The Painlevé integrability of two parameterized nonlinear evolution equations using symbolic computation

Chaos, Solitons and Fractals 33 (2007) 1652–1657 www.elsevier.com/locate/chaos

The Painleve´ integrability of two parameterized nonlinear 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 such that one can filter out Painleve´ integrable models from nonlinear equations with general forms. Then two well known nonlinear models with physical interests illustrate the effectiveness of this algorithm. Some new results are reported for the 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 in recent 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 equation passes the Painleve´ test, we call it a Painleve´ integrable model. The Painleve´ integrable models are prime candidates for being completely integrable, thus the singular manifold method of Weiss always allows the recovery of the Lax pair and Darboux tranformation (DT), and so also the Ba¨cklund 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 can be very helpful in such calculations. Various researchers have developed computer programs for the Painleve´ test of nonlinear equations [11–14]. It is much more difficult to prove the Painleve´ property for nonlinear models with parameter coefficients(pcNPDEs), which occur in many branches of physics. To our knowledge, all existed packages cannot deal 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 constraints the given parameterized nonlinear models pass the Painleve´ test. In the following Section 2, we outline the main idea 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.

E-mail addresses: xugq@staff.shu.edu.cn, [email protected] 0960-0779/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.014

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2. The algorithm Consider a system of nonlinear PDEs with parameter coefficients, say in two independent variables x and t ðiÞ ðiÞ ðiÞ H s ðuðiÞ ;uðiÞ x ;ut ;uxt ;uxx ; . . .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) are polynomials 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 ¼

1 X

ðiÞ

uj /ðx;tÞðjþai Þ ;

i ¼ 1; . . . ;m;

ð2Þ

j¼0 ðiÞ

with sufficient number of arbitrary functions as the order of (1), uj are analytic functions, ai are negative integers. In order to simplify the involved computations, we apply the Kruskal’s gauge for the singular manifold: /(x, t) = x  w(t), ðiÞ ðiÞ uj ¼ uj ðtÞ. The algorithm of the Painleve´ test for Eq. (1) is made of following three steps: 2.1. Step a. Leading order analysis ðiÞ

To determine leading order exponents ai and coefficients u0 (i = 1, . . . , m), letting ðiÞ

uðiÞ ¼ u0 /ai ;

i ¼ 1; . . . ;m ðiÞ

and inserting them into (1), then balancing the minimal power terms, one can obtain all possible ðai ;u0 Þ for all parameters constraints. For example, in the case of the coupled Schro¨dinger–KdV system [15], iut þ uxx ¼ uv;

vt þ pvvx þ qvxxx ¼ ðjuj2 Þx ;

ð3Þ

with p, q being arbitrary real parameters. Now, by applying a new variable u* = w (where the asterisk represents the complex 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 balancing 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 u0 ð6 þ v0 Þ ¼ 0;

w0 ð6 þ v0 Þ ¼ 0;

 2pv20  24qv0 þ 4u0 w0 ¼ 0;

ð5Þ

in case (ii), we obtain 2v0 ðpv0 þ 12qÞ ¼ 0;

u0 ð2 þ v0 Þ ¼ 0;

w0 ð2 þ v0 Þ ¼ 0:

Solving Eqs. (5) and (6) with respect to u0, v0, w0, p, q, we have Case ðiÞ : v0 ¼ 6; w0 ¼

18ðp 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: ðiÞ

ai þr ; uðiÞ ¼ u0 /ai þ uðiÞ r /

i ¼ 1; . . . ;m

into (1) and collecting the terms with the lowest powers of /, we get QðrÞ  ðuð1Þ ; . . . ;uðmÞ ÞT ¼ 0; where Q is an m · m matrix, whose elements depend on r. The resonances are the roots of det[Q(r)] = 0.

ð6Þ

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If all resonances are found to be integers, then one has to proceed the third step. Otherwise, we should investigate under what parameters constraints such that all resonances are integers. Then for each set of parameters constraints, the Painleve´ 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 ru20 ðr  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, r2 ðr  3Þ2 qðr  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 the truncated expansions uðiÞ ¼

rmax X

ðiÞ

uj /jþai ;

i ¼ 1; . . . ;m

j¼0

into (1), where rmax is the largest resonance. If there are compatibility conditions which can not be satisfied, one should need investigate under what parameters constraints such that the compatibility conditions are satisfied identically. Subsequently, the Painleve´ test of the original equations can be performed for all possible parameters constraints one by one. According to the above algorithm, one can perform the Painleve´ test for pcNPDEs and obtain a Painleve´ classification of nonlinear models with general forms. Once some new Painleve´ integrable models are found, other interesting integrable 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 aux u2x þ bu2 ux þ hu3x þ suux ;

ukx ¼ ok u=oxk ;

ð7Þ

1

where u = u(x, t) 2 C , 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 supposing that u = u0/s, we obtain s =  2, and 3ða  2  dÞ ; b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where d ¼ ða þ 2Þ2  40b. The corresponding resonance equation reads u0 ¼

ð8Þ

ðr þ 1Þðr  6Þðbr3  15br2 þ ð86b þ 3d  3a  6Þr  12d  6da þ 6a2 þ 24a  240b þ 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, we find from (9) that n1 þ n2 þ n3 ¼ 15; n1 n2 n3 b ¼ 12d þ 6da  6a2  24a þ 240b  24; bðn1 n2 þ n1 n3 þ n2 n3 Þ ¼ 86b þ 3d  3a  6:

ð10Þ

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

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Table 1 Parameters constraints of coefficients where all resonances are integers Case 1 2 3 4 5 6 7

Parameters constraints a = 0, a ¼ 25, a ¼ 52, a ¼ 67, a ¼ 43, a = 2, a ¼ 14,

1 b ¼ 10 18 b ¼ 125 b ¼ 15 b ¼ 10 49 5 b ¼ 18 2 b¼5 b ¼ 18

Resonances r = 1, 0, 2, 6, 13 r = 1, 0, 3, 6, 12 r = 1, 3, 5, 6, 7 r = 1, 0, 4, 6, 11 r = 1, 0, 5, 6, 10 r = 1, 0, 6, 7, 8 r = 1, 1, 2, 6, 12

Case 8 9 10 11 12 13 14

Parameters constraints a¼ a¼ a¼ a= a¼ a= a=

17 29 26, b ¼ 169 9 15 8, b ¼ 64 29 25 18, b ¼ 81 1, b ¼ 15 3 1 2, b ¼ 4 3 2, b ¼ 10 2 2, b ¼ 9

Resonances r = 1, 1, 3, 6, 11 r = 1, 1, 4, 6, 10 r = 1, 1, 5, 6, 9 r = 1, 2, 3, 6, 10 r = 1, 2, 4, 6, 9 r = 1, 2, 5, 6, 8 r = 1, 3, 4, 6, 8

Subsequently, the Painleve´ test should be performed from the Step a for all cases listed in Table 1. In the cases 1–2 and 4–6, Eq. (7) fails the Painleve´ test because the compatibility condition at r = 0 turn out to be not satisfied. In the cases 7–10 and 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 compatibility conditions 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 15, 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 second branch reduce to, u5 ð5s  2hÞ ¼ 0; ð5s  2hÞð50s2 u6  5hsu6 þ 435600u10 þ h2 u6 þ 45u6 wt Þ ¼ 0; it is obvious to see that the above conditions become identities if h ¼ 5s2 . When setting a ¼ 52, b ¼ 15 and h ¼ 5s2 , Eq. (8) is then proved to pass the Painleve´ test. In the case of a = 1, b ¼ 15, 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 second branch reduce to, ð2h  5sÞð75s2 u22 þ 600su6 þ 30su32  60su23 þ 150su2 wt  24u2 u23 þ 30u22 wt  75w2t  600hu6 þ 25u3;t þ 3u42  360u2 u6 Þ ¼ 0; ð2h  5sÞð84u25 þ 5u5;t Þ ¼ 0; it is obvious to see that the above conditions become identities if h ¼ 5s2 . When setting a = 1, b ¼ 15 and h ¼ 5s2 , Eq. (8) is then proved to pass the Painleve´ test. In the case of a = 2, b ¼ 103 , 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 second branch reduce to, ð5s  3hÞu6 ¼ 0; ð5s  3hÞð9h4  60sh3 þ 240h2 wt  50s2 h2  800hswt þ 500s3 h  2000s2 wt 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 103 and h ¼ 5s3 , 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, 1 5s ut ¼ u5x þ uu3x þ ux u2x þ u2 ux þ u3x þ suux ; 5 2 5 1 2 5s ut ¼ u5x þ uu3x þ ux u2x þ u ux þ u3x þ suux ; 2 5 2 3 2 5s ut ¼ u5x þ uu3x þ 2ux u2x þ u ux þ u3x þ suux ; 10 3

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the C-integrability and S-integrability of the above three models have been proved in Refs. [16,17]. Although we do not give a theoretical proof, it is easy to see that Eq. (7) possesses the C-integrability, S-integrability and P-integrability under the same three parameters constraints. Example 2. Next we consider the higher order nonlinear Schro¨dinger equation [18], Ex ¼ iðb1 Ett þ b2 EjEj2 Þ þ ðb3 Ettt þ b4 ðEjEj2 Þt þ b5 EðjEj2 Þt Þ;

ð11Þ pffiffiffiffiffiffiffi which describes the propagation of optical pulse of very short duration in a fibre. In Eq. (11), i ¼ 1, b1—b5 correspond 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 Painleve´ analysis. Here we restudy Eq. (11) and give more complete parameters options where the equation passes the Painleve´ test. To apply the Painleve´ analysis, we express E* = F (where the asterisk represents the complex conjugate). By applying new variable, Eq. (11) is reduced to the following coupled system, Ex  iðb1 Ett þ b2 E2 F Þ  ðb3 Ettt þ b6 E2 F t þ b7 EFEt Þ ¼ 0; F x þ iðb1 F tt þ b2 EF 2 Þ  ðb3 F ttt þ b6 F 2 Et þ b7 EFF t Þ 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, we obtain a1 ¼ a2 ¼ 1;

F0 ¼ 

6b3 ; ðb6 þ b7 ÞE0

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 2 b23 ðb6 þ b7 Þ2 rðr  3Þðr  4Þðr þ 1Þ½ðb6 þ b7 Þr2  6ðb6 þ b7 Þr 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 0 ð3b2 b3  b7 b1 Þ 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 E0 ð3b2 b3  2b1 b6 Þð18b1 b6 b2 b3  9b23 b22 þ 6b3 wx b26  þ 24b3 E0 F 2 b36 2  8b21 b26 Þ ¼ 0; b6 it is evident that the above equation holds only if 3b2b3 = 2b1b6. When we take b7 = 3b6 and 3b2b3 = 2b1b6 and perform the Painleve´ test again, Eq. (12) is proven to pass the Painleve´ test. Obviously, in Ref. [18], the authors missed one parameter 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 is also 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, 42b3 ðb1 b6 þ b2 b3 Þð576iE40 F 23 6 b23 b86 þ 216ib76 b23 E30 5 F 3;x þ 72ib76 b23 E20 F 3 5 E0;x  432b66 b23 E30 F 3 4 b2 wx  27ib66 b23 E0 4 E0;xx þ 36ib66 b23 E20;x 4 þ 192b66 E30 F 3 3 b2 b21 b3 þ 720b56 b23 E30 F 3 3 b22 b1  81b56 b23 b2 E20 3 wxx þ 54b56 b23 wx E0 3 b2 E0;x  81ib62 E20 b43  24b56 E0 b2 E0;x 2 b21 b3 þ 432b46 E30 F 3 3 b32 b33  90b46 b23 E0 b22 E0;x 2 b1  54b36 E0 b32 2 E0;x b33 þ 270ib36 b32 E20 wx b1 b23  120ib36 b32 E20 b31 b3  81ib46 b23 b22 w2x 2 E20 þ 72ib46 b22 E20 wx b21 b3  16ib46 b22 E20 b41 þ 162ib26 b42 E20 wx b33  297ib26 b42 E20 b21 b23  270ib52 E20 b1 b33 b6 Þ ¼ 0. Since E0, F3 and w are arbitrary functions, from (14) we have two possibilities: either b3 = 0 or b1b6 = b2b3.

ð14Þ

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In the case of b7 = 2b6 and b3 = 0, Eq. (12) fails the Painleve´ test in the leading order analysis. In the other case of b7 = 2b6 and b1b6 = b2b3, though all compatibility conditions at non-negative integer resonances are satisfied identically, Eq. (12) still fail the Painleve´ test because its solutions cannot admit sufficient number of arbitrary functions. 6 Case d. If b7 ¼  17b , the resonances occur at r = 1, 0, 0, 3, 4, 6. 5 Similarly, the result of leading order analysis is given by (13). From (13) it is shown that there is only one arbitrary 6 function at resonance r = 0, so Eq. (12) fail the Painleve´ test in the case of b7 ¼  17b . 5 In summary, from the above analysis we can conclude that Eq. (12) and/or Eq. (11) pass the Painleve´ test for two distinct 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 presented. The Painleve´ integrability of the parameterized fifth order KdV equation is proved for the first time here. While for the higher order nonlinear Schro¨dinger equation, its Painleve´ integrability is restudied here and some new results are obtained.

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

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