Chaos, Solitons and Fractals 33 (2007) 1652–1657

www.elsevier.com/locate/chaos

The Painleve integrability of two parameterizednonlinear evolution equations using symbolic computation

Gui-qiong Xu

Department of Information Management, College of Business and Management, Shanghai University, Shanghai 201800, China

Accepted 7 March 2006

Communicated by Professor M.S. El Naschie

Abstract

An algorithm is presented to prove the Painleve integrability of parameterized nonlinear evolution equations suchthat one can filter out Painleve integrable models from nonlinear equations with general forms. Then two well knownnonlinear models with physical interests illustrate the effectiveness of this algorithm. Some new results are reported forthe first time.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

The question of the integrability of nonlinear evolution equations has been a subject of intense investigations inrecent years. Among the various approaches followed to study the integrability of nonlinear evolution equations,the Painleve analysis has proved to be one of the most successful and widely applied tools [1–5]. If a nonlinear equationpasses the Painleve test, we call it a Painleve integrable model. The Painleve integrable models are prime candidates forbeing completely integrable, thus the singular manifold method of Weiss always allows the recovery of the Lax pair andDarboux tranformation (DT), and so also the Backlund transformation, from a truncated Painleve expansion [6–10].

It is very tedious to study whether a given PDE passes the Painleve test, thus the application of computer algebra canbe very helpful in such calculations. Various researchers have developed computer programs for the Painleve test ofnonlinear equations [11–14]. It is much more difficult to prove the Painleve property for nonlinear models with param-eter coefficients(pcNPDEs), which occur in many branches of physics. To our knowledge, all existed packages cannotdeal with such nonlinear models with general forms.

The aim of this paper is to improve the WTC algorithm such that one can analyze under what parameters con-straints the given parameterized nonlinear models pass the Painleve test. In the following Section 2, we outline the mainidea of the Painleve test algorithm for pcNPDEs. In Section 3 two examples are tested by using the improved algorithm,and some new results are obtained. Then a simple discussion is given in the final section.

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.03.014

E-mail addresses: xugq@staff.shu.edu.cn, xuguiqiong@yahoo.com

G.-q. Xu / Chaos, Solitons and Fractals 33 (2007) 1652–1657 1653

2. The algorithm

Consider a system of nonlinear PDEs with parameter coefficients, say in two independent variables x and t

H sðuðiÞ;uðiÞx ;uðiÞt ;u

ðiÞxt ;u

ðiÞxx ; . . .Þ ¼ 0; i;s ¼ 1; . . . ;m; ð1Þ

where u(i) = u(i)(x, t) (i = 1, . . . ,m) are dependent variables, the subscripts denote partial derivatives, Hs(s = 1, . . . ,m) arepolynomials about u(i) and their derivatives, maybe after a preliminary change of variables.

Eq. (1) are said to pass the Painleve test, if all solutions of Eq. (1) can be expressed as Laurent series,

uðiÞ ¼X1j¼0

uðiÞj /ðx;tÞðjþaiÞ; i ¼ 1; . . . ;m; ð2Þ

with sufficient number of arbitrary functions as the order of (1), uðiÞj are analytic functions, ai are negative integers. Inorder to simplify the involved computations, we apply the Kruskal’s gauge for the singular manifold: /(x, t) = x � w(t),uðiÞj ¼ uðiÞj ðtÞ. The algorithm of the Painleve test for Eq. (1) is made of following three steps:

2.1. Step a. Leading order analysis

To determine leading order exponents ai and coefficients uðiÞ0 (i = 1, . . . ,m), letting

uðiÞ ¼ uðiÞ0 /ai ; i ¼ 1; . . . ;m

and inserting them into (1), then balancing the minimal power terms, one can obtain all possible ðai;uðiÞ0 Þ for all param-

eters constraints.For example, in the case of the coupled Schrodinger–KdV system [15],

iut þ uxx ¼ uv; vt þ pvvx þ qvxxx ¼ ðjuj2Þx; ð3Þ

with p, q being arbitrary real parameters. Now, by applying a new variable u* = w (where the asterisk represents thecomplex conjugate), Eq. (3) are reduced to a three coupled system,

iut þ uxx ¼ uv; � iwt þ wxx ¼ wv; vt þ pvvx þ qvxxx ¼ ðuwÞx: ð4Þ

Substituting u = u0/a, v = v0/

b, w = w0/c into Eq. (4), the following two cases are found to be possible from bal-

ancing the highest order derivative terms with the nonlinear terms:

Case ðiÞ : a ¼ b ¼ c ¼ �2; Case ðiiÞ : a ¼ c ¼ �1;b ¼ �2:

In case (i), the leading coefficients u0, v0 and w0 satisfy

u0ð�6þ v0Þ ¼ 0; w0ð�6þ v0Þ ¼ 0; � 2pv20 � 24qv0 þ 4u0w0 ¼ 0; ð5Þ

in case (ii), we obtain

�2v0ðpv0 þ 12qÞ ¼ 0; u0ð�2þ v0Þ ¼ 0; w0ð�2þ v0Þ ¼ 0: ð6Þ

Solving Eqs. (5) and (6) with respect to u0, v0, w0, p, q, we have

Case ðiÞ : v0 ¼ 6; w0 ¼18ðp þ 2qÞ

u0

; Case ðiiÞ : v0 ¼ 2; p ¼ �6q:

2.2. Step b. To find the resonances

Substituting the following truncated Painleve expansion:

uðiÞ ¼ uðiÞ0 /ai þ uðiÞr /aiþr; i ¼ 1; . . . ;m

into (1) and collecting the terms with the lowest powers of /, we get

QðrÞ � ðuð1Þ; . . . ;uðmÞÞT ¼ 0;

where Q is an m · m matrix, whose elements depend on r. The resonances are the roots of det[Q(r)] = 0.

1654 G.-q. Xu / Chaos, Solitons and Fractals 33 (2007) 1652–1657

If all resonances are found to be integers, then one has to proceed the third step. Otherwise, we should investigateunder what parameters constraints such that all resonances are integers. Then for each set of parameters constraints, thePainleve test will be performed from the Step a.

As for Eq. (4), the resonance equation can be obtained by inserting u = u0/a + ur/

r+a, v = v0/b + vr/

r+b,w = w0/

c + wr/r+c into Eq. (4) and collecting all terms with the lowest powers. In case (i), the resonance equation is

�ru20ðr � 4Þðr � 5Þðr � 6Þðr þ 1Þðqr2 � 5qr þ 6p þ 12qÞ ¼ 0;

which on solving yields four distinct cases where all resonance are integers, namely, if p = �q, the resonances occur at�1,0,2,3,4,5,6; when p = �4q/3, the resonances occur at �1,0,1,4,4,5,6; if p = �2q, the resonances occur at�1,0,0,4,5,5,6; if p = �3q, the resonances occur at �1,�1,0,4,5,6,6.

While for case (ii), the resonance equation reads,

�r2ðr � 3Þ2qðr � 4Þðr � 6Þðr þ 1Þ ¼ 0;

so the resonances occur at r = �1,4,6,0,0,3,3.

2.3. Step c. To verify compatibility conditions

The compatibility conditions should be verified for every non-negative integer resonance. To this end, we insert thetruncated expansions

uðiÞ ¼Xrmax

j¼0

uðiÞj /jþai ; i ¼ 1; . . . ;m

into (1), where rmax is the largest resonance. If there are compatibility conditions which can not be satisfied, one shouldneed investigate under what parameters constraints such that the compatibility conditions are satisfied identically. Sub-sequently, the Painleve test of the original equations can be performed for all possible parameters constraints one byone.

According to the above algorithm, one can perform the Painleve test for pcNPDEs and obtain a Painleve classifi-cation of nonlinear models with general forms. Once some new Painleve integrable models are found, other interestingintegrable properties can be further investigated.

3. Applications

Example 1. Let us first consider the parameterized fifth order KdV equation [16,17],

ut ¼ u5x þ uu3x þ auxu2x þ bu2ux þ hu3x þ suux; ukx ¼ oku=oxk ; ð7Þ

where u = u(x, t) 2 C1, a, b, h and s are arbitrary real constants.

In order to make the calculations simpler, we choose / = x � w(t). Looking at the leading order behavior and sup-posing that u = u0/

s, we obtain s = � 2, and

u0 ¼3ð�a� 2� dÞ

b; ð8Þ

where d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaþ 2Þ2 � 40b

q. The corresponding resonance equation reads

ðr þ 1Þðr � 6Þðbr3 � 15br2 þ ð86bþ 3d� 3a� 6Þr � 12d� 6daþ 6a2 þ 24a� 240bþ 24Þ ¼ 0: ð9Þ

Due to (9), two resonances lies in the positions n = �1, 6. Denoting the positions of other resonances as n1, n2, n3, wefind from (9) that

n1 þ n2 þ n3 ¼ 15; n1n2n3b ¼ 12dþ 6da� 6a2 � 24aþ 240b� 24;

bðn1n2 þ n1n3 þ n2n3Þ ¼ 86bþ 3d� 3a� 6:ð10Þ

We require that the considered branch is principle, i.e. four resonances lie in nonnegative positions. Taking into accountthe possible values of n1, n2 and n3, we have to study 14 distinct cases. The resonances of principle branch are listed inTable 1 for all possible parameters constraints where all resonance are integers.

Table 1Parameters constraints of coefficients where all resonances are integers

Case Parameters constraints Resonances Case Parameters constraints Resonances

1 a = 0, b ¼ 110 r = �1,0,2,6,13 8 a ¼ 17

26, b ¼ 29169 r = �1,1,3,6,11

2 a ¼ 25, b ¼ 18

125 r = �1,0,3,6,12 9 a ¼ 98, b ¼ 15

64 r = �1,1,4,6,10

3 a ¼ 52, b ¼ 1

5 r = �1,3,5,6,7 10 a ¼ 2918, b ¼ 25

81 r = �1,1,5,6,9

4 a ¼ 67, b ¼ 10

49 r = �1,0,4,6,11 11 a = 1, b ¼ 15 r = �1,2,3,6,10

5 a ¼ 43, b ¼ 5

18 r = �1,0,5,6,10 12 a ¼ 32, b ¼ 1

4 r = �1,2,4,6,9

6 a = 2, b ¼ 25 r = �1,0,6,7,8 13 a = 2, b ¼ 3

10 r = �1,2,5,6,8

7 a ¼ 14, b ¼ 1

8 r = �1,1,2,6,12 14 a = 2, b ¼ 29 r = �1,3,4,6,8

G.-q. Xu / Chaos, Solitons and Fractals 33 (2007) 1652–1657 1655

Subsequently, the Painleve test should be performed from the Step a for all cases listed in Table 1. In the cases 1–2 and4–6, Eq. (7) fails the Painleve test because the compatibility condition at r = 0 turn out to be not satisfied. In the cases 7–10and 12, Eq. (7) fails the Painleve test because there are complex resonance points in the secondary branch. In the case 14,the resonances in secondary branch occur at�1,�5,6,8,10. After tedious computations it is shown that the compatibilityconditions at r = 8 for principle branch and r = 10 for secondary branch cannot hold for any parameters constraints.

In the case of a ¼ 52, b ¼ 1

5, the result of leading order analysis is given by (8), the resonances occur at �1,3,5,6,7 and

�1,�7,6,10,12. For the first branch, u3, u5, u6 are arbitrary functions with respect to t. For the second branch, u6, u10

are arbitrary functions. However, the compatibility condition at r = 7 for the first branch and r = 12 for the secondbranch reduce to,

u5ð5s� 2hÞ ¼ 0;

ð5s� 2hÞð�50s2u6 � 5hsu6 þ 435600u10 þ h2u6 þ 45u6wtÞ ¼ 0;

it is obvious to see that the above conditions become identities if h ¼ 5s2. When setting a ¼ 5

2, b ¼ 1

5and h ¼ 5s

2, Eq. (8) is

then proved to pass the Painleve test.In the case of a = 1, b ¼ 1

5, the result of leading order analysis is given by (8), the resonances occur at �1,2,3,6,10

and �1,�2,5,6,12. For the first branch, u2, u3, u6 are arbitrary functions with respect to t. For the second branch, u5, u6

are arbitrary functions. However, the compatibility condition at r = 10 for the first branch and r = 12 for the secondbranch reduce to,

ð2h� 5sÞð75s2u22 þ 600su6 þ 30su3

2 � 60su23 þ 150su2wt � 24u2u2

3 þ 30u22wt

� 75w2t � 600hu6 þ 25u3;t þ 3u4

2 � 360u2u6Þ ¼ 0;

ð2h� 5sÞð84u25 þ 5u5;tÞ ¼ 0;

it is obvious to see that the above conditions become identities if h ¼ 5s2. When setting a = 1, b ¼ 1

5and h ¼ 5s

2, Eq. (8) is

then proved to pass the Painleve test.In the case of a = 2, b ¼ 3

10, the result of leading order analysis is given by (8), the resonances occur at �1,2,5,6,8

and �1,�3,6,8,10, For the first branch, u2, u5, u6 are arbitrary functions with respect to t. For the second branch, u6, u8

are arbitrary functions. However, the compatibility condition at r = 8 for the first branch and r = 10 for the secondbranch reduce to,

ð5s� 3hÞu6 ¼ 0;

ð5s� 3hÞð9h4 � 60sh3 þ 240h2wt � 50s2h2 � 800hswt þ 500s3h� 2000s2wt

þ 525s4 þ 56448000u8 þ 1600w2t Þ ¼ 0;

it is obvious to see that the above conditions become identities if h ¼ 5s3. When setting a = 2, b ¼ 3

10and h ¼ 5s

3, Eq. (8) is

then proved to pass the Painleve test.It is shown that the fifth order KdV Eq. (7) passes the Painleve test for integrability in three distinct cases of its

coefficients,

ut ¼ u5x þ uu3x þ uxu2x þ1

5u2ux þ

5s2

u3x þ suux;

ut ¼ u5x þ uu3x þ5

2uxu2x þ

1

5u2ux þ

5s2

u3x þ suux;

ut ¼ u5x þ uu3x þ 2uxu2x þ3

10u2ux þ

5s3

u3x þ suux;

1656 G.-q. Xu / Chaos, Solitons and Fractals 33 (2007) 1652–1657

the C-integrability and S-integrability of the above three models have been proved in Refs. [16,17]. Although we do notgive a theoretical proof, it is easy to see that Eq. (7) possesses the C-integrability, S-integrability and P-integrabilityunder the same three parameters constraints.

Example 2. Next we consider the higher order nonlinear Schrodinger equation [18],

Ex ¼ iðb1Ett þ b2EjEj2Þ þ �ðb3Ettt þ b4ðEjEj2Þt þ b5EðjEj2ÞtÞ; ð11Þ

which describes the propagation of optical pulse of very short duration in a fibre. In Eq. (11), i ¼ffiffiffiffiffiffiffi�1p

, b1—b5 corre-spond to the effect of group velocity dispersion (GVD), self-phase modulation (SMP), third-order dispersion (TOD),self-steepening (SS), and simulated Raman scattering (SRS). This equation has been investigated by means of the Pain-leve analysis. Here we restudy Eq. (11) and give more complete parameters options where the equation passes the Pain-leve test.

To apply the Painleve analysis, we express E* = F (where the asterisk represents the complex conjugate). By applyingnew variable, Eq. (11) is reduced to the following coupled system,

Ex � iðb1Ett þ b2E2F Þ � �ðb3Ettt þ b6E2F t þ b7EFEtÞ ¼ 0;

F x þ iðb1F tt þ b2EF 2Þ � �ðb3F ttt þ b6F 2Et þ b7EFF tÞ ¼ 0:ð12Þ

Looking at the leading order behavior, we substitute E ¼ E0/a1 , F ¼ F 0/

a2 into Eq. (12), upon balancing terms, weobtain

a1 ¼ a2 ¼ �1; F 0 ¼ �6b3

ðb6 þ b7ÞE0

; ð13Þ

where E0 is an arbitrary function with respect to x. For the simplicity of computation, we suppose /(t,x) = t � w(x).Subsequently the resonance points of Eq. (12) are determined by

E20�

2b23ðb6 þ b7Þ

2rðr � 3Þðr � 4Þðr þ 1Þ½ðb6 þ b7Þr2 � 6ðb6 þ b7Þr þ 17b6 þ 5b7� ¼ 0;

solving it we obtain four distinct parameters options where all resonances are integers.Case a. If b6 = 0, the resonances occur at r = �1,0,1,3,4,5.In this case, set b6 = 0 in Eq. (12), the result of leading order analysis is given by (13). The compatibility condition at

resonance r = 1 reduces to

�4iF 0ð3b2b3 � b7b1Þ ¼ 0;

which is satisfied identically If b1b7 = 3b2b3. When b6 = 0 and b1b7 = 3b2b3, Eq. (12) is proven to pass the Painleve test,which is coincided with the parameters constraint (31) given by Ref. [18] when one takes b7 = 1.

Case b. If b7 = 3b6, the resonances occur at r = �1,0,2,3,4,4.In this case, the result of leading order analysis is given by (13). E0 and F2 are arbitrary functions with respect to x,

and the compatibility condition r = 3 reduces to

E0ð3b2b3 � 2b1b6Þð18b1b6b2b3 � 9b23b

22 þ 6b3wxb

26�þ 24b3E0F 2b

36�

2 � 8b21b

26Þ

b6

¼ 0;

it is evident that the above equation holds only if 3b2b3 = 2b1b6. When we take b7 = 3b6 and 3b2b3 = 2b1b6 and per-form the Painleve test again, Eq. (12) is proven to pass the Painleve test. Obviously, in Ref. [18], the authors missed oneparameter constraint of coefficients.

Case c. If b7 = �2b6, the resonances occur at r = �1,�1,0,3,4,7. In this case, the result of leading order analysis isalso given by (13). The compatibility conditions at resonances r = 0,3,4 turn out to be satisfied identically. However,the condition at resonance r = 7 reduces to,

42b3ðb1b6 þ b2b3Þð576iE40F 2

3�6b2

3b86 þ 216ib7

6b23E3

0�5F 3;x þ 72ib7

6b23E2

0F 3�5E0;x � 432b6

6b23E3

0F 3�4b2wx

� 27ib66b

23E0�

4E0;xx þ 36ib66b

23E2

0;x�4 þ 192b6

6E30F 3�

3b2b21b3 þ 720b5

6b23E3

0F 3�3b2

2b1 � 81b56b

23b2E2

0�3wxx

þ 54b56b

23wxE0�

3b2E0;x � 81ib62E2

0b43 � 24b5

6E0b2E0;x�2b2

1b3 þ 432b46E3

0F 3�3b3

2b33 � 90b4

6b23E0b

22E0;x�

2b1

� 54b36E0b

32�

2E0;xb33 þ 270ib3

6b32E2

0wx�b1b23 � 120ib3

6b32E2

0b31b3 � 81ib4

6b23b

22w

2x�

2E20 þ 72ib4

6b22E2

0wx�b21b3

� 16ib46b

22E2

0b41 þ 162ib2

6b42E2

0wx�b33 � 297ib2

6b42E2

0b21b

23 � 270ib5

2E20b1b

33b6Þ ¼ 0. ð14Þ

Since E0, F3 and w are arbitrary functions, from (14) we have two possibilities: either b3 = 0 or b1b6 = �b2b3.

G.-q. Xu / Chaos, Solitons and Fractals 33 (2007) 1652–1657 1657

In the case of b7 = �2b6 and b3 = 0, Eq. (12) fails the Painleve test in the leading order analysis. In the other case ofb7 = �2b6 and b1b6 = �b2b3, though all compatibility conditions at non-negative integer resonances are satisfied iden-tically, Eq. (12) still fail the Painleve test because its solutions cannot admit sufficient number of arbitrary functions.

Case d. If b7 ¼ � 17b6

5, the resonances occur at r = �1,0,0,3,4,6.

Similarly, the result of leading order analysis is given by (13). From (13) it is shown that there is only one arbitraryfunction at resonance r = 0, so Eq. (12) fail the Painleve test in the case of b7 ¼ � 17b6

5.

In summary, from the above analysis we can conclude that Eq. (12) and/or Eq. (11) pass the Painleve test for twodistinct parameters constraints of coefficients: b6 = 0, b1b7 = 3b2b3 and b7 = 3b6, 2b1b6 = 3b2b3.

4. Summary

In this paper, an algorithm of the Painleve test for nonlinear evolution equations with parameter coefficients is pre-sented. The Painleve integrability of the parameterized fifth order KdV equation is proved for the first time here. Whilefor the higher order nonlinear Schrodinger equation, its Painleve integrability is restudied here and some new results areobtained.

Acknowledgement

The author would like to thank Prof. Sen-yue Lou for his valuable suggestions and discussions. This work has beensupported by the National Natural Science Foundations of China (No. 10547123).

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