- Email: [email protected]
On properties of new parameterized 5th-order nonlinear evolution equation
Chaos, Solitons and Fractals 21 (2004) 1145–1152 www.elsevier.com/locate/chaos
On properties of new parameterized 5th-order nonlinear evolution equation Ruo-Xia Yao a
a,b
, Chang-Zheng Qu c, Zhi-Bin Li
a,*
Department of Computer Science, East China Normal University, Shanghai 200062, PR China b Department of Computer Science, Weinan Teachers College, Shaanxi 714000, PR China c Department of Mathematics, Northwest University, Xi’an 710069, PR China Accepted 8 December 2003
Abstract By means of direct algebraic methods, generalized symmetries and conservation laws are obtained for a new parameterized 5th-order nonlinear evolution equation under the different parameters constraints. The multi-soliton solutions are also constructed by combining some nonlinear transformations and the simplified Hirota approach with the assistance of symbolic manipulation system Maple. 2004 Elsevier Ltd. All rights reserved.
1. Introduction As is well known that some partial differential equations, such as the Korteweg–de Vries equation and the vortex filament equation [1], are completely integrable systems; in particular, they posses infinite many sequences of higher order symmetries and conservation laws. Some other equations, such as the 5th-order Ito equation and the lubrication equation, posses finite numbers of local conservation laws which can be exploited in the analysis of the equation [2]. The aim of this paper is first to analysis under what compatibility conditions such that the following parameterized 5thorder KdV-type equation with some additional terms possesses polynomial type higher order symmetries and conservation laws (PClaws), which reads ut ¼ u5x þ uu3x þ aux u2x þ bu2 ux þ hu3x þ cuux KðuÞ;
ukx ¼ ok u=xk
ð1:1Þ
with a, b, h, c reals and u ¼ uðx; tÞ 2 C 1 . After the integrable cases of (1.1) are filtered, the one-, two- and three-soliton solutions are constructed respectively by using some nonlinear transformations. It is noticeable that (1.1) is not uniform in rank [3]. To construct the PClaws of (1.1) by using the Maple procedure CONSLAW presented in [4] we should, first of all, treat all parameters appearing in (1.1) as extra variables with (unknown) weights. Then from (1.1), xðvÞ denotes the weight of v [3,4], we arrive at xðo=otÞ þ xðuÞ ¼ 5 þ xðuÞ ¼ 3 þ 2xðuÞ ¼ 3 þ xðaÞ þ 2xðuÞ ¼ 1 þ xðbÞ þ 3xðuÞ ¼ 3 þ xðhÞ þ xðuÞ ¼ 1 þ xðcÞ þ 2xðuÞ;
ð1:2Þ
which on solving yields xðo=otÞ ¼ 7;
xðuÞ ¼ xðhÞ ¼ xðcÞ ¼ 2;
xðaÞ ¼ xðbÞ ¼ 0:
*
Corresponding author. E-mail addresses: [email protected] (R.-X. Yao), [email protected] (Z.-B. Li).
0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.078
ð1:3Þ
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Once the weights of the variables and the differentials in hand, one can construct the polynomial type generalized symmetries and conservation laws for (1.1) immediately by using the following direct algebraic methods. Definition 1. A function r ¼ rðt; x; u; ux ; u2x ; . . .Þ, is a generalized symmetry of (1.1), if and only if rt ¼ K 0 r
ð1:4Þ
for every solution of (1.1), where K 0 r ¼ oo Kðu þ rÞj¼0 ¼ lim!0 Kðt; x; ðu þ rÞ; ðu þ rÞx ; ðu þ rÞ2x ; . . .Þ. Definition 2. A conservation law for (1.1) is a divergence expression qt þ Jx ¼ 0;
ð1:5Þ
which vanishes for all solutions of (1.1), where q and J are respectively conserved density and flux and depend only on u and uix ’s.
2. Generalized symmetries of the 5th-order nonlinear evolution equation Before computing the generalized symmetries of (1.1), we describe the algorithm briefly. • Determine the general form of r for a fixed rank R, i.e. find the building blocks in u and Uix such that the monomials just have the same rank R. • Linear combination of the above monomials with constant coefficients ci ’s directly yields the general form of r. • Compute the derivative of r with respect to t, i.e. rt , and eliminate all the t-derivatives of u using ut ¼ KðuÞ, like ut ; utx , and then compute oo Kðu þ rÞj¼o . Thus from (1.4), the ci ’s can be determined. Now, we list some results of the generalized symmetries of (1.1) as follows: (A) RankðrÞ ¼ 3: a 1st-order symmetry reads r ¼ ux , for which without any parameter constraint. (B) RankðrÞ ¼ 5: a 3rd-order symmetry reads r ¼ u3x þ 83 cux þ 35 uux , for which the parameters constraints are 3 3 ð2:1Þ a ¼ 2; b ¼ ; c ¼ h : 10 5 (C) RankðrÞ ¼ 7: a 5th-order symmetry reads r ¼ u5x þ uu3x þ 49 h2 ux þ 103 u2 ux þ 2ux u2x , for which the parameters con25 straints are just the same as (2.1). (D) RankðrÞ ¼ 9: three 7th-order generalized symmetries are obtained under the different parameters constraints. The first one is 49 2 272 3 7 21 7 7 7 8 4 c uux þ c ux þ u3 ux þ 7u2x u3x þ ux u4x þ uu5x þ u3x þ u2 u3x þ cu5x þ cu2 ux 9 27 50 5 5 10 10 3 5 16 14 245 2 8 c u3x þ cuu3x þ cux u2x þ uux u2x þ 3 5 27 3
r ¼ u7x þ
for which the parameters constraints are 3 3 a ¼ 2; b ¼ ; c ¼ h : 10 5 The second one is 7 203 3 28 3 21 14 7 7 14 42 7 c ux þ u ux þ u2x u3x þ ux u4x þ uu5x þ u3x þ u2 u3x þ uux u2x þ c2 u3x r ¼ u7x þ cux u2x þ 2 8 375 5 5 5 25 25 25 4 14 21 21 2 7 2 þ cu5x þ cuu3x þ cu ux þ c uux 5 10 50 10
ð2:2Þ
R.-X. Yao et al. / Chaos, Solitons and Fractals 21 (2004) 1145–1152
for which the parameters constraints are 1 2 a ¼ 1; b ¼ ; c ¼ h : 5 5
1147
ð2:3Þ
The third one is 21 2 203 3 7 7 21 14 21 7 42 49 c uux þ c ux þ cuu3x þ cu2 ux þ c2 u3x þ u2 u3x þ cu5x þ u3x þ u2x u3x þ ux u4x 10 8 2 10 4 25 5 10 5 10 28 3 7 63 161 þ u ux þ uu5x þ uux u2x þ cux u2x 375 5 25 20
r ¼ u7x þ
for which the parameters constraints are 5 1 2 a¼ ; b¼ ; c¼ h : 2 5 5
ð2:4Þ
In [5], Sanders and Wang have proved the S-integrability for an equation of order 5 needs a symmetry of order 7 whence the S-integrability of (1.1). The above results shows that we filter out three sets of parameters constraints which guaranteeing the S-integrability of (1.1). Then in Sections 3 and 4 we will discuss the C-integrability and search the multi-soliton solutions of (1.1) under the parameters constraints (2.2)–(2.4) respectively.
3. PClaws of the 5th-order nonlinear evolution equation Prom the case (D) in Section 2 we see that (1.1) includes three cases that are of particular importance. By using the same procedure presented in [4], we compute the polynomial type conservation laws under those three cases directly and list some results below. (I) Case a ¼ 2, b ¼ 3=10, c ¼ 3h=5. (A) In the case of RankðqÞ ¼ 2, the conserved density–flux pair reads ðuÞt þ
1 2 1 3 ux þ u4x þ hu2x þ uu2x þ u3 þ hu2 2 10 10
¼ 0:
ð3:1Þ
x
(B) In the case of RankðqÞ ¼ 4, the conserved density–flux pair reads 1 3 1 3 ðu2 þ uhÞt þ hu2x þ hu4x þ h2 u2x þ 2u2 u2x þ u4 þ 2uu4x þ 3huu2x þ hu3 þ h2 u2 2ux u3x þ u22x 2 20 2 10 x ¼ 0:
ð3:2Þ
(C) In the case of RankðqÞ ¼ 6, the conserved density–flux pair reads 1 1 ð5u2x þ u3 þ uh2 þ u2 hÞt þ 3u2x u2 þ h2 u3 12hux u3x h2 u2x þ 3u2 u4x þ 2huu4x þ h2 u4x þ h3 u2x þ 8uu22x 2 2 9 3 3 þ 5u2 hu2x þ 6hu22 x þ u5 þ 3uh2 u2x þ 3u3 u2x þ h3 u2 16ux uu3x 10ux u5x 6huu2x þ hu4 14u2x u2x 50 10 5 þ 10u2x u4x 5u23x
¼ 0:
ð3:3Þ
x
(D) In the case of RankðqÞ ¼ 8, the conserved density, from now on the conserved fluxes no longer listed due to that containing more than 30 terms, reads q ¼ u4 þ u2 h2 þ uh3 þ u3 h 5hu2x 20uu2x þ 20u22x :
ð3:4Þ
(E) In the case of RankðqÞ ¼ 10, the conserved density reads q ¼ 100uu22x þ 20hu22x 5h2 u2x
500 2 u þ u5 20huu2x 50u2x u2 þ hu4 þ h2 u3 þ h3 u2 þ h4 u: 7 3x
ð3:5Þ
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(F) In the case of RankðqÞ ¼ 12, the conserved density reads 500 2 5000 2 250 4 10000 3 hu3x þ 20h2 u22x þ u u þ u þ h5 u þ hu5 þ h2 u4 þ h3 u3 7 21 4x 3 x 21 2x 3000 2 þ h4 u2 þ u6 þ 300u2 u22x u u 100u3 u2x 5h3 u2x þ 100huu22x : 7 3x
q ¼ 20h2 uu2x 50hu2x u2
ð3:6Þ
(G) In the case of RankðqÞ ¼ 14, the conserved density reads q¼
5000 2 5000 2 500 2 2 25000 10000 3 25000 2 uu4x þ 20u22x h3 þ hu4x u h þ 3500u2x u22x u2x u23x þ u h u 3 21 7 3x 3 21 2x 33 5x 10000 3 1750 4 250 4 uu2x 1500u2 u23x þ u4 h3 uux u h 5u2x h4 þ 700u3 u22x 175u4 u2x þ uh6 þ u3 h4 þ 3 3 3 x 3000 uhu23x þ 300u2 hu22x þ 100uh2 u22x : þ u5 h2 þ u7 þ u2 h5 þ u6 h 50u2 h2 u2x 100u3 hu2x 20uh3 u2x 7
ð3:7Þ
In the case of RankðqÞ ¼ 16 and 18 the conserved densities contain 42 and 64 terms respectively and not listed here. (II) Case a ¼ 1, b ¼ 1=5, c ¼ 2h=5. (A) In the case of RankðqÞ ¼ 2, the conserved density–flux pair reads 1 1 ðuÞt þ u4x þ hu2x þ uu2x þ u3 þ u2 h ¼ 0: ð3:8Þ 15 5 x (B) In the case of RankðqÞ ¼ 4, the conserved density–flux pair reads 1 3 1 2 ¼ 0: ðuhÞt þ h u4x þ hu2x þ uu2x þ u þ u h 15 5 x
ð3:9Þ
Bear in mind that h is in fact an arbitrary constant, hence (3.9) is the same as (3.8). (C) In the case of RankðqÞ ¼ 6, the conserved density reads q ¼ 15u2x þ u3 þ 3u2 h þ uh2 :
ð3:10Þ
(D) In the case of RankðqÞ ¼ 8, the conserved density reads q ¼ u4 45uu2x þ u3 h 3u2 h2 þ uh3 þ 75u22x :
ð3:11Þ
(E) In the case of RankðqÞ ¼ 10, the conserved density reads 1 q ¼ 25hu22x 15uhu2x þ u2 h3 þ u3 h2 þ u4 h þ uh4 10u2x h2 : 3
ð3:12Þ
(F) In the case of RankðqÞ ¼ 12, the conserved density reads 1 2 6 4 5 8 8 17 7 16 u u h þ u3 h3 þ u4 h2 þ uh5 þ u2 h4 u2 u22x þ u4x þ uu23x uu22x h q ¼ u3 u2x 3 1125 375 25 5 30 2 5 73 2 2 7 2 122 2 2 8 3 5 2 2 2 2 3 u h 10ux h u2x u4x : uux h þ u ux h þ u3x h þ 5 2 5 2x 3 2
ð3:13Þ
(G) In the case of RankðqÞ ¼ 14, the conserved density reads q¼
1125 2 34787 2 3 115 2 1011 2 2 900 2 2 2000 107 3 625 2 4075 3 uu þ u h hu þ u h þ uu u2x u23x þ u h u þ uu 286 4x 1430 2x 286 4x 286 3x 143 x 2x 143 858 2x 286 5x 858 2x
725 2 2 6829 4 3 105 4 41 4 665 3 2 35 4 2 1109 5 2 u u3x þ uh uux þ ux h 10u2x h4 þ u u2x u ux þ uh6 þ u3 h4 uh 286 21450 143 165 858 286 107250
þ
1 7 1 6 38 3 2 10404 3 2 643 2 2 4611 2 2 u þ u2 h5 u h þ u2 h2 u2x þ u hux uh ux þ uhu23x u hu2x uh u2x : 2574 2750 429 715 1430 1430 ð3:14Þ
R.-X. Yao et al. / Chaos, Solitons and Fractals 21 (2004) 1145–1152
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In the case of RankðqÞ ¼ 16, 18 and 20 the conserved densities contain 28, 64 and 96 terms respectively and not listed here. (III) Case a ¼ 5=2, b ¼ 1=5, c ¼ 2h=5. (A) In the case of RankðqÞ ¼ 2, the conserved density–flux pair reads 3 2 1 1 ux þ u4x þ hu2x þ uu2x þ u3 þ hu2 ¼ 0: ðuÞt þ ð3:15Þ 4 15 5 x (B) In the case of RankðqÞ ¼ 4, the conserved density–flux pair reads 3 2 1 3 1 2 2 2 ¼ 0: hu þ hu4x þ h u2x þ huu2x þ hu þ h u ðuhÞt þ 4 x 15 5 x
ð3:16Þ
(C) In the case of RankðqÞ ¼ 6, the conserved density–flux pair reads 15 27 1 27 3 3 u2x þ uh2 þ 3u2 h þ u3 þ 3huu2x hux u3x þ 9u2 hu2x þ 3u3 u2x þ h3 u2 þ uu22x þ u5 u2x u2 4 2 5 4 25 2 t 27 2 27 15 9 2 2 13 2 3 51 2 3 2 2 þ hu2x ux uu3x ux u5x þ 3u u4x þ h u2x þ h u4x h ux þ h u ux u2x 4 2 2 4 15 4 3 4 15 15 2 2 þ 7uh u2x þ 6huu4x þ hu þ u2x u4x u3x ¼ 0: ð3:17Þ 5 2 4 x (D) In the case of RankðqÞ ¼ 8, the conserved density reads 25 1 25 15 q ¼ u22x þ u4 hu2x þ uh3 þ u2 h2 þ u3 h uu2x : 4 3 4 2 (E) In the case of RankðqÞ ¼ 10, the conserved density reads q ¼ h3 u2
25 2 2 1 25 15 h ux þ h2 u3 þ hu4 þ hu22x huu2x þ uh4 : 4 3 4 2
ð3:18Þ
ð3:19Þ
(F) In the case of RankðqÞ ¼ 12, the conserved density reads q¼
4 6 8 5 4 283 2 4 25 2 185 3 31 4 25 2 1 u þ u h þ u3 h3 u4 h2 þ uh5 þ u h þ u4x þ u u uu þ u2 u2 3825 1275 85 85 68 204 2x 408 x 34 3x 2 2x 25 23 227 3 2 7 7 h ux þ huu22x þ h2 uu2x hu2 u2x u3 u2x : hu23x h2 u22x 34 34 68 17 51
ð3:20Þ
(G) In the case of RankðpÞ ¼ 14, the conserved density reads q¼
244 5 2 4187 3 2 11170 2 2 4575 2 173195 3 68385 2 2 12455 2 2 uh h u2x h u3x hu u u uu þ u u 1995 1064 931 1862 4x 5586 2x 3724 x 2x 931 3x 141 4 2 2162 3 2 1927 4 4 6 1175 815 3 1823 4 3 þ u ux þ u2x h4 þ hu4x u u2x þ uu þ u hþ u2x u23x þ hu uh 133 399 532 x 855 14 2793 2x 3990 14100 2 1285 2428 2 2 66 3 2 1044 2 2 2 4765 3 2 188 7 uu4x þ huu23x þ h uu2x þ hu2 u22x u hux u h ux þ uh ux u 931 931 133 133 133 532 41895 34 5875 2 þ u2 h5 þ u þ u3 h4 þ uh6 : 5 931 5x
ð3:21Þ
We see that in every level (1.1) possesses PClaws under the three parameters constraints and in view of the conclusion in Section 2 we state that, although we do not give a theoretical proof, (1.1) possesses infinitely many PClaws and is Cintegrability. 4. Multi-soliton solutions of the 5th-order nonlinear evolution equation This section is devoted to construct the multi-soliton solutions to (1.1) under the above three sets of parameters constraints. (I) Case a ¼ 1, b ¼ 1=5, c ¼ 2h=5. To obtain the soliton solutions of (1.1) under this case, we introduce a nonlinear transformation of dependent variable [6], uðx; tÞ ¼ r þ p½lnð/ðx; tÞÞxx ;
ð4:1Þ
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where pð6¼ 0Þ and r are real parameters to be determined, which allows us to transform the corresponding equation into a 9th-degree homogeneous equation, denoted by Q. Then, we seek a solution of type /ðx; tÞ ¼ 1 þ expðnÞ;
n ¼ kx þ wt;
ð4:2Þ
where kð6¼ 0Þ and w are parameters to be determined. Substituting of (4.2) into Q and equating the coefficients of different powers of expðnÞ to zero, one obtains a nonlinear system, which on solving can determine the possible relations between the parameters r, p, k and w. Solving the nonlinear system of algebraic equations we arrive at 2 1 p ¼ 30; w ¼ hk 3 rhk k 5 r2 k rk 3 ; k ¼ k; r ¼ r; h ¼ h : ð4:3Þ 5 5 Substitution of (4.3) into (4.2) and (4.1) yields uðx; tÞ ¼ r þ 30
k 2 expðkx þ wtÞ ½1 þ expðkx þ wtÞ2
ð4:4Þ
;
where w ¼ hk 3 25 rhk k 5 15 r2 k rk 3 , and k, r, h are arbitrary reals, which can also be written as uðx; tÞ ¼ r þ
15 sech2 ðkx þ wtÞ: 2
ð4:5Þ
If setting r ¼ 0, (4.5) is in fact a one-soliton solution of (1.1) under this case and can also be obtained using the methods presented in [7–14]. Alternatively, if we seek a solution of type /ðx; tÞ ¼ s þ expðnÞ þ expð2nÞ;
n ¼ kx þ wt;
ð4:6Þ
then we get uðx; tÞ ¼ 5k 2 h þ 30
k 2 ½expðkx þ wtÞs þ expðkx þ wtÞ expð2kx þ 2wtÞ þ 4 expð2kx þ 2wtÞs ½s þ expðkx þ wtÞ þ expð2kx þ 2wtÞ2
;
ð4:7Þ
where w ¼ 15 h2 k k 5 , and h, s, k free. Now, to construct the two-soliton solution, one can use the nonlinear transformation uðx; tÞ ¼ 30½lnð/ðx; tÞÞxx ;
ð4:8Þ
which reduces the equation into a 7th-degree homogeneous one ð30/2x;t þ 30/7x þ 30h/5x Þ/6 þ ð60/x /xt þ 270/5x /2x 30/2x /t 210/6x /x þ 60h/3x /2x 150/4x /3x 150h/4x /x Þ/5 þ ð600/23x /x þ 240h/3x /2x þ 60/2x /t þ 360/5x /2x 900/4x /x /2x 180/22x /x Þ/4 ¼ 0: The linear and nonlinear operators are thus defined as 3 o o7 o5 þ ; þ h L ¼ 30 ox2 ot ox7 ox5 N1 ðf ; gÞ ¼ 60fx gxt þ 270f5x g2x 30f2x gt 210f6x gx þ 60hf3x g2x 150f4x g3x 150hf4x gx ; N2 ðf ; g; hÞ ¼ 600f3x g3x hx þ 240hf3x gx hx þ 60fx gx ht þ 360f5x gx hx 900f4x gx h2x 180hf2x g2x hx :
ð4:9Þ
ð4:10Þ
To solve (4.9), one can use the -expansion method [15,16]. Suppose that the solution of (4.9) is of the form /ðx; tÞ ¼ 1 þ f1 þ f2 2 þ f3 3 þ f4 4 þ ;
ð4:11Þ
where fi (i ¼ 1; 2; 3; . . .) to be determined later, be a small parameter (for simplicity we may take ¼ 1). Substituting (4.11) into (4.9), collecting all terms with the same order of together, and setting each coefficient of k (k ¼ 1; 2; 3; . . .) to zero give a hierarchy of equations for fk , 1 : Lf1 ¼ 0; 2
ð4:12Þ
: Lf2 ¼ N1 ðf1 ; f1 Þ;
ð4:13Þ
3 : Lf3 ¼ 6f1 Lf2 5f1 N1 ðf1 ; f1 Þ N1 ðf1 ; f2 Þ N1 ðf2 ; f1 Þ N2 ðf1 ; f1 ; f1 Þ;
ð4:14Þ
and so on to be solved.
R.-X. Yao et al. / Chaos, Solitons and Fractals 21 (2004) 1145–1152
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To obtain the one-soliton solution, suppose that f1 ¼ expðnÞ, with n ¼ kx þ wt. Computation of (4.12) gives the dispersion law w ¼ k 3 ðh þ k 2 Þ. It is easy to check that the right-hand side of (4.13) is equal to zero, and then we can set fn ¼ 0 for n P 2. Therefore the series (4.11) truncates. One-soliton solution is thus got and is just the same as (4.5) with r ¼ 0. To obtain the two-soliton solution, one can take f1 ¼ expðn1 Þ þ expðn2 Þ;
ni ¼ ki x þ wi t;
i ¼ 1; 2:
ð4:15Þ
It is straightforward to check that fn ¼ 0 for n P 3. The series (4.11) therefore truncates. By means of -expansion method, and after long tedious computation one obtain / ¼ 1 þ expðn1 Þ þ expðn2 Þ þ a expðn1 þ n2 Þ;
ð4:16Þ
where wi ¼ ki3 ðh þ ki2 Þ;
a¼
ðk1 k2 Þ2 ð3h þ 5k12 5k1 k2 þ 5k22 Þ ðk1 þ k2 Þ2 ð3h þ 5k12 þ 5k1 k2 þ 5k22 Þ
:
The explicit two-soliton solution uðx; tÞ can be got by substituting (4.16) into (4.8). To obtain the three-soliton solution, we take f1 ¼ expðn1 Þ þ expðn2 Þ þ expðn3 Þ;
ni ¼ ki x þ wi t;
i ¼ 1–3:
ð4:17Þ
By checking we know that fn ¼ 0 for n P 4. The series (4.11) therefore truncates. Likewise, we get /¼1þ
3 X
X
expðni Þ þ
aij expðni þ nj Þ þ a12 a13 a23 expðn1 þ n2 þ n3 Þ
ð4:18Þ
1 6 i
i¼1
with wi ¼ ki3 ðh þ ki2 Þ;
a¼
ðki kj Þ2 ð3h þ 5ki2 5ki kj þ 5kj2 Þ ðki þ kj Þ2 ð3h þ 5ki2 þ 5ki kj þ 5kj2 Þ
ð1 6 i < j 6 3Þ:
Then, the three-soliton solution is obtained from (4.8). (II) Case a ¼ 2, b ¼ 3=10, c ¼ 3h=5. Similarly, the one-, two- and three-soliton solutions can be generated by the following nonlinear transformation of dependent variable, uðx; tÞ ¼ 20½lnð/ðx; tÞÞxx ;
ð4:19Þ
along with the following expressions, / ¼ 1 þ expðkx k 3 ðh þ k 2 ÞtÞ;
ð4:20Þ
/ ¼ 1 þ expðn1 Þ þ expðn2 Þ þ a12 expðn1 þ n2 Þ;
ð4:21Þ
/¼1þ
3 X
X
expðni Þ þ
i¼1
aij expðni þ nj Þ þ a12 a13 a23 expðn1 þ n2 þ n3 Þ
ð4:22Þ
1 6 i
with ni ¼ ki x þ ki3 ðh þ ki2 Þt;
aij ¼
ðki ki Þ2 ðki þ kj Þ2
ð1 6 i < j 6 3Þ:
(III) Case a ¼ 5=2, b ¼ 1=5, c ¼ 2h=5. Similarly, the one- and two-soliton solutions can be generated by the following nonlinear transformation of dependent variable, uðx; tÞ ¼ 15½lnð/ðx; tÞÞxx ;
ð4:23Þ
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along with the following expressions, 5k 2 þ 4h expð2nÞ; n ¼ kx k 3 ðh þ k 2 Þt; 16ð5k 2 þ hÞ 2 2 X X 4h þ 5ki2 expðni Þ þ expð2ni Þ þ a12 expðn1 þ n2 Þ /¼1þ 16ðh þ 5ki2 Þ i¼1 i¼1 / ¼ 1 þ expðnÞ þ
ð4:24Þ
þ b12 expð2h1 þ h2 Þ þ b21 expðh1 þ 2h2 Þ þ B expð2n1 þ 2n2 Þ with a12 ¼ bij ¼
10k14 þ 6k12 h 5k22 k12 þ 10k24 þ 6k22 h 2ðk1 þ k2 Þ2 ð5k22 þ 5k1 k2 þ 5k12 þ 3hÞ
;
ðki kj Þ2 ð4h þ 5ki2 Þð5ki2 5ki kj þ 5kj2 þ 3hÞ 16ðki þ kj Þ2 ðh þ 5ki2 Þð5ki2 þ 5ki kj þ 5kj2 þ 3hÞ
;
i; j ¼ 1; 2; i 6¼ j; B ¼ b12 b21 :
However, due to the computation expands dramatically when constructing the three-soliton solution, so we does not got it by now.
5. Summary In this paper, a parameterized 5th-order KdV-type equation with some additional terms are analyzed with respect to the S-integrability and the C-integrability by giving rich of the explicit generalized symmetries and polynomial type conservation laws. Also, we show that there exists three sets of parameters constraints which guarantee (1.1) is completely integrable, under which the multi-soliton solutions are derived by introducing some nonlinear transformations.
Acknowledgements Project supported by the State Key Program of Basic Research of China (grant no. G1998030600) and the Research Fund for the Doctoral Program of the Higher Education of China (grant no. 20020269003).
References [1] Langer J, Perline R. J Nonlinear Sci 1991;1:71. [2] Bertozzi AL. SIAM J Appl Math 1996;56(3):681. € Hereman W. Symbolic computation of conserved densities for systems of non linear evolution equations. Technical [3] G€ oktas U, Report MCS-96-06, Colorado School of Mines, 1996. [4] Yao RX, Li ZB. Conservation laws and new exact solutions for the generalized seventh order KdV equation. Chaos, Solitons & Fractals 2004;20(2):259–66. [5] Sanders JA, Wang JP. J Diff Eq 2000;166:132. [6] Li ZB, Wang ML. J Phys A: Math Gen 1993;26:6027. [7] Sanders JA, Wang JP. Math Comput Simulat 1997;44:471. [8] Malfliet M. J Phys 1992;60:650. [9] Parkes EJ. J Phys A 1994;27:497. [10] Fan EG. Phys Lett A 2000;277:212. [11] Li ZB, Zhang SQ. Acta Math Sin 1997;17:81 (in Chinese). [12] Fu ZT, Liu SK, Liu SD, Zhao Q. Phys Lett A 2001;290:72. [13] Liu YP, Li ZB. Comp Phys Comm 2002;148:256. [14] Yao RX, Li ZB. Phys Lett A 2002;297:196. [15] Wang ML, Wang YM, Zhou YB. Phys Lett A 2002;303:45. [16] Matsuno Y. Bilinear transformation method. Orlando: Academic Press; 1984.
On properties of new parameterized 5th-order nonlinear evolution equation Ruo-Xia Yao a
a,b
, Chang-Zheng Qu c, Zhi-Bin Li
a,*
Department of Computer Science, East China Normal University, Shanghai 200062, PR China b Department of Computer Science, Weinan Teachers College, Shaanxi 714000, PR China c Department of Mathematics, Northwest University, Xi’an 710069, PR China Accepted 8 December 2003
Abstract By means of direct algebraic methods, generalized symmetries and conservation laws are obtained for a new parameterized 5th-order nonlinear evolution equation under the different parameters constraints. The multi-soliton solutions are also constructed by combining some nonlinear transformations and the simplified Hirota approach with the assistance of symbolic manipulation system Maple. 2004 Elsevier Ltd. All rights reserved.
1. Introduction As is well known that some partial differential equations, such as the Korteweg–de Vries equation and the vortex filament equation [1], are completely integrable systems; in particular, they posses infinite many sequences of higher order symmetries and conservation laws. Some other equations, such as the 5th-order Ito equation and the lubrication equation, posses finite numbers of local conservation laws which can be exploited in the analysis of the equation [2]. The aim of this paper is first to analysis under what compatibility conditions such that the following parameterized 5thorder KdV-type equation with some additional terms possesses polynomial type higher order symmetries and conservation laws (PClaws), which reads ut ¼ u5x þ uu3x þ aux u2x þ bu2 ux þ hu3x þ cuux KðuÞ;
ukx ¼ ok u=xk
ð1:1Þ
with a, b, h, c reals and u ¼ uðx; tÞ 2 C 1 . After the integrable cases of (1.1) are filtered, the one-, two- and three-soliton solutions are constructed respectively by using some nonlinear transformations. It is noticeable that (1.1) is not uniform in rank [3]. To construct the PClaws of (1.1) by using the Maple procedure CONSLAW presented in [4] we should, first of all, treat all parameters appearing in (1.1) as extra variables with (unknown) weights. Then from (1.1), xðvÞ denotes the weight of v [3,4], we arrive at xðo=otÞ þ xðuÞ ¼ 5 þ xðuÞ ¼ 3 þ 2xðuÞ ¼ 3 þ xðaÞ þ 2xðuÞ ¼ 1 þ xðbÞ þ 3xðuÞ ¼ 3 þ xðhÞ þ xðuÞ ¼ 1 þ xðcÞ þ 2xðuÞ;
ð1:2Þ
which on solving yields xðo=otÞ ¼ 7;
xðuÞ ¼ xðhÞ ¼ xðcÞ ¼ 2;
xðaÞ ¼ xðbÞ ¼ 0:
*
Corresponding author. E-mail addresses: [email protected] (R.-X. Yao), [email protected] (Z.-B. Li).
0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.078
ð1:3Þ
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Once the weights of the variables and the differentials in hand, one can construct the polynomial type generalized symmetries and conservation laws for (1.1) immediately by using the following direct algebraic methods. Definition 1. A function r ¼ rðt; x; u; ux ; u2x ; . . .Þ, is a generalized symmetry of (1.1), if and only if rt ¼ K 0 r
ð1:4Þ
for every solution of (1.1), where K 0 r ¼ oo Kðu þ rÞj¼0 ¼ lim!0 Kðt; x; ðu þ rÞ; ðu þ rÞx ; ðu þ rÞ2x ; . . .Þ. Definition 2. A conservation law for (1.1) is a divergence expression qt þ Jx ¼ 0;
ð1:5Þ
which vanishes for all solutions of (1.1), where q and J are respectively conserved density and flux and depend only on u and uix ’s.
2. Generalized symmetries of the 5th-order nonlinear evolution equation Before computing the generalized symmetries of (1.1), we describe the algorithm briefly. • Determine the general form of r for a fixed rank R, i.e. find the building blocks in u and Uix such that the monomials just have the same rank R. • Linear combination of the above monomials with constant coefficients ci ’s directly yields the general form of r. • Compute the derivative of r with respect to t, i.e. rt , and eliminate all the t-derivatives of u using ut ¼ KðuÞ, like ut ; utx , and then compute oo Kðu þ rÞj¼o . Thus from (1.4), the ci ’s can be determined. Now, we list some results of the generalized symmetries of (1.1) as follows: (A) RankðrÞ ¼ 3: a 1st-order symmetry reads r ¼ ux , for which without any parameter constraint. (B) RankðrÞ ¼ 5: a 3rd-order symmetry reads r ¼ u3x þ 83 cux þ 35 uux , for which the parameters constraints are 3 3 ð2:1Þ a ¼ 2; b ¼ ; c ¼ h : 10 5 (C) RankðrÞ ¼ 7: a 5th-order symmetry reads r ¼ u5x þ uu3x þ 49 h2 ux þ 103 u2 ux þ 2ux u2x , for which the parameters con25 straints are just the same as (2.1). (D) RankðrÞ ¼ 9: three 7th-order generalized symmetries are obtained under the different parameters constraints. The first one is 49 2 272 3 7 21 7 7 7 8 4 c uux þ c ux þ u3 ux þ 7u2x u3x þ ux u4x þ uu5x þ u3x þ u2 u3x þ cu5x þ cu2 ux 9 27 50 5 5 10 10 3 5 16 14 245 2 8 c u3x þ cuu3x þ cux u2x þ uux u2x þ 3 5 27 3
r ¼ u7x þ
for which the parameters constraints are 3 3 a ¼ 2; b ¼ ; c ¼ h : 10 5 The second one is 7 203 3 28 3 21 14 7 7 14 42 7 c ux þ u ux þ u2x u3x þ ux u4x þ uu5x þ u3x þ u2 u3x þ uux u2x þ c2 u3x r ¼ u7x þ cux u2x þ 2 8 375 5 5 5 25 25 25 4 14 21 21 2 7 2 þ cu5x þ cuu3x þ cu ux þ c uux 5 10 50 10
ð2:2Þ
R.-X. Yao et al. / Chaos, Solitons and Fractals 21 (2004) 1145–1152
for which the parameters constraints are 1 2 a ¼ 1; b ¼ ; c ¼ h : 5 5
1147
ð2:3Þ
The third one is 21 2 203 3 7 7 21 14 21 7 42 49 c uux þ c ux þ cuu3x þ cu2 ux þ c2 u3x þ u2 u3x þ cu5x þ u3x þ u2x u3x þ ux u4x 10 8 2 10 4 25 5 10 5 10 28 3 7 63 161 þ u ux þ uu5x þ uux u2x þ cux u2x 375 5 25 20
r ¼ u7x þ
for which the parameters constraints are 5 1 2 a¼ ; b¼ ; c¼ h : 2 5 5
ð2:4Þ
In [5], Sanders and Wang have proved the S-integrability for an equation of order 5 needs a symmetry of order 7 whence the S-integrability of (1.1). The above results shows that we filter out three sets of parameters constraints which guaranteeing the S-integrability of (1.1). Then in Sections 3 and 4 we will discuss the C-integrability and search the multi-soliton solutions of (1.1) under the parameters constraints (2.2)–(2.4) respectively.
3. PClaws of the 5th-order nonlinear evolution equation Prom the case (D) in Section 2 we see that (1.1) includes three cases that are of particular importance. By using the same procedure presented in [4], we compute the polynomial type conservation laws under those three cases directly and list some results below. (I) Case a ¼ 2, b ¼ 3=10, c ¼ 3h=5. (A) In the case of RankðqÞ ¼ 2, the conserved density–flux pair reads ðuÞt þ
1 2 1 3 ux þ u4x þ hu2x þ uu2x þ u3 þ hu2 2 10 10
¼ 0:
ð3:1Þ
x
(B) In the case of RankðqÞ ¼ 4, the conserved density–flux pair reads 1 3 1 3 ðu2 þ uhÞt þ hu2x þ hu4x þ h2 u2x þ 2u2 u2x þ u4 þ 2uu4x þ 3huu2x þ hu3 þ h2 u2 2ux u3x þ u22x 2 20 2 10 x ¼ 0:
ð3:2Þ
(C) In the case of RankðqÞ ¼ 6, the conserved density–flux pair reads 1 1 ð5u2x þ u3 þ uh2 þ u2 hÞt þ 3u2x u2 þ h2 u3 12hux u3x h2 u2x þ 3u2 u4x þ 2huu4x þ h2 u4x þ h3 u2x þ 8uu22x 2 2 9 3 3 þ 5u2 hu2x þ 6hu22 x þ u5 þ 3uh2 u2x þ 3u3 u2x þ h3 u2 16ux uu3x 10ux u5x 6huu2x þ hu4 14u2x u2x 50 10 5 þ 10u2x u4x 5u23x
¼ 0:
ð3:3Þ
x
(D) In the case of RankðqÞ ¼ 8, the conserved density, from now on the conserved fluxes no longer listed due to that containing more than 30 terms, reads q ¼ u4 þ u2 h2 þ uh3 þ u3 h 5hu2x 20uu2x þ 20u22x :
ð3:4Þ
(E) In the case of RankðqÞ ¼ 10, the conserved density reads q ¼ 100uu22x þ 20hu22x 5h2 u2x
500 2 u þ u5 20huu2x 50u2x u2 þ hu4 þ h2 u3 þ h3 u2 þ h4 u: 7 3x
ð3:5Þ
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(F) In the case of RankðqÞ ¼ 12, the conserved density reads 500 2 5000 2 250 4 10000 3 hu3x þ 20h2 u22x þ u u þ u þ h5 u þ hu5 þ h2 u4 þ h3 u3 7 21 4x 3 x 21 2x 3000 2 þ h4 u2 þ u6 þ 300u2 u22x u u 100u3 u2x 5h3 u2x þ 100huu22x : 7 3x
q ¼ 20h2 uu2x 50hu2x u2
ð3:6Þ
(G) In the case of RankðqÞ ¼ 14, the conserved density reads q¼
5000 2 5000 2 500 2 2 25000 10000 3 25000 2 uu4x þ 20u22x h3 þ hu4x u h þ 3500u2x u22x u2x u23x þ u h u 3 21 7 3x 3 21 2x 33 5x 10000 3 1750 4 250 4 uu2x 1500u2 u23x þ u4 h3 uux u h 5u2x h4 þ 700u3 u22x 175u4 u2x þ uh6 þ u3 h4 þ 3 3 3 x 3000 uhu23x þ 300u2 hu22x þ 100uh2 u22x : þ u5 h2 þ u7 þ u2 h5 þ u6 h 50u2 h2 u2x 100u3 hu2x 20uh3 u2x 7
ð3:7Þ
In the case of RankðqÞ ¼ 16 and 18 the conserved densities contain 42 and 64 terms respectively and not listed here. (II) Case a ¼ 1, b ¼ 1=5, c ¼ 2h=5. (A) In the case of RankðqÞ ¼ 2, the conserved density–flux pair reads 1 1 ðuÞt þ u4x þ hu2x þ uu2x þ u3 þ u2 h ¼ 0: ð3:8Þ 15 5 x (B) In the case of RankðqÞ ¼ 4, the conserved density–flux pair reads 1 3 1 2 ¼ 0: ðuhÞt þ h u4x þ hu2x þ uu2x þ u þ u h 15 5 x
ð3:9Þ
Bear in mind that h is in fact an arbitrary constant, hence (3.9) is the same as (3.8). (C) In the case of RankðqÞ ¼ 6, the conserved density reads q ¼ 15u2x þ u3 þ 3u2 h þ uh2 :
ð3:10Þ
(D) In the case of RankðqÞ ¼ 8, the conserved density reads q ¼ u4 45uu2x þ u3 h 3u2 h2 þ uh3 þ 75u22x :
ð3:11Þ
(E) In the case of RankðqÞ ¼ 10, the conserved density reads 1 q ¼ 25hu22x 15uhu2x þ u2 h3 þ u3 h2 þ u4 h þ uh4 10u2x h2 : 3
ð3:12Þ
(F) In the case of RankðqÞ ¼ 12, the conserved density reads 1 2 6 4 5 8 8 17 7 16 u u h þ u3 h3 þ u4 h2 þ uh5 þ u2 h4 u2 u22x þ u4x þ uu23x uu22x h q ¼ u3 u2x 3 1125 375 25 5 30 2 5 73 2 2 7 2 122 2 2 8 3 5 2 2 2 2 3 u h 10ux h u2x u4x : uux h þ u ux h þ u3x h þ 5 2 5 2x 3 2
ð3:13Þ
(G) In the case of RankðqÞ ¼ 14, the conserved density reads q¼
1125 2 34787 2 3 115 2 1011 2 2 900 2 2 2000 107 3 625 2 4075 3 uu þ u h hu þ u h þ uu u2x u23x þ u h u þ uu 286 4x 1430 2x 286 4x 286 3x 143 x 2x 143 858 2x 286 5x 858 2x
725 2 2 6829 4 3 105 4 41 4 665 3 2 35 4 2 1109 5 2 u u3x þ uh uux þ ux h 10u2x h4 þ u u2x u ux þ uh6 þ u3 h4 uh 286 21450 143 165 858 286 107250
þ
1 7 1 6 38 3 2 10404 3 2 643 2 2 4611 2 2 u þ u2 h5 u h þ u2 h2 u2x þ u hux uh ux þ uhu23x u hu2x uh u2x : 2574 2750 429 715 1430 1430 ð3:14Þ
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1149
In the case of RankðqÞ ¼ 16, 18 and 20 the conserved densities contain 28, 64 and 96 terms respectively and not listed here. (III) Case a ¼ 5=2, b ¼ 1=5, c ¼ 2h=5. (A) In the case of RankðqÞ ¼ 2, the conserved density–flux pair reads 3 2 1 1 ux þ u4x þ hu2x þ uu2x þ u3 þ hu2 ¼ 0: ðuÞt þ ð3:15Þ 4 15 5 x (B) In the case of RankðqÞ ¼ 4, the conserved density–flux pair reads 3 2 1 3 1 2 2 2 ¼ 0: hu þ hu4x þ h u2x þ huu2x þ hu þ h u ðuhÞt þ 4 x 15 5 x
ð3:16Þ
(C) In the case of RankðqÞ ¼ 6, the conserved density–flux pair reads 15 27 1 27 3 3 u2x þ uh2 þ 3u2 h þ u3 þ 3huu2x hux u3x þ 9u2 hu2x þ 3u3 u2x þ h3 u2 þ uu22x þ u5 u2x u2 4 2 5 4 25 2 t 27 2 27 15 9 2 2 13 2 3 51 2 3 2 2 þ hu2x ux uu3x ux u5x þ 3u u4x þ h u2x þ h u4x h ux þ h u ux u2x 4 2 2 4 15 4 3 4 15 15 2 2 þ 7uh u2x þ 6huu4x þ hu þ u2x u4x u3x ¼ 0: ð3:17Þ 5 2 4 x (D) In the case of RankðqÞ ¼ 8, the conserved density reads 25 1 25 15 q ¼ u22x þ u4 hu2x þ uh3 þ u2 h2 þ u3 h uu2x : 4 3 4 2 (E) In the case of RankðqÞ ¼ 10, the conserved density reads q ¼ h3 u2
25 2 2 1 25 15 h ux þ h2 u3 þ hu4 þ hu22x huu2x þ uh4 : 4 3 4 2
ð3:18Þ
ð3:19Þ
(F) In the case of RankðqÞ ¼ 12, the conserved density reads q¼
4 6 8 5 4 283 2 4 25 2 185 3 31 4 25 2 1 u þ u h þ u3 h3 u4 h2 þ uh5 þ u h þ u4x þ u u uu þ u2 u2 3825 1275 85 85 68 204 2x 408 x 34 3x 2 2x 25 23 227 3 2 7 7 h ux þ huu22x þ h2 uu2x hu2 u2x u3 u2x : hu23x h2 u22x 34 34 68 17 51
ð3:20Þ
(G) In the case of RankðpÞ ¼ 14, the conserved density reads q¼
244 5 2 4187 3 2 11170 2 2 4575 2 173195 3 68385 2 2 12455 2 2 uh h u2x h u3x hu u u uu þ u u 1995 1064 931 1862 4x 5586 2x 3724 x 2x 931 3x 141 4 2 2162 3 2 1927 4 4 6 1175 815 3 1823 4 3 þ u ux þ u2x h4 þ hu4x u u2x þ uu þ u hþ u2x u23x þ hu uh 133 399 532 x 855 14 2793 2x 3990 14100 2 1285 2428 2 2 66 3 2 1044 2 2 2 4765 3 2 188 7 uu4x þ huu23x þ h uu2x þ hu2 u22x u hux u h ux þ uh ux u 931 931 133 133 133 532 41895 34 5875 2 þ u2 h5 þ u þ u3 h4 þ uh6 : 5 931 5x
ð3:21Þ
We see that in every level (1.1) possesses PClaws under the three parameters constraints and in view of the conclusion in Section 2 we state that, although we do not give a theoretical proof, (1.1) possesses infinitely many PClaws and is Cintegrability. 4. Multi-soliton solutions of the 5th-order nonlinear evolution equation This section is devoted to construct the multi-soliton solutions to (1.1) under the above three sets of parameters constraints. (I) Case a ¼ 1, b ¼ 1=5, c ¼ 2h=5. To obtain the soliton solutions of (1.1) under this case, we introduce a nonlinear transformation of dependent variable [6], uðx; tÞ ¼ r þ p½lnð/ðx; tÞÞxx ;
ð4:1Þ
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R.-X. Yao et al. / Chaos, Solitons and Fractals 21 (2004) 1145–1152
where pð6¼ 0Þ and r are real parameters to be determined, which allows us to transform the corresponding equation into a 9th-degree homogeneous equation, denoted by Q. Then, we seek a solution of type /ðx; tÞ ¼ 1 þ expðnÞ;
n ¼ kx þ wt;
ð4:2Þ
where kð6¼ 0Þ and w are parameters to be determined. Substituting of (4.2) into Q and equating the coefficients of different powers of expðnÞ to zero, one obtains a nonlinear system, which on solving can determine the possible relations between the parameters r, p, k and w. Solving the nonlinear system of algebraic equations we arrive at 2 1 p ¼ 30; w ¼ hk 3 rhk k 5 r2 k rk 3 ; k ¼ k; r ¼ r; h ¼ h : ð4:3Þ 5 5 Substitution of (4.3) into (4.2) and (4.1) yields uðx; tÞ ¼ r þ 30
k 2 expðkx þ wtÞ ½1 þ expðkx þ wtÞ2
ð4:4Þ
;
where w ¼ hk 3 25 rhk k 5 15 r2 k rk 3 , and k, r, h are arbitrary reals, which can also be written as uðx; tÞ ¼ r þ
15 sech2 ðkx þ wtÞ: 2
ð4:5Þ
If setting r ¼ 0, (4.5) is in fact a one-soliton solution of (1.1) under this case and can also be obtained using the methods presented in [7–14]. Alternatively, if we seek a solution of type /ðx; tÞ ¼ s þ expðnÞ þ expð2nÞ;
n ¼ kx þ wt;
ð4:6Þ
then we get uðx; tÞ ¼ 5k 2 h þ 30
k 2 ½expðkx þ wtÞs þ expðkx þ wtÞ expð2kx þ 2wtÞ þ 4 expð2kx þ 2wtÞs ½s þ expðkx þ wtÞ þ expð2kx þ 2wtÞ2
;
ð4:7Þ
where w ¼ 15 h2 k k 5 , and h, s, k free. Now, to construct the two-soliton solution, one can use the nonlinear transformation uðx; tÞ ¼ 30½lnð/ðx; tÞÞxx ;
ð4:8Þ
which reduces the equation into a 7th-degree homogeneous one ð30/2x;t þ 30/7x þ 30h/5x Þ/6 þ ð60/x /xt þ 270/5x /2x 30/2x /t 210/6x /x þ 60h/3x /2x 150/4x /3x 150h/4x /x Þ/5 þ ð600/23x /x þ 240h/3x /2x þ 60/2x /t þ 360/5x /2x 900/4x /x /2x 180/22x /x Þ/4 ¼ 0: The linear and nonlinear operators are thus defined as 3 o o7 o5 þ ; þ h L ¼ 30 ox2 ot ox7 ox5 N1 ðf ; gÞ ¼ 60fx gxt þ 270f5x g2x 30f2x gt 210f6x gx þ 60hf3x g2x 150f4x g3x 150hf4x gx ; N2 ðf ; g; hÞ ¼ 600f3x g3x hx þ 240hf3x gx hx þ 60fx gx ht þ 360f5x gx hx 900f4x gx h2x 180hf2x g2x hx :
ð4:9Þ
ð4:10Þ
To solve (4.9), one can use the -expansion method [15,16]. Suppose that the solution of (4.9) is of the form /ðx; tÞ ¼ 1 þ f1 þ f2 2 þ f3 3 þ f4 4 þ ;
ð4:11Þ
where fi (i ¼ 1; 2; 3; . . .) to be determined later, be a small parameter (for simplicity we may take ¼ 1). Substituting (4.11) into (4.9), collecting all terms with the same order of together, and setting each coefficient of k (k ¼ 1; 2; 3; . . .) to zero give a hierarchy of equations for fk , 1 : Lf1 ¼ 0; 2
ð4:12Þ
: Lf2 ¼ N1 ðf1 ; f1 Þ;
ð4:13Þ
3 : Lf3 ¼ 6f1 Lf2 5f1 N1 ðf1 ; f1 Þ N1 ðf1 ; f2 Þ N1 ðf2 ; f1 Þ N2 ðf1 ; f1 ; f1 Þ;
ð4:14Þ
and so on to be solved.
R.-X. Yao et al. / Chaos, Solitons and Fractals 21 (2004) 1145–1152
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To obtain the one-soliton solution, suppose that f1 ¼ expðnÞ, with n ¼ kx þ wt. Computation of (4.12) gives the dispersion law w ¼ k 3 ðh þ k 2 Þ. It is easy to check that the right-hand side of (4.13) is equal to zero, and then we can set fn ¼ 0 for n P 2. Therefore the series (4.11) truncates. One-soliton solution is thus got and is just the same as (4.5) with r ¼ 0. To obtain the two-soliton solution, one can take f1 ¼ expðn1 Þ þ expðn2 Þ;
ni ¼ ki x þ wi t;
i ¼ 1; 2:
ð4:15Þ
It is straightforward to check that fn ¼ 0 for n P 3. The series (4.11) therefore truncates. By means of -expansion method, and after long tedious computation one obtain / ¼ 1 þ expðn1 Þ þ expðn2 Þ þ a expðn1 þ n2 Þ;
ð4:16Þ
where wi ¼ ki3 ðh þ ki2 Þ;
a¼
ðk1 k2 Þ2 ð3h þ 5k12 5k1 k2 þ 5k22 Þ ðk1 þ k2 Þ2 ð3h þ 5k12 þ 5k1 k2 þ 5k22 Þ
:
The explicit two-soliton solution uðx; tÞ can be got by substituting (4.16) into (4.8). To obtain the three-soliton solution, we take f1 ¼ expðn1 Þ þ expðn2 Þ þ expðn3 Þ;
ni ¼ ki x þ wi t;
i ¼ 1–3:
ð4:17Þ
By checking we know that fn ¼ 0 for n P 4. The series (4.11) therefore truncates. Likewise, we get /¼1þ
3 X
X
expðni Þ þ
aij expðni þ nj Þ þ a12 a13 a23 expðn1 þ n2 þ n3 Þ
ð4:18Þ
1 6 i
i¼1
with wi ¼ ki3 ðh þ ki2 Þ;
a¼
ðki kj Þ2 ð3h þ 5ki2 5ki kj þ 5kj2 Þ ðki þ kj Þ2 ð3h þ 5ki2 þ 5ki kj þ 5kj2 Þ
ð1 6 i < j 6 3Þ:
Then, the three-soliton solution is obtained from (4.8). (II) Case a ¼ 2, b ¼ 3=10, c ¼ 3h=5. Similarly, the one-, two- and three-soliton solutions can be generated by the following nonlinear transformation of dependent variable, uðx; tÞ ¼ 20½lnð/ðx; tÞÞxx ;
ð4:19Þ
along with the following expressions, / ¼ 1 þ expðkx k 3 ðh þ k 2 ÞtÞ;
ð4:20Þ
/ ¼ 1 þ expðn1 Þ þ expðn2 Þ þ a12 expðn1 þ n2 Þ;
ð4:21Þ
/¼1þ
3 X
X
expðni Þ þ
i¼1
aij expðni þ nj Þ þ a12 a13 a23 expðn1 þ n2 þ n3 Þ
ð4:22Þ
1 6 i
with ni ¼ ki x þ ki3 ðh þ ki2 Þt;
aij ¼
ðki ki Þ2 ðki þ kj Þ2
ð1 6 i < j 6 3Þ:
(III) Case a ¼ 5=2, b ¼ 1=5, c ¼ 2h=5. Similarly, the one- and two-soliton solutions can be generated by the following nonlinear transformation of dependent variable, uðx; tÞ ¼ 15½lnð/ðx; tÞÞxx ;
ð4:23Þ
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R.-X. Yao et al. / Chaos, Solitons and Fractals 21 (2004) 1145–1152
along with the following expressions, 5k 2 þ 4h expð2nÞ; n ¼ kx k 3 ðh þ k 2 Þt; 16ð5k 2 þ hÞ 2 2 X X 4h þ 5ki2 expðni Þ þ expð2ni Þ þ a12 expðn1 þ n2 Þ /¼1þ 16ðh þ 5ki2 Þ i¼1 i¼1 / ¼ 1 þ expðnÞ þ
ð4:24Þ
þ b12 expð2h1 þ h2 Þ þ b21 expðh1 þ 2h2 Þ þ B expð2n1 þ 2n2 Þ with a12 ¼ bij ¼
10k14 þ 6k12 h 5k22 k12 þ 10k24 þ 6k22 h 2ðk1 þ k2 Þ2 ð5k22 þ 5k1 k2 þ 5k12 þ 3hÞ
;
ðki kj Þ2 ð4h þ 5ki2 Þð5ki2 5ki kj þ 5kj2 þ 3hÞ 16ðki þ kj Þ2 ðh þ 5ki2 Þð5ki2 þ 5ki kj þ 5kj2 þ 3hÞ
;
i; j ¼ 1; 2; i 6¼ j; B ¼ b12 b21 :
However, due to the computation expands dramatically when constructing the three-soliton solution, so we does not got it by now.
5. Summary In this paper, a parameterized 5th-order KdV-type equation with some additional terms are analyzed with respect to the S-integrability and the C-integrability by giving rich of the explicit generalized symmetries and polynomial type conservation laws. Also, we show that there exists three sets of parameters constraints which guarantee (1.1) is completely integrable, under which the multi-soliton solutions are derived by introducing some nonlinear transformations.
Acknowledgements Project supported by the State Key Program of Basic Research of China (grant no. G1998030600) and the Research Fund for the Doctoral Program of the Higher Education of China (grant no. 20020269003).
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