a second-ord er four th-degree e quation. Let y = D − 1 x " u x R n II u + n − 1 X j =1 κ j R j II u !# − 1 2 xu 2 − αu. (3.26) Dif feren tiating (3.2 6) and usin g (3.1), w e f ind u 2 + 2 y x = 0 . (3.27) Equations (3.26) and (3.27) def ine a B¨ ackl und transformation b etw een the second P ainlev ´ e hierarc h y ( 3.1) and a new hierarc h y of dif ferentia l equations for y . In o rder to obtain the new h ierarc h y , w e will pro v e that D − 1 x  u x R j II u  = − D − 1 x  y xx y x S j II y x  , (3.28) where S II is the recursion o p erator S II = D 2 x − y xx y x D x − y xxx 2 y x + 3 y 2 xx 4 y 2 x + 8 y x − 4 y x D − 1 x y xx y x . First o f a ll, w e will use indu ction to pro v e that R j II u = − 2 u S j II y x . (3.29) Using (3 .27), w e f ind u x = − y xx u , u xx = − 1 u  y xxx − y 2 xx 2 y x  . (3.30) Hence R II u = u xx − 2 u 3 = − 1 u  y xxx − y 2 xx 2 y x + 8 y 2 x  = − 2 u S II y x . Th us (3.29) is tru e for j = 1. 8 A.H. S akk a Assume i t is true for j = k . Then R k +1 II u = − 2 R II 1 u S k II y x = − 2 u  D 2 x − 2 u x u D x − u xx u + 2 u 2 x u 2 − 4 u 2 + 4 u 2 D − 1 x u x u  S k II y x . Using (3 .30) to substitute u x and u xx and usin g (3.27) to sub stitute u 2 , we f ind the result. As a second s tep, we use (3 .29) to f ind D − 1 x  u x R k II u  = − 2 D − 1 x  u x u S k II y x  . Th us u s ing (3.30) to su bstitute u x and usin g (3.27) to sub stitute u 2 w e f ind (3.28). Therefore (3.26 ) implies αu = − y + x y x − D − 1 x " y xx y x S n II y x + n − 1 X j =1 κ j S j II y x !# . (3.31) If α 6 = 0, then su b stituting u from (3.31) into (3.27) w e obtain the follo wing h ierarc h y of dif ferentia l equations for y D − 1 x " y xx y x S n II y x + n − 1 X j =1 κ j S j II y x !# − xy x + y ! 2 + 2 α 2 y x = 0 . (3.32) If α = 0, then y satisf ies the hierarc hy D − 1 x " y xx y x S n II y x + n − 1 X j =1 κ j S j II y x !# − xy x + y = 0 . The f irst mem b er of the hierarc h y (3.3 2) is a fourth-degree equation, wher eas the other mem b ers are second-degree equatio ns. No w we giv e some examples. Example 6 ( n = 1) . In the presen t case, ( 3.26) r eads 2 y = u 2 x − u 4 − xu 2 − 2 αu. (3.33) Eliminating u b et w een (3.27) and (3.33) yields the follo wing second-order four th-degree equation for y  y 2 xx + 8 y 3 x − 4 y x ( xy x − y )  2 + 32 α 2 y 3 x = 0 . (3 .34) The change of v ariables w = 2 y transform (3.34) into the follo win g equ ation  w 2 xx + 4 w 3 x − 4 w x ( xw x − w )  2 + 16 α 2 y 3 x = 0 . (3.35) Equation (3 .35) w as deriv ed b efore [23]. Example 7 ( n = 2) . When n = 2, (3.2 6) reads 2 y = 2 u x u xxx − u 2 xx − 10 u 2 u 2 x + 2 u 6 − xu 2 − 2 αu + κ 1  u 2 x − u 4  . (3.36) Equations (3.27) and ( 3.36) giv e a B¨ ac klund transformation b et w een the second mem b er o f PI I hierarc h y ( 3.14) and the foll o wing fourth-order second-degree equation for y " y xx y xxxx − 3 y 2 xx 2 y x  y xxx − y 2 xx 2 y x  − 1 2  y xxx − y 2 xx 2 y x  2 + 10 y x y 2 xx + 16 y 4 x − 2 y x ( xy x − y ) + 1 2 κ 1  y 2 xx + 8 y 3 x  # 2 + 8 α 2 y 3 x = 0 . (3.37) Equation (3 .37) w as giv en b efore [19]. B¨ ac klund T ransformations f or First and Second Painlev ´ e Hierarchies 9 Example 8 ( n = 3) . In this case, (3.26) rea d 2 y = 2 u x u xxxxx − 2 u xx u xxxx + u 2 xxx − 28 u 2 u x u xxx + 14 u 2 u 2 xx − 56 uu 2 x u xx − 21 u 4 x (3.38) + 70 u 4 u 2 x − 5 u 8 + κ 2  2 u x u xxx − u 2 xx − 10 u 2 u 2 x + 2 u 6  + κ 1  u 2 x − u 4  − xu 2 − 2 αu, and ( 3.32) has the f orm " 2 y xx y xxxxxx −  2 y xxx + 3 y 2 xx y x   y xxxxx + 5 y xx y xxxx y x  +  y xxxx − 3 y xx y xxx 2 y x + 3 y 3 xx 4 y 2 x  2 +  2 y xxx − y 2 xx y x   2 y xx y xxxx y x + 3 y 2 xxx 2 y x − 9 y 2 xx y xxx 2 y 2 x + 15 y 2 xx 8 y 3 x − 7 y 2 xx − 14 y x y xxx  + 15 y 2 xx 2 y 2 x  3 y 2 xxx − 5 y 2 xx y xxx y x + 7 y 4 xx 4 y 3 x  + 21 y 4 xx 2 y x + 280 y 2 x y 2 xx − 150 y 5 x − 4 y x ( xy x − y ) + 2 κ 2 " y xx y xxxx − 3 y 2 xx 2 y x  y xxx − y 2 xx 2 y x  − 1 2  y xxx − y 2 xx 2 y x  2 + 10 y x y 2 xx + 16 y 4 x # + κ 1  y 2 xx + 8 y 3 x  # 2 + 32 α 2 y 3 x = 0 . (3.39) The B¨ ac klund transformation b et w een the third memb er of PI I hierarch y (3.22) and the new equation (3.39 ) is give n b y (3.27) and (3.38). 3.3 Auto-B¨ ac klund transformations for PI I hierarc h y In t his subsection, we will use the SD-I.d h ierarc h y (3.12) to d eriv e auto-B¨ ac klund trans f orma- tions for PI I h ierarc h y (3.1 ). Let u b e solution o f (3.1) with p arameter α and le t ¯ u b e solution of (3.1) w ith parameter ¯ α. Since (3.12) is in v arian t under the transformation 2 α − ǫ = − 2 ¯ α + ǫ , a solution y of (3.12) corresp onds to t w o sol utions u and ¯ u of (3.1). T h e relat ion b et w een y and u is giv en b y (3 .11) and t he r elation betw een y and ¯ u is given by − x + H n [ y x ] + n − 1 X j =1 κ j H j [ y x ] ! ¯ u 2 − (2 ¯ α − ǫ ) ¯ u + 2 D − 1 x y x H n x [ y x ] + n − 1 X j =1 κ j H j x [ y x ] ! − 2 y = 0 . (3.40) Subtracting ( 3.11) from (3.40), w e obtain − x + H n [ y x ] + n − 1 X j =1 κ j H j [ y x ] !  ¯ u 2 − u 2  − (2 ¯ α − ǫ ) ¯ u + (2 α − ǫ ) u = 0 . (3.41) Using 2 α − ǫ = − 2 ¯ α + ǫ and dividing by ¯ u + u , (3.41) yields − x + H n [ y x ] + n − 1 X j =1 κ j H j [ y x ] ! ( ¯ u − u ) + (2 α − ǫ ) = 0 . No w usin g (3.3) to sub stitute y x , we obtain the follo wing tw o auto-B¨ ac klund transformations for PI I hierarc hy (3.1) ¯ α = − α + ǫ, ǫ = ± 1 , 10 A.H. S akk a ¯ u = u − (2 α − ǫ )  − x + H n [ 1 2 ( ǫu x − u 2 )] + n − 1 P j =1 κ j H j [ 1 2 ( ǫu x − u 2 )]  . (3.42) These auto-B¨ ac klund transformations and the discrete symmetry ¯ u = − u , ¯ α = − α ca n b e used to d eriv e the auto-B¨ ac klun d transformations g iv en in [6, 16]. The au to-B¨ ac klund transformations (3.42) can b e us ed to obta in inf inite hierarc h ies of solu- tions of the PI I hierarch y (3.1). 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