Bäcklund Transformations for First and Second Painlevé Hierarchies

We give Bäcklund transformations for first and second Painlevé hierarchies. These Bäcklund transformations are generalization of known Bäcklund transformations of the first and second Painlevé equations and they relate the considered hierarchies to new hierarchies of Painlevé-type equations.

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Datum:	2009
1. Verfasser:	Sakka, A.H.
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Sprache:	English
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Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1492522019-02-20T01:26:55Z Bäcklund Transformations for First and Second Painlevé Hierarchies Sakka, A.H. We give Bäcklund transformations for first and second Painlevé hierarchies. These Bäcklund transformations are generalization of known Bäcklund transformations of the first and second Painlevé equations and they relate the considered hierarchies to new hierarchies of Painlevé-type equations. 2009 Article Bäcklund Transformations for First and Second Painlevé Hierarchies / A.H. Sakka // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 23 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 34M55; 33E17 http://dspace.nbuv.gov.ua/handle/123456789/149252 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We give Bäcklund transformations for first and second Painlevé hierarchies. These Bäcklund transformations are generalization of known Bäcklund transformations of the first and second Painlevé equations and they relate the considered hierarchies to new hierarchies of Painlevé-type equations.
format	Article
author	Sakka, A.H.
spellingShingle	Sakka, A.H. Bäcklund Transformations for First and Second Painlevé Hierarchies Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Sakka, A.H.
author_sort	Sakka, A.H.
title	Bäcklund Transformations for First and Second Painlevé Hierarchies
title_short	Bäcklund Transformations for First and Second Painlevé Hierarchies
title_full	Bäcklund Transformations for First and Second Painlevé Hierarchies
title_fullStr	Bäcklund Transformations for First and Second Painlevé Hierarchies
title_full_unstemmed	Bäcklund Transformations for First and Second Painlevé Hierarchies
title_sort	bäcklund transformations for first and second painlevé hierarchies
publisher	Інститут математики НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/149252
citation_txt	Bäcklund Transformations for First and Second Painlevé Hierarchies / A.H. Sakka // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 23 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT sakkaah backlundtransformationsforfirstandsecondpainlevehierarchies
first_indexed	2025-07-12T21:41:47Z
last_indexed	2025-07-12T21:41:47Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 024, 11 pages Bäcklund Transformations for First and Second Painlevé Hierarchies Ayman Hashem SAKKA Department of Mathematics, Islamic University of Gaza, P.O. Box 108, Rimal, Gaza, Palestine E-mail: asakka@iugaza.edu.ps Received November 25, 2008, in final form February 24, 2009; Published online March 02, 2009 doi:10.3842/SIGMA.2009.024 Abstract. We give Bäcklund transformations for first and second Painlevé hierarchies. These Bäcklund transformations are generalization of known Bäcklund transformations of the first and second Painlevé equations and they relate the considered hierarchies to new hierarchies of Painlevé-type equations. Key words: Painlevé hierarchies; Bäcklund transformations 2000 Mathematics Subject Classification: 34M55; 33E17 1 Introduction One century ago Painlevé and Gambier have discovered the six Painlevé equations, PI–PVI. These equations are the only second-order ordinary differential equations whose general solutions can not be expressed in terms of elementary and classical special functions; thus they define new transcendental functions. Painlevé transcendental functions appear in many areas of modern mathematics and physics and they paly the same role in nonlinear problems as the classical special functions play in linear problems. In recent years there is a considerable interest in studying hierarchies of Painlevé equations. This interest is due to the connection between these hierarchies of Painlevé equations and com- pletely integrable partial differential equations. A Painlevé hierarchy is an infinite sequence of nonlinear ordinary differential equations whose first member is a Painlevé equation. Airault [1] was the first to derive a Painlevé hierarchy, namely a second Painlevé hierarchy, as the similari- ty reduction of the modified Korteweg–de Vries (mKdV) hierarchy. A first Painlevé hierarchy was given by Kudryashov [2]. Later on several hierarchies of Painlevé equations were introdu- ced [3, 4, 5, 6, 7, 8, 9, 10, 11]. As it is well known, Painlevé equations possess Bäcklund transformations; that is, mappings between solutions of the same Painlevé equation or between solutions of a particular Painlevé equation and other second-order Painlevé-type equations. Various methods to derive these Bäcklund transformations can be found for example in [12, 13, 14, 15]. Bäcklund transformations are nowadays considered to be one of the main properties of integrable nonlinear ordinary differential equations, and there is much interest in their derivation. In the present article, we generalize known Bäcklund transformations of the first and se- cond Painlevé equations to the first and second Painlevé hierarchies given in [6, 11]. We give a Bäcklund transformation between the considered first Painlevé hierarchy and a new hierarchy of Painlevé-type equations. In addition, we give two new hierarchies of Painlevé-type equations related, via Bäcklund transformations, to the considered second Painlevé hierarchy. Then we derive auto-Bäcklund transformations for this second Painlevé hierarchy. Bäcklund transforma- tions of the second Painlevé hierarchy have been studied in [6, 16]. mailto:asakka@iugaza.edu.ps http://dx.doi.org/10.3842/SIGMA.2009.024 2 A.H. Sakka 2 Bäcklund transformations for PI hierarchy In this section, we will derive a Bäcklund transformation for the first Painlevé hierarchy (PI hierarchy) [6] n+1∑ j=2 γjL j [u] = γx, (2.1) where the operator Lj [u] satisfies the Lenard recursion relation DxLj+1[u] = ( D3 x − 4uDx − 2ux ) Lj [u], L1[u] = u. (2.2) The special case γj = 0, 2 ≤ j ≤ n, of this hierarchy is a similarity reduction of the Schwarz– Korteweg–de Vries hierarchy [2, 4]. Moreover its members may define new transcendental func- tions. The PI hierarchy (2.1) can be written in the following form [11] Rn I u + n∑ j=2 κjRn−j I u = x, (2.3) where RI is the recursion operator RI = D2 x − 8u + 4D−1 x ux. In [17, 18], it is shown that the Bäcklund transformation u = −yx, y = 1 2 ( u2 x − 4u3 − 2xu ) , (2.4) defines a one-to-one correspondence between the first Painlevé equation uxx = 6u2 + x. (2.5) and the SD-I.e equation of Cosgrove and Scoufis [17] y2 xx = −4y3 x − 2(xyx − y). (2.6) We will show that there is a generalization of this Bäcklund transformation to all members of the PI hierarchy (2.3). Let y = −xu + D−1 x ux [ Rn I u + n∑ j=2 κjRn−j I u ] . (2.7) Differentiating (2.7) and using (2.3), we find u = −yx. (2.8) Substituting u = −yx into (2.7), we obtain the following hierarchy of differential equation for y D−1 x yxx [ Sn I yx + n∑ j=2 κjSn−j I yx ] + (xyx − y) = 0, (2.9) where SI is the recursion operator SI = D2 x + 8yx − 4D−1 x yxx. Bäcklund Transformations for First and Second Painlevé Hierarchies 3 The first member of the hierarchy (2.9) is the SD-I.e equation (2.6). Thus we shall call this hierarchy SD-I.e hierarchy. Therefor we have derived the Bäcklund transformation (2.7)–(2.8) between solutions u of the first Painlevé hierarchy (2.3) and solutions y of the SD-I.e hierarchy (2.9). When n = 1, the Bäcklund transformation (2.7)–(2.8) gives the Bäcklund transformation (2.4) between the first Painlevé equation (2.5) and the SD-I.e equation (2.6). Next we will consider the cases n = 2 and n = 3. Example 1 (n = 2). The second member of the PI hierarchy (2.3) is the fourth-order equation uxxxx = 20uuxx + 10u2 x − 40u3 − κ2u + x. (2.10) In this case, the Bäcklund transformation (2.7) reads y = 1 2 ( 2uxuxxx − u2 xx − 20uu2 x + 20u4 + κ2u 2 − 2xu ) . (2.11) Equations (2.11) and (2.8) give one-to-one correspondence between (2.10) and the following equation 2yxxyxxxx − y2 xxx + 20yxy2 xx + 20y4 x + κ2y 2 x + 2(xyx − y) = 0. (2.12) Equation (2.12) and the Bäcklund transformation (2.8) and (2.11) were given before [19]. Example 2 (n = 3). The third member of the PI hierarchy (2.3) reads uxxxxxx = 28uuxxxx + 56uxuxxx + 42u2 xx − 280u2uxx − 280uu2 x + 280u4 − κ2 ( uxx − 6u2 ) − κ3u + x. (2.13) In this case, the Bäcklund transformation (2.7) has the form y = 1 2 [ 2uxuxxxxx − 2uxxuxxxx + u2 xxx − 56uuxuxxx + 28uu2 xx − 56u2 xuxx + 280u2u2 x − 112u5 + κ2 ( u2 x − 4u3 ) + κ3u 2 − 2xu ] . (2.14) Equations (2.8) and (2.14) give one-to-one correspondence between solutions u of (2.13) and solutions y of the following equation 2yxxyxxxxxx − 2yxxxyxxxxx + y2 xxxx + 56yxyxxyxxxx − 28yxy2 xxx + 56y2 xxyxxx + 280y2 xy2 xxx + 112y5 x + κ2 ( y2 xx + 4y3 x ) + κ3y 2 x + 2(xyx − y) = 0. (2.15) Equation (2.15) is a new sixth-order Painlevé-type equation. 3 Bäcklund transformations for second Painlevé hierarchy In the present section, we will study Bäcklund transformations of the second Painlevé hierarchy (PII hierarchy) [6] (Dx − 2u) n∑ j=1 γjL j [ ux + u2 ] + 2γxu− γ − 4δ = 0, where the operator Lj [u] is defined by (2.2). The special case γj = 0, 1 ≤ j ≤ n − 1, of this hierarchy is a similarity reduction of the modified Korteweg–de Vries hierarchy [2, 4]. The members of this hierarchy may define new transcendental functions. This hierarchy can be written in the following alternative form [11] Rn II u + n−1∑ j=1 κjRj II u− (xu + α) = 0, (3.1) where RII is the recursion operator RII = D2 x − 4u2 + 4uD−1 x ux. 4 A.H. Sakka 3.1 A hierarchy of SD-I.d equation As a first Bäcklund transformation for the PII hierarchy (3.1), we will generalize the Bäcklund transformation between the second Painlevé equation and the SD-I.d equation of Cosgrove and Scoufis [17, 18]. Let y = D−1 x [ ux ( Rn II u + n−1∑ j=1 κjRj II u )] − 1 2xu2 − 1 2(2α− ε)u, (3.2) where ε = ±1. Differentiating (3.2) and using (3.1), we find ux = ε ( u2 + 2yx ) . (3.3) Now we will show that D−1 x ( uxRj II u ) = 1 2 ( u2Hj [yx] + D−1 x yxHj x[yx] ) , (3.4) where the operator Hj [p] satisfies the Lenard recursion relation DxHj+1[p] = ( D3 x + 8pDx + 4px ) Hj [p], H1[p] = 4p. (3.5) Firstly, we will use induction to show that for any j = 1, 2, . . . , Rj II u = 1 2(εDx + 2u)Hj [yx]. (3.6) For j = 1, RIIu = uxx − 2u3. Using (3.3), we find that uxx = 2u3 + 4yxu + 2εyxx. (3.7) Thus RIIu = 4uyx + 2εyxx = 1 2(εDx + 2u)H1[yx]. Assume that it is true for j = k. Then 2Rk+1 II u = RII(εDx + 2u)Hk[yx] = εHk xxx[yx] + 2uHk xx[yx] + 4uxHk x [yx] + 2uxxHk[yx] − 4u2 ( εHk x [yx] + 2uHk[yx] ) + 4uD−1 x ( εuxHk x [yx] + 2uuxHk[yx] ) . (3.8) Integration by parts gives D−1 x ( εuxHk x [yx] + 2uuxHk[yx] ) = u2Hk[yx] + D−1 x [( εux − u2 ) Hk x [yx] ] . Hence (3.8) can be written as 2Rk+1 II u = εHk xxx[yx] + 2uHk xx[yx] + 4uxHk x [yx] + 2uxxHk[yx]− 4u2 ( εHk x [yx] + 2uHk[yx] ) + 4u ( u2Hk[yx] + D−1 x [( εux − u2 ) Hk x [yx] ]) . (3.9) Using (3.3) to substitute ux and (3.7) to substitute uxx, (3.9) becomes 2Rk+1 II u = ε ( Hk xxx[yx] + 8yxHk x [yx] + 4yxxHk[yx] ) + 2u ( Hk xx[yx] + 4yxHk[yx] + 4D−1 x yxHk x [yx] ) = (εDx + 2u) ( Hk xx[yx] + 4yxHk[yx] + 4D−1 x yxHk x [yx] ) . Bäcklund Transformations for First and Second Painlevé Hierarchies 5 Since Dx ( Hk xx[yx] + 4yxHk[yx] + 4D−1 x yxHk x [yx] ) = Hk xxx[yx] + 8yxHk x [yx] + 4yxxHk[yx], we have Hk xx[yx] + 4yxHk[yx] + 4D−1 x yxHk x [yx] = Hk+1[yx], see (3.5), and hence the proof by induction is finished. Now using (3.6) we find 2uxRk II (u) = ( εux − u2 ) Hk x [yx] + Dx ( u2Hk[yx] ) . (3.10) Using (3.3) to substitute ux into (3.10) and then integrating, we obtain (3.4). Therefore (3.2) can be used to obtain the following quadratic equation for u( − x + Hn[yx] + n−1∑ j=1 κjH j [yx] ) u2 − (2α− ε)u + 2D−1 x yx ( Hn x [yx] + n−1∑ j=1 κjH j x[yx] ) − 2y = 0. (3.11) Eliminating u between (3.3) and (3.11) gives a one-to-one correspondence between the second Painlevé hierarchy (3.1) and the following hierarchy of second-degree equations( Hn x [yx] + n−1∑ j=1 κjH j x[yx]− 1 )2 + 8 ( Hn[yx] + n−1∑ j=1 κjH j [yx]− x ) × ( D−1 x yxHn x [yx] + n−1∑ j=1 κjD −1 x yxHj x[yx]− y ) = (2α− ε)2. (3.12) Therefore we have derived the Bäcklund transformation (3.2) and (3.11) between the PII hierarchy (3.1) and the new hierarchy (3.12). Next we will give the explicit forms of the above results when n = 1, 2, 3. Example 3 (n = 1). The first member of the second Painlevé hierarchy (3.1) is the second Painlevé equation uxx = 2u3 + xu + α. In this case, (3.2) and (3.11) read y = 1 2 [ u2 x − u4 − xu2 − (2α− ε)u ] and (4yx − x)u2 − (2α− ε)u + 4y2 x − 2y = 0, respectively. The second-degree equation for y is (4yxx − 1)2 + 8(4yx − x) ( 2y2 x − y ) = (2α− ε)2. (3.13) The change of variables w = y− 1 8x2 transforms (3.13) into the SD-I.d equation of Cosgrove and Scoufis [17] w2 xx + 4w3 x + 2wx(xwx − w) = 1 16(2α− ε)2. Thus when n = 1, the Bäcklund transformation (3.2) and (3.11) is the known Bäcklund transformation between the second Painlevé equation and the SD-I.d equation (3.12). Since the first member of the hierarchy (3.12) is the SD-I.d equation, we shall call it SD-I.d hierarchy. 6 A.H. Sakka Example 4 (n = 2). The second member of the second Painlevé hierarchy (3.1) reads uxxxx = 10u2uxx + 10uu2 x − 6u5 − κ1 ( uxx − 2u3 ) + xu + α. (3.14) Equation (3.14) is labelled in [20, 21] as F-XVII. In this case, (3.2) and (3.11) read y = 1 2 [ 2uxuxxx − u2 xx − 10u2u2 x + 2u6 + κ1 ( u2 x − u4 ) − xu2 − (2α− ε)u ] (3.15) and ( 4yxxx+ 24y2 x+ 4κ1yx − x ) u2− (2α− ε)u + 8yxyxxx− 4y2 xx+ 32y3 x+ 4κ1y 2 x− 2y = 0, (3.16) respectively. Equations (3.15) and (3.16) give one-to-one correspondence between (3.14) and the following fourth-order second-degree equation [4yxxxx + 48yxyxx + 4κ1yxx − 1]2 (3.17) + 8 [ 4yxxx + 24y2 x + 4κ1yx − x ][ 4yxyxxx − 2y2 xx + 16y3 x + 2κ1y 2 x − y ] = (2α− ε)2. Equation (3.17) is a first integral of the following fifth-order equation yxxxxx = −20yxyxxx − 10y2 xx − 40y3 x − κ1yxxx − 6κ1y 2 x + xyx + y. (3.18) The transformation y = −(w + 1 2γz + 5γ3), z = x + 30γ2 transforms (3.18) into the equation wzzzzz = 20wzwzzz + 10w2 zz − 40w3 z + zwz + w + γz. (3.19) The Bäcklund transformation [22] v = wz, w = vzzzz − 20vvzz − 10v2 z + 40v3 − zv − γz, (3.20) gives a one-to-one correspondence between (3.19) and Cosgrove’s Fif-III equation [20] vzzzzz = 20vvzzz + 40vzvzz − 120v2vz + zvz + 2v + γ. (3.21) Therefore we have rederived the known relation v = −1 2 ( εux − u2 + γ ) , u = −ε[vzzz − 12vvz + 4γvz + ε 2α] 2[vzz − 6v2 + 4γv + 1 4z − 4γ2] . between Cosgrove’s equations Fif-III (3.21) and F-XVII (3.14) [20]. Example 5 (n = 3). The third member of the second Painlevé hierarchy (3.1) reads uxxxxxx = 14u2uxxxx + 56uuxuxxx + 42uu2 xx + 70u2 xuxx − 70u4uxx − 140u3u2 x + 20u7 − κ2(uxxxx − 10u2uxx − 10uu2 x + 6u5)− κ1(uxx − 2u3) + xu + α. (3.22) In this case, (3.2) and (3.11) have the following forms respectively 2y = 2uxuxxxxx − 2uxxuxxxx + u2 xxx − 28u2uxuxxx + 14u2u2 xx− 56uu2 xuxx − 21u4 x+ 70u4u2 x − 5u8 + κ2(2uxuxxx − u2 xx − 10u2u2 x + 2u6) + κ1(u2 x − u4)− xu2 − (2α− ε)u (3.23) and 4 [ yxxxxx + 20yxyxx + 10y2 xx + 40y3 x + κ2 ( yxxx + 6y2 x ) + κ1yx − 1 4x ] u2 Bäcklund Transformations for First and Second Painlevé Hierarchies 7 − (2α− ε)u + 4 ( 2yxyxxxxx − 2yxxyxxxx + y2 xxx + 40y2 xyxxx + 60y4 x ) + 4κ2 ( 2yxyxxx − y2 xx + 8y3 x ) + 4κ1y 2 x − 2y = 0. (3.24) Equations (3.23) and (3.24) give one-to-one correspondence between (3.22) and the following six-order second-degree equation[ yxxxxxx + 20yxyxxxx + 40yxxyxxx + 120y2 xyxx + κ2(yxxxx + 12yxyxx) + κ1yxx − 1 4 ]2 + 2 [ yxxxxx + 20yxyxx + 10y2 xx + 40y3 x + κ2 ( yxxx + 6y2 x ) + κ1yx − 1 4x ] × [ 4yxyxxxxx − 4yxxyxxxx + 2y2 xxx + 80y2 xyxxx + 120y4 x + 2κ2 ( 2yxyxxx − y2 xx + 8y3 x ) + 2κ1y 2 x − y ] = 1 16(2α− ε)2. (3.25) The Bäcklund transformation (3.23), (3.24) and the equation (3.25) are not given before. 3.2 A hierarchy of a second-order fourth-degree equation In this subsection, we will generalize the Bäcklund transformation given in [23] between the second Painlevé equation and a second-order fourth-degree equation. Let y = D−1 x [ ux ( Rn II u + n−1∑ j=1 κjRj II u )] − 1 2xu2 − αu. (3.26) Differentiating (3.26) and using (3.1), we find u2 + 2yx = 0. (3.27) Equations (3.26) and (3.27) define a Bäcklund transformation between the second Painlevé hierarchy (3.1) and a new hierarchy of differential equations for y. In order to obtain the new hierarchy, we will prove that D−1 x ( uxRj II u ) = −D−1 x ( yxx yx Sj II yx ) , (3.28) where SII is the recursion operator SII = D2 x − yxx yx Dx − yxxx 2yx + 3y2 xx 4y2 x + 8yx − 4yxD−1 x yxx yx . First of all, we will use induction to prove that Rj II u = −2 u Sj II yx. (3.29) Using (3.27), we find ux = −yxx u , uxx = −1 u ( yxxx − y2 xx 2yx ) . (3.30) Hence RIIu = uxx − 2u3 = −1 u ( yxxx − y2 xx 2yx + 8y2 x ) = −2 u SIIyx. Thus (3.29) is true for j = 1. 8 A.H. Sakka Assume it is true for j = k. Then Rk+1 II u = −2RII 1 u Sk II yx = −2 u { D2 x − 2ux u Dx − uxx u + 2u2 x u2 − 4u2 + 4u2D−1 x ux u } Sk II yx. Using (3.30) to substitute ux and uxx and using (3.27) to substitute u2, we find the result. As a second step, we use (3.29) to find D−1 x ( uxRk II u ) = −2D−1 x (ux u Sk II yx ) . Thus using (3.30) to substitute ux and using (3.27) to substitute u2 we find (3.28). Therefore (3.26) implies αu = −y + xyx −D−1 x [ yxx yx ( Sn II yx + n−1∑ j=1 κjSj II yx )] . (3.31) If α 6= 0, then substituting u from (3.31) into (3.27) we obtain the following hierarchy of differential equations for y( D−1 x [ yxx yx ( Sn II yx + n−1∑ j=1 κjSj II yx )] − xyx + y )2 + 2α2yx = 0. (3.32) If α = 0, then y satisfies the hierarchy D−1 x [ yxx yx ( Sn II yx + n−1∑ j=1 κjSj II yx )] − xyx + y = 0. The first member of the hierarchy (3.32) is a fourth-degree equation, whereas the other members are second-degree equations. Now we give some examples. Example 6 (n = 1). In the present case, (3.26) reads 2y = u2 x − u4 − xu2 − 2αu. (3.33) Eliminating u between (3.27) and (3.33) yields the following second-order fourth-degree equation for y [ y2 xx + 8y3 x − 4yx(xyx − y) ]2 + 32α2y3 x = 0. (3.34) The change of variables w = 2y transform (3.34) into the following equation[ w2 xx + 4w3 x − 4wx(xwx − w) ]2 + 16α2y3 x = 0. (3.35) Equation (3.35) was derived before [23]. Example 7 (n = 2). When n = 2, (3.26) reads 2y = 2uxuxxx − u2 xx − 10u2u2 x + 2u6 − xu2 − 2αu + κ1 ( u2 x − u4 ) . (3.36) Equations (3.27) and (3.36) give a Bäcklund transformation between the second member of PII hierarchy (3.14) and the following fourth-order second-degree equation for y[ yxxyxxxx − 3y2 xx 2yx ( yxxx − y2 xx 2yx ) − 1 2 ( yxxx − y2 xx 2yx )2 + 10yxy2 xx + 16y4 x − 2yx(xyx − y) + 1 2 κ1 ( y2 xx + 8y3 x )]2 + 8α2y3 x = 0. (3.37) Equation (3.37) was given before [19]. Bäcklund Transformations for First and Second Painlevé Hierarchies 9 Example 8 (n = 3). In this case, (3.26) read 2y = 2uxuxxxxx − 2uxxuxxxx + u2 xxx − 28u2uxuxxx + 14u2u2 xx − 56uu2 xuxx − 21u4 x (3.38) + 70u4u2 x − 5u8 + κ2 ( 2uxuxxx − u2 xx − 10u2u2 x + 2u6 ) + κ1 ( u2 x − u4 ) − xu2 − 2αu, and (3.32) has the form[ 2yxxyxxxxxx − ( 2yxxx + 3y2 xx yx )( yxxxxx + 5yxxyxxxx yx ) + ( yxxxx − 3yxxyxxx 2yx + 3y3 xx 4y2 x )2 + ( 2yxxx − y2 xx yx )( 2yxxyxxxx yx + 3y2 xxx 2yx − 9y2 xxyxxx 2y2 x + 15y2 xx 8y3 x − 7y2 xx − 14yxyxxx ) + 15y2 xx 2y2 x ( 3y2 xxx − 5y2 xxyxxx yx + 7y4 xx 4y3 x ) + 21y4 xx 2yx + 280y2 xy2 xx − 150y5 x − 4yx(xyx − y) + 2κ2 [ yxxyxxxx − 3y2 xx 2yx ( yxxx − y2 xx 2yx ) − 1 2 ( yxxx − y2 xx 2yx )2 + 10yxy2 xx + 16y4 x ] + κ1 ( y2 xx + 8y3 x )]2 + 32α2y3 x = 0. (3.39) The Bäcklund transformation between the third member of PII hierarchy (3.22) and the new equation (3.39) is given by (3.27) and (3.38). 3.3 Auto-Bäcklund transformations for PII hierarchy In this subsection, we will use the SD-I.d hierarchy (3.12) to derive auto-Bäcklund transforma- tions for PII hierarchy (3.1). Let u be solution of (3.1) with parameter α and let ū be solution of (3.1) with parameter ᾱ. Since (3.12) is invariant under the transformation 2α − ε = −2ᾱ + ε, a solution y of (3.12) corresponds to two solutions u and ū of (3.1). The relation between y and u is given by (3.11) and the relation between y and ū is given by( − x + Hn[yx] + n−1∑ j=1 κjH j [yx] ) ū2 − (2ᾱ− ε)ū + 2D−1 x yx ( Hn x [yx] + n−1∑ j=1 κjH j x[yx] ) − 2y = 0. (3.40) Subtracting (3.11) from (3.40), we obtain( − x + Hn[yx] + n−1∑ j=1 κjH j [yx] )( ū2 − u2 ) − (2ᾱ− ε)ū + (2α− ε)u = 0. (3.41) Using 2α− ε = −2ᾱ + ε and dividing by ū + u, (3.41) yields( − x + Hn[yx] + n−1∑ j=1 κjH j [yx] ) (ū− u) + (2α− ε) = 0. Now using (3.3) to substitute yx, we obtain the following two auto-Bäcklund transformations for PII hierarchy (3.1) ᾱ = −α + ε, ε = ±1, 10 A.H. Sakka ū = u− (2α− ε)( − x + Hn[12(εux − u2)] + n−1∑ j=1 κjHj [12(εux − u2)] ) . (3.42) These auto-Bäcklund transformations and the discrete symmetry ū = −u, ᾱ = −α can be used to derive the auto-Bäcklund transformations given in [6, 16]. The auto-Bäcklund transformations (3.42) can be used to obtain infinite hierarchies of solu- tions of the PII hierarchy (3.1). For example, starting by the solution u = 0, α = 0 of (3.1), the auto-Bäcklund transformations (3.42) yields the new solution ū = − ε x , ᾱ = ε. Now applying the auto-Bäcklund transformations (3.42) with ε = 1 to the solution ū = 1 x , ᾱ = −1, we obtain the new solution ¯̄u = −2(x3−2κ1) x(x3+4κ1) , ¯̄α = 2. References [1] Airault H., Rational solutions of Painlevé equations, Stud. Appl. Math. 61 (1979), 31–53. [2] Kudryashov N.A., The first and second Painlevé equations of higher order and some relations between them, Phys. Lett. A 224 (1997), 353–360. [3] Hone A.N.W., Non-autonomous Hénon–Heiles systems, Phys. D 118 (1998), 1–16, solv-int/9703005. [4] Kudryashov N.A., Soukharev M.B., Uniformization and transcendence of solutions for the first and second Painlevé hierarchies, Phys. Lett. A 237 (1998), 206–216. [5] Kudryashov N.A., Fourth-order analogies of the Painlevé equations, J. Phys. A: Math. Gen. 35 (2002), 4617–4632. [6] Kudryashov N.A., Amalgamations of the Painlevé equations, J. Math. Phys. 44 (2003), 6160–6178. [7] Muğan U., Jrad F., Painlevé test and the first Painlevé hierarchy, J. Phys. A: Math. Gen. 32 (1999), 7933–7952. [8] Muğan U., Jrad F., Painlevé test and higher order differntial equations, J. Nonlinear Math. Phys. 9 (2002), 282–310, nlin.SI/0301043. [9] Gordoa P.R., Pickering A., Nonisospectral scattering problems: a key to integrable hierarchies, J. Math. Phys. 40 (1999), 5749–5786. [10] Gordoa P.R., Joshi N., Pickering A., On a generalized 2+1 dispersive water wave hierarchy, Publ. Res. Inst. Math. Sci. 37 (2001), 327–347. [11] Sakka A., Linear problems and hierarchies of Painlevé equations, J. Phys. A: Math. Theor. 42 (2009), 025210, 19 pages. [12] Fokas A.S., Ablowitz M.J., On a unified approach to transformations and elementary solutions of Painlevé equations, J. Math. Phys. 23 (1982), 2033–2042. [13] Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, PII and PIV, Math. Ann. 275 (1986), 221–255. [14] Gordoa P. R., Joshi N., Pickering A., Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations, Nonlinearity 12 (1999), 955–968, solv-int/9904023. [15] Gromak V., Laine I., Shimomura S., Painlevé differential equations in the complex plane, de Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2002. [16] Clarkson P.A., Joshi N., Pickering A., Bäcklund transformations for the second Pianlevé hierarchy: a modi- fied truncation approach, Inverse Problems 15 (1999), 175–187, solv-int/9811014. [17] Cosgrove C.M., Scoufis G., Painlevé classification of a class of differential equations of the second order and second degree, Stud. Appl. Math. 88 (1993), 25–87. [18] Muğan U., Sakka A., Second-order second-degree Painlevé equations related with Painlevé I–VI equations and Fuchsian-type transformations, J. Math. Phys. 40 (1999), 3569–3587. [19] Sakka A., Elshamy S., Bäcklund transformations for fourth-order Painlevé-type equations, The Islamic University Journal, Natural Science Series 14 (2006), 105–120. http://arxiv.org/abs/solv-int/9703005 http://arxiv.org/abs/nlin.SI/0301043 http://arxiv.org/abs/solv-int/9904023 http://arxiv.org/abs/solv-int/9811014 Bäcklund Transformations for First and Second Painlevé Hierarchies 11 [20] Cosgrove C. M., Higher-order Painlevé equations in the polynomial class. I. Bureau symbol P2, Stud. Appl. Math. 104 (2000), 1–65. [21] Cosgrove C.M., Higher-order Painlevé equations in the polynomial class. II. Bureau symbol P1, Stud. Appl. Math. 116 (2006), 321–413. [22] Sakka A., Bäcklund transformations for fifth-order Painlevé equations, Z. Naturforsch. A 60 (2005), 681– 686. [23] Sakka A., Second-order fourth-degree Painlevé-type equations, J. Phys. A: Math. Gen. 34 (2001), 623–631. 1 Introduction 2 Bäcklund transformations for PI hierarchy 3 Bäcklund transformations for second Painlevé hierarchy 3.1 A hierarchy of SD-I.d equation 3.2 A hierarchy of a second-order fourth-degree equation 3.3 Auto-Bäcklund transformations for PII hierarchy References

Bäcklund Transformations for First and Second Painlevé Hierarchies

Institution

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