Special polynomials associated with some hierarchies

Physics Letters A 372 (2008) 1945–1956 www.elsevier.com/locate/pla

Special polynomials associated with some hierarchies Nikolai A. Kudryashov Department of Applied Mathematics, Moscow Engineering and Physics Institute (State university), 31 Kashirskoe Shosse, 115409 Moscow, Russian Federation Received 9 January 2007; received in revised form 29 October 2007; accepted 31 October 2007 Available online 6 November 2007 Communicated by A.P. Fordy

Abstract Special polynomials associated with rational solutions of a hierarchy of equations of Painlevé type are introduced. The hierarchy arises by similarity reduction from the Fordy–Gibbons hierarchy of partial differential equations. Some relations for these special polynomials are given. Differential-difference hierarchies for finding special polynomials are presented. These formulae allow us to obtain special polynomials associated with the hierarchy studied. It is shown that rational solutions of members of the Schwarz–Sawada–Kotera, the Schwarz–Kaup–Kupershmidt, the Fordy–Gibbons, the Sawada–Kotera and the Kaup–Kupershmidt hierarchies can be expressed through special polynomials of the hierarchy studied. © 2007 Elsevier B.V. All rights reserved. PACS: 02.30.Hq Keywords: Special polynomials; Painlevé equations; Special solutions; Differential-difference equations

1. Introduction In this Letter our interest is in special polynomials associated with rational solutions of the hierarchy [1–5]     d 1 + w HN wz − w 2 − zw − β = 0. dz 2

(1.1)

Here the nonlinear operator HN is determined by the following recursion formula [2–6] HN+2 = J [v]Ω[v]HN

(1.2)

under the conditions H0 [v] = 1,

H1 [v] = vzz + 4v 2 ,

(1.3)

where the operators Ω[v] and J [v] take the form [3–6] Ω = D 3 + 2vD + vz ,

D=

d dz

    J = D 3 + 3(vD + Dv) + 2 D 2 vD −1 + D −1 vD 2 + 8 v 2 D −1 + D −1 v 2 ,  D −1 = dz.

E-mail address: [email protected]. 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.10.087

(1.4)

(1.5)

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N.A. Kudryashov / Physics Letters A 372 (2008) 1945–1956

Hierarchy (1.1) is important because special solutions of the Fordy–Gibbons hierarchy [7,8], the Sawada–Kotera hierarchy (Caudrey–Dodd–Gibbon hierarchy) [9,10] and the Kaup–Kupershmidt hierarchy [11] can be expressed through solutions of the hierarchy (1.1). Apparently hierarchy (1.1) defines new transcendental functions like the Painlevé equations do. Hierarchy (1.1) can be written as well in the form  

1 d GN −2wz − 2w 2 − zw − β = 0. w− (1.6) 2 dz The recursion relations GN is determined by the nonlinear operator [2–6] GN+2 = J1 [u]Ω[u]GN

(1.7)

under the conditions 1 G1 [u] = uzz + u2 . 4 The operator J1 [u] takes the form G0 [u] = 1,

(1.8)

 1  1  2 −1 D uD + D −1 uD 2 + u2 D −1 + D −1 u2 . (1.9) 2 8 Hierarchy (1.1) is similar in appearance to the second Painlevé hierarchy but hierarchy (1.1) is distinguished from the P2 hierarchy by the nonlinear operator HN [12–14]. Special polynomials associated with the rational solutions of the P2 hierarchy were considered in papers [13,14]. Recently it was shown [15] that rational solutions of hierarchy (1.1) at N = 1 can be found using special polynomials. These special polynomials were obtained in [15] taking into consideration the power expansions near infinity [14, 16,17]. This approach is not simple to calculate special polynomials. This raises the question of whether the recursion formulae to find the special polynomials. (N ) (N ) The aim of this Letter is to introduce the special polynomials qn (z) and Rn (z) associated with the rational solutions of hierarchy (1.1) and to derive the recursion formulae to search for special polynomials. For N = 1 these polynomials were found in [1] as tau-functions of (1.1) at N = 1. The main result of this Letter is the differential-difference hierarchies    3 d2 ln Rm (m = 1, 2, . . .), qm+1 qm−1 = Rm z − HN (1.10) 2 dz2    d2 4 z − GN 12 2 ln qm (m = 1, 2, . . .) Rm+1 Rm−1 = qm (1.11) dz J1 = D 3 +

(N )

(N )

for the polynomials qn (z) and Rn (z)). These formulae allow us to obtain special polynomials associated with rational solutions (N ) (N ) (N ) (N ) of hierarchy (1.1) using q0 (z) = R0 (z) = 1 and q1 (z) = R1 (z) = z. In the case N = 1 formulae (1.10) and (1.11) were presented without derivation in [1] as equations for tau-functions of (1.1) at N = 1. This Letter is organized as follows. The general properties of the hierarchy considered are presented in Section 2. Some relations (N ) (N ) (N ) for the special polynomials qn (z) and Rn (z) and the differential-difference hierarchies for finding polynomials qn (z) and (N ) (N ) (N ) Rn (z) are given in Section 3. The special polynomials qn (z) and Rn (z) associated with the rational solutions of the first and the second members of hierarchy (1.1) are introduced in Sections 4, 5. Formulae for the rational solutions of members of the Schwarz–Sawada–Kotera, the Schwarz–Kaup–Kupershmidt, the Fordy–Gibbons, the Sawada–Kotera and the Kaup–Kupershmidt are presented in Sections 6–10. 2. Some properties of hierarchy (1.1) Let us briefly review some facts concerning equations (1.1) needed later. Suppose w(z) ≡ w(z; β) is a solution of (1.1). Assuming ϕzz w(z) = ϕz

(2.1)

and taking into account 2 1 ϕzzz 3ϕzz − wz − w 2 = {ϕ; z} = 2 ϕzz 2ϕz2

we have the equation in the form [4,5]

ϕ HN {ϕ; z} − z − (β − 1) = 0. ϕz

(2.2)

(2.3)

N.A. Kudryashov / Physics Letters A 372 (2008) 1945–1956

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Using w(z) = −

ψzz 2ψz

(2.4)

and taking into consideration −2wz − 2w 2 = {ψ; z} =

2 ψzzz 3ψzz − ψzz 2ψz2

(2.5)

we obtain the equation in the form [4,5]

ψ GN {ψ; z} − z − (β − 1) = 0. ψz

(2.6)

Constants of integration in (2.3) and (2.6) we choose zero because we can remove them using transformations of variables [4]. The Backlund transformations for solutions of hierarchy (1.1) can be written in the form [4,5] w(z, 2 − β) = w(z, β) −

2β − 2

(2.7)

HN [wz − 12 w 2 ] − z

and w(z, −1 − β) = w(z, β) −

2β + 1 . GN [−2wz − 2w 2 ] − z

(2.8)

These transformations allow us to find the rational solutions of hierarchy (1.1) [4,5]. These formulae can be presented in the form

GN −2wz − 2w 2 − z = − and

2β + 1 w(z; −1 − β) − w(z; β)

(2.9)

  1 2(β − 1) . HN wz − w 2 − z = − 2 w(z; 2 − β) − w(z; β)

(2.10)

At N = 1 transformations (2.7) and (2.8) were first found in [1]. The rational solutions of hierarchy (1.1) are classified in the following theorem. Theorem 2.1. Eq. (1.1) possesses rational solutions if and only if β ∈ Z\{1 ± 3k, k ∈ N ∪ 0}. They are unique and have the form (N )

 (q )2  d , w z; βn(1) = − ln n−1 (N ) dz Rn

(N )

  Rn−1 d w z; βn(2) = , ln (N dz (qn ) )2

(2.11)

where qn(N ) (z) and Rn(N ) (z) are polynomials, n ∈ N and 1 3n (−1)n 1 3n (−1)n − − , βn(2) = + − . 4 2 4 4 2 4 The only remaining rational solution is the trivial solution w(z; 0) = 0. βn(1) =

(2.12)

Proof. This theorem can be proved by the analogy with corresponding theorem of the works [15,18] of the case N = 1, making use of standard techniques that were applied extensively to the Painlevé equations by Gromak and Lukashevich [19,20]. However there are some technical issues of the general case of Eq. (1.1). The main part of the proof has to make use of the fact that the local Laurent expansions around poles of solutions of each Eq. (1.1) begin with a simple pole them. This follows from general properties of Painlevé analysis applied to the Fordy–Gibbons hierarchy, and can be proved directly using the recursions for the nonlinear operators GN and HN . The other important aspect of the proof is to consider rational solutions and their Laurent expansion around z = ∞. They show that such solutions will be regular and also vanishing at infinity. (N ) Rational solutions of (1.1) can be described with the help of two families of polynomials. The polynomials {qn (z)} we call the (N ) (1) (2) (N ) (N ) (N ) first family and {Rn (z)} the second. By pn (pn ) denote the degree of qn (z) (Rn (z)). The special polynomials qn (z) and (N ) Rn (z) can be defined as monic polynomials and each polynomial can be presented in the form (1)

qn(N ) (z) =

pn  k=0

(1,N )

(1)

An,k zpn

−k

,

(1,N )

An,0

= 1,

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N.A. Kudryashov / Physics Letters A 372 (2008) 1945–1956

Table 1 (j ) (j ) Values of βn and pn (j = 1, 2) 1

2

3

4

5

6

7

8

9

10

11

12

−1

−3

−4

−6

−7

−9

−10

−12

−13

−15

−16

−18

1

2

5

7

12

15

22

26

35

40

51

57

n (1) βn (1) pn (2) βn (2) pn

2

3

5

6

8

9

11

12

14

15

17

18

1

5

8

16

21

33

40

56

65

85

96

120

(2)

Rn(N ) (z) =

pn 

(2)

(2,N )

An,k zpn

−k

,

(2,N )

An,0

= 1.

(2.13)

k=0

The first non-trivial solutions for all members of (1.1) are w(z; −1) = 1/z, w(z; 2) = −2/z and w(z; 3) = −3/z. It follows from hierarchy (1.1) and from the Backlund transformations (2.7) and (2.8). Hence it can be set q0(N ) (z) = R0(N ) (z) = 1, q1(N ) (z) = (N ) R1 (z) = z. We also observe that degree of polynomials can be rewritten in terms 6n2 + 6n + 1 (−1)n (2n + 1) − 16 16 2 + 6n − 1 n (2n + 1) 6n (−1) pn(2) = + 8 8 pn(1) =

(j )

(j )

(n = 0, 1, . . .), (n = 0, 1, . . .).

The first few βn and pn are given in Table 1.

(2.14)

2

3. Some relations and differential-difference hierarchies for special polynomials (N )

(N )

In this section we derive some relations and the differential-difference hierarchies for special polynomials qn (z) and Rn (z). Later in this section when it does cause any contradiction the index N will be omitted. Theorem 3.1. Special polynomials qn (z) and Rn (z) satisfy the following relations   3m (−1)m 3 Dz qm+1 • qm−1 = + + Rm (m = 1, 2, . . .), 2 4 4   (−1)m 3 4 Dz Rm+1 • Rm−1 = 3m − (m = 1, 2, . . .) + q 2 2 m where Dz is the Hirota operator defined by    d d − . Dz f (z) • g(z) = f (z1 )g(z2 ) dz1 dz2 z1 =z2 =z

(3.1) (3.2)

(3.3)

Proof. Without loss of generality let us prove formulae (3.2). Comparison of Eqs. (2.3) and (2.9), (2.6) and (2.10) yields ψ=

Rm+1 , Rm−1

ϕ=

qm+2 . qm

(3.4)

Solutions of Eqs. (2.3) and (2.10) can be found taking into account the following recursion formulae [4–6] ψz =

ϕ4 , ϕz2

ψ

ϕz =

1

.

(3.5)

ψz2

Substituting (3.4) into the second equation (3.5) yields 4 2 (qm+2,z qm − qm+2 qm,z )2 (Rm+1,z Rm−1 − Rm+1 Rm−1,z ) = qm Rm+1 .

(3.6)

Assuming ψ=

Rm+1 , Rm−1

ϕ=

qm qm−2

(3.7)

N.A. Kudryashov / Physics Letters A 372 (2008) 1945–1956

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in the first equation (3.5) we get 2 4 (Rm+1,z Rm−1 − Rm+1 Rm−1 )(qm,z qm−2 − qm qm−2 )2 = Rm−1 qm .

(3.8)

From (3.6) and (3.8) we have (qm+2,z qm − qm+2 qm,z )2 (qm,z qm−2 − qm qm−2,z )2 = 2 2 Rm+1 Rm−1

(m = 2, 3, . . .).

(3.9)

Taking into account m = 2 and q2 (z) = z2 + C1 (where C1 is constant) in the right-hand side (3.9) we get q2,z q0 − q2 q0,z = 2R1 .

(3.10)

By the induction suppose at m = k there are qk,z qk−2 − qk qk−2,z = F1 (k)Rk−1

(3.11)

then from (3.9) the equality at m = k + 1 satisfies the form qk+2,z qk − qk+2 qk,z = F1 (k)Rk+1 .

(3.12)

Using (3.12) at k = m we find from (3.8) 4 Rm+1,z Rm−1 − Rm+1 Rm−1,z = F2 (m)qm .

(3.13)

Constants F1 (m) and F2 (m) are determined taking into account monic polynomials. Finally the relations for qn (z) and Rn (z) can be written in the form   3m (−1)m 3 + + Rm (m = 1, 2, . . .), qm+1,z qm−1 − qm+1 qm−1,z = (3.14) 2 4 4   (−1)m 3 4 (m = 1, 2, . . .). + q Rm+1,z Rm−1 − Rm+1 Rm+1,z = 3m − (3.15) 2 2 m Now if we formulate equations (3.14), and (3.15) using the Hirota operator we get Eqs. (3.1) and (3.2).

2

Theorem 3.2. The following relations are valid for the special polynomials qn (z) and Rn (z) 2 Dz2 Rm • qm−1 d2 ln R + =0 m 2 dz2 Rm qm−1

(m = 1, 2, . . .),

(3.16)

2 Dz2 Rm • qm+1 d2 ln R + =0 m 2 dz2 Rm qm+1

(m = 0, 1, . . .),

(3.17)

2 •R Dz2 qm d2 m−1 ln q + =0 m 2R dz2 2qm m−1

(m = 1, 2, . . .),

(3.18)

2 •R Dz2 qm d2 m+1 ln q + =0 m 2R dz2 2qm m+1

(m = 0, 1, . . .)

(3.19)

where Dz is the Hirota operator defined by (3.3). Proof. To prove theorem (3.2) we use the relations for solutions w(z; β), w(z; 2 − β) and w(z; −1 − β) in the form [5] 1 1 wz (z; β) − w 2 (z; β) = wz (z; 2 − β) − w 2 (z; 2 − β), 2 2 wz (z; β) + w 2 (z; β) = wz (z; −1 − β) + w2 (z; −1 − β).

(3.20) (3.21)

Assuming w(z; β) = +

Rm+1,z 2qm,z − , Rm+1 qm

w(z; 2 − β) =

Rm+1,z 2qm+2,z − Rm+1 qm+2

(3.22)

in equality (3.20) we have qm+2,zz qm,zz Rm+1,z qm+2,z qm,z Rm+1,z − − + = 0. qm+2 qm Rm+1 qm+2 qm Rm+1

(3.23)

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N.A. Kudryashov / Physics Letters A 372 (2008) 1945–1956

Adding

Rm+1,zz Rm+1

to (3.23), it can be obtained

2 2 Dz2 Rm+1 • qm+2 Dz2 Rm+1 • qm d2 d2 ln q + = ln q + . m+2 m 2 2 dz2 dz2 2Rm+1 qm 2Rm+1 qm+2

(3.24)

Substituting w(z; β) =

Rm−1,z 2qm,z − , Rm−1 qm

w(z; −1 − β) =

Rm+1,z 2qm,z − Rm+1 qm

(3.25)

in equality (3.21) yields Rm−1,zz 4qm,z Rm−1,z Rm+1,zz 4qm,z Rm+1,z − = − . Rm−1 qm Rm−1 Rm+1 qm Rm+1

(3.26)

From equality (3.26) we get 2 2 •R Dz2 qm Dz2 Rm−1 • qm m+1 = . 2 2R Rm−1 qm qm m+1

Adding

4qm,zz qm

(3.27)

to both parts of equality (3.26) yields

2 •R 2 •R Dz2 qm Dz2 qm d2 d2 m−1 m+1 2 ln q + = ln q + . m m 2R 2 2R dz2 2qm dz 2q m−1 m m+1

(3.28)

We have Dz2 q02 • R1 d2 ln q + = 0. 0 dz2 2q02 R1

(3.29)

By the induction assuming m = k we get Dz2 qk2 • Rk+1 d2 ln q + =0 k dz2 2qk2 Rk+1

(3.30)

then it can be obtained from (3.24) 2 Dz2 qk+2 • Rk+1 d2 ln q + = 0. k+2 2 2 dz 2qk+2 Rk+1

(3.31)

From (3.28) it follows 2 • Rk+3 Dz2 qk+2 d2 ln q + = 0. k+2 2 2 dz 2qk+2 Rk+3

This completes the proof.

(3.32)

2

Theorem 3.3. The following differential-difference hierarchies are valid for the special polynomials qn (z) and Rn (z)    3 d2 ln(R ) (m = 0, 1, . . .), qm+2 qm = Rm+1 z − HN m+1 2 dz2    d2 4 (m = 0, 1, . . .). z − GN 12 2 ln(qm+1 ) Rm+2 Rm = qm+1 dz

(3.33) (3.34)

Proof. Without loss of generality let us obtain the formula (3.33). From (3.19) we have the equality 2qm,z,z 2qm,z Rm+1,z Rm+1,zz − =− . qm qm Rm+1 2Rm+1

(3.35)

Assuming w(z; β) = −

d q2 ln m , dz Rm+1

w(z; 2 − β) =

d Rm+1 ln 2 dz qm+2

(3.36)

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and using (3.35) we get 2 1 Rm+1,zz 3Rm+1,z 2qm,z,z 2qm,z Rm+1,z 3 d 2 − − + = ln Rm+1 . wz − w 2 = 2 2 Rm+1 qm qm Rm+1 2 dz2 2Rm+1

(3.37)

Taking into account (3.1), (3.36) and (3.37) we obtain from Eq. (2.10)  HN

 3 d2 qm qm+2 ln Rm+1 − z = − . 2 2 dz Rm+1

This gives equality (3.33).

(3.38)

2

The differential-difference hierarchies (3.33) and (3.34) are useful to obtain the special polynomials qn (z) and Rn (z) associated with rational solutions of hierarchy (1.1). 4. Special polynomials associated with the first member of hierarchy (1.1) The expressions H1 [v] and G1 [u] take the form H1 [v] = vzz + 4v 2 ,

1 G1 [u] = uzz + u2 . 4

(4.1)

Substituting (4.1) into hierarchies (1.1) and (1.6) at N = 1 we have the fourth-order equation [1,18,21–23] wzzzz + 5wz wzz − 5w 2 wzz − 5wwz2 + w 5 − zw − β1 = 0. (1)

(1)

(1)

(1)

(4.2)

Assuming q0 (z) = R0 (z) = 1, q1 (z) = R1 (z) = z, and using the differential-difference hierarchies (3.33) and (3.34) we find (1) (1) (1) (1) the special polynomials qn (z) and Rn (z) at n  2. The first seven special polynomials qn (z) and Rn (z) are given in Tables 2 and 3. Table 2 (1) Polynomials qn (z) (1)

q0 = 1, (1)

q1 = z, (1)

q2 = z2 , (1)

q3 = z5 − 144, (1)

q4 = z2 (z5 − 504), (1)

q5 = z2 (z10 − 3168z5 − 3193344), (1)

q6 = (z15 − 6552z10 − 13208832z5 − 951035904), (1)

q7 = z2 (z20 − 22176z15 − 95001984z10 − 902898855936z5 + 303374015594496)

Table 3 (1) Polynomials Rn (z) (1)

R0 = 1, (1)

R1 = z, (1)

R2 = z5 + 36, (1)

R3 = z8 , (1)

R4 = z(z15 − 1152z10 + 1824768z5 + 131383296), (1)

R5 = z(z20 − 3276z15 + 6604416z10 + 3328625664z5 − 119830523904), (1)

R6 = z8 (z25 − 15840z20 + 63866880z15 + 708155965440z10 + 1922177762806726656), (1)

R7 = z40 − 29952z35 + 203793408z30 + 3066139754496z25 + 5234197284126720z20 + 36006491762989203456z15 − 3574462636834928197632z10 − 7206116675859215246426112z5 + 129710100165465874435670016

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N.A. Kudryashov / Physics Letters A 372 (2008) 1945–1956

Using the special polynomials qn(1) (z) and Rn(1) (z) we find the rational solutions of Eq. (4.2) by means of formulae (1.10). 3 5z4 (z5 + 216) 2 , w(z; 3) = − , w(z; 5) = − 5 w(z; 2) = − , z z (z + 36)(z5 − 144) 6(z5 + 336) w(z; 6) = − 5 , z(z − 504) 1 4 3(z5 − 24) w(z; −1) = , , w(z; −4) = , w(z; −3) = z z z(z5 + 36) 20 15 10 5 6(z − 576z − 912384z − 459841536z − 3153199104) . w(z; −6) = z(z5 − 144)(z15 − 1152z10 + 1824768z5 + 131383296) Some of these rational solutions were found before in works [1,3,4,15,18].

(4.3)

(4.4)

5. Special polynomials associated with the second member of hierarchy (1.1) From (1.2) and (1.7) we have 32 3 v , 3 3 3 1 G2 [u] = uzzzz + uuzz + u2z + u3 . 2 4 6 Substituting (5.1) and (5.2) into (1.1) and (1.6) at N = 2 yields the sixth-order equation H2 [v] = vzzzz + 12vvzz + 6vz2 +

2 wzzzzzz + 7wz wzzzz − 20wz2 wzz − 21wwzz + 14wzz wzzz − 7w 3 wzzzz −

(5.1) (5.2)

28 wwz2 + 14w 4 wzz + 28w 3 wz2 − 28wwz wzzz 3

4 − 14w 2 wz wzz − w 7 − zw − β2 = 0. 3

(5.3)

Taking q0(2) (z) = R0(2) (z) = 1 and q1(2) (z) = R1(2) (z) = z from the differential-difference hierarchies (3.33) and (3.34) we get the (2) (2) special polynomials qn (z) and Rn (z) at n  2. The first seven special polynomials are given in Tables 4 and 5. Table 4 (2) Polynomials qn (z) (2)

q0 = 1, (2)

q1 = z, (2)

q2 = z 2 , (2)

q3 = z5 , (2)

q4 = z7 + 4320, (2)

q5 = z5 (z7 + 95040), (2)

q6 = z(z14 + 280800z7 + 4852224000), (2)

q7 = z2 (z21 + 1615680z14 − 383885568000z7 + 3316771307520000)

Table 5 (2) Polynomials Rn (z) (2)

R0 = 1, (2)

R1 = z, (2)

R2 = z5 , (2)

R3 = z(z7 − 1728), (2)

R4 = z16 , (2)

R5 = z21 + 43200z14 − 2426112000z7 + 2620200960000, (2)

R6 = z5 (z28 + 646272z21 + 307108454400z14 − 14593793753088000z7 − 126090378026680320000), (2)

R7 = z5 (z35 + 1684800z28 + 1610938368000z21 − 632202087628800000z14 − 46523097626443776000000z7 + 803919126984948449280000000)

N.A. Kudryashov / Physics Letters A 372 (2008) 1945–1956

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Using the special polynomials qn(2) (z) and Rn(2) (z) we get the rational solutions of Eq. (5.3) in the form 2 3 5 w(z; 2) = − , w(z; 3) = − , w(z; 5) = − , z z z 14 7 6(z − 9504z + 1244160) 8(z7 − 71280) w(z; 6) = − , w(z; 8) = − , (z7 − 1728)z(z7 + 4320) z(z7 + 95040)

(5.4)

3 1 w(z; −3) = , w(z; −1) = , z z 7 6 4(z + 1296) , w(z; −6) = , w(z; −4) = z(z7 − 1728) z w(z; −7) =

7(z21 + 12960z14 + 2799360000z7 − 15721205760000)z6 . (z7 + 4320)(z21 + 43200z14 − 2426112000z7 + 2620200960000) (2)

(5.5)

(2)

Special polynomials qn (z) and Rn (z) for these solutions were taken from Tables 4 and 5. 6. Rational solutions of members of the Schwarz–Sawada–Kotera hierarchy (N )

(N )

Using the polynomials qn (z) and Rn (z) associated with rational solutions of hierarchy (1.1) it is easy to find rational solutions of some partial differential hierarchies. Let us demonstrate our approach to obtain rational solutions of members of the Schwarz– Sawada–Kotera hierarchy. This hierarchy can be written in the form [3–6]

ϕt + ϕx HN {ϕ, x} = 0. (6.1) (N )

Rational solutions of each equation (6.1) can be found taking into account polynomials qn (z)   rn(1) qn(N ) (z) ϕ(x, t) = (2N + 1)t 2N+1 (N ) , qn−2 (z)

z=

x 1

((2N + 1)t) 2N+1

(6.2)

.

(1)

Here the degree rn is rn(1) =

6n − 3 − (−1)n . 2

(6.3)

Remark 1. Hierarchy (6.1) is invariant under the Moebius group [6] ϕ=

aω + b , cω + d

{ϕ; x} = {ω; x}.

(6.4)

Using this property one can search for other rational solutions of each equation of hierarchy (6.1). 7. Rational solutions of members of the Schwarz–Kaup–Kupershmidt hierarchy Consider the Schwarz–Kaup–Kupershmidt hierarchy that takes the form [3–6]

ψt + ψx GN {ψ, x} = 0.

(7.1)

Rational solutions of each member of this hierarchy can be obtained in the form   rn(2) Rn(N ) (z) , ψ(x, t) = (2N + 1)t 2N+1 (N ) Rn−2 (z)

z=

x 1

((2N + 1)t) 2N+1

(7.2)

,

(2)

where degree rn is determined by the formula rn(2) =

6n − 3 + (−1)n . 2

(7.3) (N )

Remark 2. Rational solutions (7.2) can be expressed through special polynomials qn (z) if we use relation (3.1).

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N.A. Kudryashov / Physics Letters A 372 (2008) 1945–1956

8. Rational solutions of members of the Fordy–Gibbons hierarchy Let us present formulae to search for rational solutions of members of the Fordy–Gibbons hierarchy that are the modified Kaup–Kupershmidt hierarchy and the modified Sawada–Kotera hierarchy. The Fordy–Gibbons hierarchy can be written in the form [3–6,10]     ∂ 1 ∂ ωt + (8.1) + ω HN ωx − ω2 = 0. ∂x ∂x 2 Hierarchy (8.1) can be presented by means of the nonlinear operator GN [u] as well  

∂ 1 ∂ ωt − − ω GN −2ωx − 2ω2 = 0. ∂x 2 ∂x

(8.2)

Taking into account the self-similar solutions of each equation of hierarchies (8.1) and (8.2) ω(x, t) =

1 ((2N + 1)t)

1 2N+1

w(z),

z=

x

(8.3)

1

((2N + 1)t) 2N+1

we have nonlinear ordinary differential equations (1.1) and (1.6). We obtain that rational solutions of each equation of hierarchies (N ) (N ) (8.1) and (8.2) can be found using special polynomials qn (z) and Rn (z) by the formula (N )

ω(x, t) = −

(q (z))2 ∂ ln n−1 , (N ) ∂x Rn (z)

z=

x 1

((2N + 1)t) 2N+1

(N )

ω(x, t) =

Rn−1 (z) ∂ , ln (N ∂x (qn ) (z))2

z=

x 1

((2N + 1)t) 2N+1

,

(8.4)

.

(N )

Remark 3. Rational solutions (8.4) of members of the Fordy–Gibbons hierarchy can be written using special polynomials qn (z) by relation (3.1). 9. Rational solutions of members of the Sawada–Kotera hierarchy The Sawada–Kotera hierarchy has the form [3–6,9,10]  3  ∂ ∂ ∂v + 2v + HN [v] = 0. vt + ∂x ∂x ∂x 3

(9.1)

Taking into account self-similar solutions of each equation of hierarchy (9.1) in the form v(x, t) =

1 ((2N + 1)t)

2 2N+1

V (z),

z=

x 1

((2N + 1)t) 2N+1

(9.2)

,

we obtain the nonlinear ordinary differential equations as  3  d d + 2V + Vz HN [V ] − zVz − 2V = 0. dz dz3

(9.3) (N )

Solutions of each member of hierarchy (9.3) is found taking into consideration the special polynomials Rn (z) by the formula V (z) =

3 d2 ln Rn(N ) (z). 2 dz2

(9.4) (N )

Substituting (9.4) into Eq. (9.3) leads to equation for the polynomials Rn (z). Certainly we can rewrite rational solutions (9.2) and (N ) (9.4) using special polynomials qn (z) by formula (3.1). 10. Rational solutions of members of the Kaup–Kupershmidt hierarchy The Kaup–Kupershmidt hierarchy has the form [3–6,11]  3  ∂ ∂ ∂u + 2u ut + + GN [u] = 0. ∂x ∂x ∂x 3

(10.1)

N.A. Kudryashov / Physics Letters A 372 (2008) 1945–1956

1955

Taking into account self-similar solutions u(x, t) =

1 ((2N + 1)t)

2 2N+1

U (z),

z=

x 1

((2N + 1)t) 2N+1

(10.2)

,

we obtain from hierarchy (10.1) the nonlinear ordinary differential equations as   3 d d + 2U + Uz GN [U ] − zUz − 2U = 0. dz dz3

(10.3) (N )

Rational solutions of each equation of hierarchy (10.3) can be found using special polynomials qn (z) by the formula U (z) = 12

d2 ln qn(N ) (z). dz2

(10.4)

Substituting (10.4) Eq. (10.3) we obtain equation for the polynomials qn(N ) (z). It needs to study this equation in future as example of integrable equation with the polynomial solution. 11. Conclusion We have introduced the special polynomials associated with the rational solutions of hierarchy (1.1). This hierarchy arises if we look for the special solutions taking into account scaling reductions of members for the Fordy–Gibbons, the Sawada–Kotera and the Kaup–Kupershmidt hierarchies. We have found some recursion relations for the special polynomials of members of the hierarchy studied. Using these relations we have obtained the differential-difference hierarchies to obtain the special polynomials of the hierarchy considered. These recursion formulae were used for recursively finding the special polynomials in terms of two previously known two polynomials. The special polynomials of the first and the second members of hierarchy (1.1) were presented. For the N th member of the ODE hierarchy, these special polynomials generalize the polynomials for N = 1, which were presented in [1,15]. They are similar to the Yablonskii–Vorob’ev polynomials [24–27], to the Okamoto polynomials [28,29] and to the Umemura polynomials [30] associated with the rational solutions of the Painlevé equations. Using the special polynomials found the rational solutions for the first and the second members of hierarchy (1.1) were given. It was shown that rational solutions of the Schwarz– Sawada–Kotera, the Schwarz–Kaup–Kupershmidt, the Fordy–Gibbons, the Sawada–Kotera and the Kaup–Kupershmidt hierarchies (N ) (N ) can be expressed through special polynomials qn (z) and Rn (z). We can observe that all the mentioned above rational solutions (N ) can be expressed through special polynomials qn (z). We hope that the recursion relations (3.1), (3.2), (3.16)–(3.19) and the differential-difference hierarchies (3.33) and (3.34) will be useful in proving various properties of the hierarchy (1.1). For every member of the hierarchy (1.1) the recursion relations in Section 3 can actually be generalized to all solutions, so they will be useful for proving properties of the tau-functions for general solution (and not just the special polynomials). Acknowledgements We are grateful to the referees for helpful comments and suggestions. This work was supported by the International Science and Technology Center under Project B 1213. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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