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Symbolic computation of exact solutions for a nonlinear evolution equation
Chaos, Solitons and Fractals 31 (2007) 1173–1180 www.elsevier.com/locate/chaos
Symbolic computation of exact solutions for a nonlinear evolution equation Liu Yinping *, Li Zhibin, Wang Kuncheng Department of Computer Science, East China Normal University, Shanghai 200062, China Accepted 14 September 2005
Abstract In this paper, by means of the Jacobi elliptic function method, exact double periodic wave solutions and solitary wave solutions of a nonlinear evolution equation are presented. It can be shown that not only the obtained solitary wave solutions have the property of loop-shaped, cusp-shaped and hump-shaped for different values of parameters, but also different types of double periodic wave solutions are possible, namely periodic loop-shaped wave solutions, periodic hump-shaped wave solutions or periodic cusp-shaped wave solutions. Furthermore, periodic loop-shaped wave solutions will be degenerated to loop-shaped solitary wave solutions for the same values of parameters. So do cuspshaped solutions and hump-shaped solutions. All these solutions are new and first reported here. Ó 2006 Elseiver Ltd. All rights reserved.
1. Introduction Consider the nonlinear evolution equation o 1 o o D2 u þ pu2 þ bu þ qDu ¼ 0; D :¼ þ u ; ox 2 ot ox
ð1:1Þ
where p, q and b are arbitrary constants. To construct exact solutions of Eq. (1.1) is all-important. For much research has been conducted on special cases of Eq. (1.1). For example, when p = b = 0 and q = 1, (1.1) is reduced to the well known Vakhnenko equation (VE) [1], which governs the propagation of waves in a relaxing medium. In [2] it was shown that the VE has a multi-loop soliton solution. When p = q = 1 and b an arbitrary non-zero constant, (1.1) is reduced as the GVE (generalization of the VE), in [3] it was shown that GVE has N-soliton solutions. When p = 2q and b an arbitrary non-zero constant, (1.1) is reduced to mGVE (modified generalization of the VE), in [4] it was shown that mGVE not only has loop soliton solution, but also has hump-like and cusp-like soliton solution. Some work for other special cases: the case p = 2q, b = 1 was discussed by Ablowitz et al. [5] and was shown to be integrable by inverse scattering, the case p = q, b = 1 was discussed by Hirota and Satsuma [6] and was shown to be integrable using HirotaÕs bilinear technique, etc.
*
Corresponding author.
0960-0779/$ - see front matter Ó 2006 Elseiver Ltd. All rights reserved. doi:10.1016/j.chaos.2005.09.055
1174
Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
Much of the above work is closely related to the following equation: # " dx uxx uxt þ sgn ¼ 0. ds ð1 þ u2x Þ3=2 xx
ð1:2Þ
This equation shares the similar properties with VE, and its loop soliton solutions have been discussed successively in articles [7–10]. Especially Ishimori [9] found a transformation of the dependent and independent variables to Eq. (1.2) and obtained multiple loop solutions. The methods used in studying VE, GVE and mGVE are similar, although it is only the independent variables that are transformed. Our interesting is to construct exact double periodic wave solutions as well as solitary wave solutions of Eq. (1.1) by the Jacobi elliptic function method [11–13] in this paper. In Section 2, we briefly introduce the Jacobi elliptic function method. In Section 3, we construct exact solutions of Eq. (1.1) by this method, not only new loop-like, hump-like and cusp-like soliton solutions, but also more general double periodic wave solutions, were obtained. It is very interesting that different types of double periodic wave solutions are possible, namely, periodic loop-shaped wave solutions, periodic cusp-shaped wave solutions or periodic hump-shaped wave solutions, for different values of the parameters p, q, b, k. A conclusion is then given in the final Section 4.
2. The Jacobi elliptic function method Consider nonlinear evolution equation H ðu; ut ; ux ; uxx ; . . .Þ ¼ 0;
ð2:3Þ
where u = u(x, t). H is a polynomial about u and its derivatives. We seek its explicit solution in the form u ¼ uðnÞ;
n ¼ kðx ctÞ;
ð2:4Þ
where k and c are the wave number and wave speed, respectively. Using the Jacobi elliptic function method, u(n) can be expressed as a polynomial of different Jacobi elliptic functions. Such as uðnÞ ¼
n X
ai snðnÞi ;
ð2:5Þ
i¼0
P P similarly, uðnÞ ¼ ni¼0 ai cnðnÞi , or uðnÞ ¼ ni¼0 ai dnðnÞi , where sn(n) is the Jacobi elliptic sine function, cn(n) the Jacobi elliptic cosine function and dn(n) the Jacobi elliptic function of the third kind. It is apparent that the highest degree of u(n) is O(u(n)) = n. Since cnðnÞ2 ¼ 1 snðnÞ2 ;
dnðnÞ2 ¼ 1 m2 snðnÞ2 ;
ð2:6Þ
and d snðnÞ ¼ cnðnÞ dnðnÞ; dn
d cnðnÞ ¼ snðnÞ dnðnÞ; dn
d dnðnÞ ¼ m2 snðnÞ cnðnÞ; dn
where m (0 < m < 1) is the modulus, the highest degree of dpu/dnp is taken as p p du qd u ¼ n þ p; O u ¼ ðq þ 1Þn þ p; p ¼ 1; 2; 3; . . . ; q ¼ 0; 1; 2; . . . . O dnp dnp
ð2:7Þ
ð2:8Þ
Thus n can be obtained by balancing the linear term(s) of highest order with the highest-order nonlinear term(s) in Eq. (2.3). Here k, c, a0, . . . , an are parameters to be determined later. Substituting Eq. (2.5) into Eq. (2.3) yields a set of algebraic equations for k, c, a0, . . . , an. WuÕs elimination method[14,15] is a powerful tool for solving nonlinear algebraic equations, and k, c, a0, . . . , an can be obtained. Therefore, the exact double periodic wave solutions are obtained. When m ! 1, the Jacobi elliptic functions degenerate to the hyperbolic functions, i.e., snðnÞ ! tanhðnÞ;
cnðnÞ ! sechðnÞ;
dnðnÞ ! sechðnÞ.
So, besides double periodic wave solutions, solitary wave solutions can also be obtained by this method.
ð2:9Þ
Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
1175
3. Symbolic computation of exact solutions For the nonlinear evolution Eq. (1.1), introducing new independent variables X and T, defined by Z X U ðX 0 ; T Þ dX 0 þ x0 ; t ¼ X ; x¼T þ
ð3:10Þ
1
where u(x, t) = U(X, T) and x0 is a constant [4]. Let Z X W ðX ; T Þ ¼ U ðX 0 ; T Þ dX 0 ;
ð3:11Þ
1
we have W X ðX ; T Þ ¼ U ðX ; T Þ.
ð3:12Þ
From (3.10) it follows that o o o ¼ þu ; oX ot ox
o o ¼w ; oT ox
where w(X, T) = 1 + WT. Use of (3.10) transforms (1.1) into the following equation Z 1 U XXT þ pUU T qU X U T ðX 0 ; T Þ dX 0 þ bU T þ qU X ¼ 0;
ð3:13Þ
ð3:14Þ
X
substituting (3.12) into (3.14) yields W XXXT þ pW X W XT þ qW XX W T þ bW XT þ qW XX ¼ 0.
ð3:15Þ
Now, introducing a new independent variable n ¼ kðX cT Þ;
ð3:16Þ
substituting (3.16) into Eq. (3.15) and integrating once yields k 4 cW 3n 1=2ðp þ qÞk 3 cW 2n þ ðqk 2 bk 2 cÞW n ¼ C;
ð3:17Þ
where C is an integral constant. By taking V(n) = Wn, we obtain k 4 cV 2n 1=2ðp þ qÞk 3 cV ðnÞ2 þ ðqk 3 bk 2 cÞV ðnÞ ¼ C. Suppose that Eq. (3.18) has solutions with the form n X ai snðnÞi ; V ðnÞ ¼
ð3:18Þ
ð3:19Þ
i¼0
By balancing the linear term(s) of highest order with the highest-order nonlinear term in Eq. (3.18) we obtain n = 2. So the ansatz solution of Eq. (3.18) can be rewritten as V ðnÞ ¼ a0 þ a1 snðnÞ þ a2 snðnÞ2 .
ð3:20Þ
Substituting (3.20) into Eq. (3.18) and collecting each power of sn yields algebraic equations about parameters aj (j = 0, 1, 2) and k, c, i.e., k 3 cpa20 2qk 3 a0 þ 2bk 2 ca0 þ k 3 cqa20 þ 4k 4 ca2 ¼ 0; k 4 ca1 m2 k 3 cpa1 a0 þ k 4 ca1 bk 2 ca1 þ qk 3 a1 k 3 cqa1 a0 ¼ 0; 8k 4 ca2 m2 þ 2k 3 qa2 k 3 cqa21 þ 8k 4 ca2 2k 3 cqa2 a0 2k 2 bca2 2k 3 cpa2 a0 þ k 3 cpa21 ¼ 0; 4
2
3
3
2k ca1 m þ k cqa1 a2 þ k cpa1 a2 ¼ 0; k 3 cqa22 þ k 3 cpa22 þ 12k 4 ca2 m2 ¼ 0;
ð3:21Þ
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Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
from which we have (
12km2 ; pþq
a1 0;
a2 ¼
a1 ¼ 0;
12km2 ; a2 ¼ pþq
(
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k m2 þ 1 þ m4 m2 þ 1 ; pþq pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k m2 þ 1 m4 m2 þ 1 a0 ¼ ; pþq
a0 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) kðq 4kc m4 m2 þ 1Þ ; c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) k q þ 4kc m4 m2 þ 1 b¼ . c
b¼
Fig. 1. Hump-shaped curve of solution (3.25) at p = 4, q = 1 and t = 0.
Fig. 2. Loop-shaped curve of solution (3.25) at p = 1, q = 1 and t = 0.
Fig. 3. Cusp-shaped curve of solution (3.25) at p = 2, q = 1 and t = 0.
ð3:22Þ
Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
Substituting (3.22) into the ansatz solution (3.20), we obtain two non-trivial solutions of Eq. (3.18) i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k h 2 V 1 ðn1 Þ ¼ m þ 1 þ m4 m2 þ 1 3m2 snðn1 Þ2 ; p þ q kq p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which n1 ¼ k X T , and bþ4k 2 m4 m2 þ1 h i p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k m2 þ 1 m4 m2 þ 1 3m2 snðn2 Þ2 ; V 2 ðn2 Þ ¼ p þq kq p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi T . where n2 ¼ k X 2 4 2 b4k
m m þ1
Fig. 4. Cusp-shaped curve of solution (3.25) and (3.26) at k = 1, b = 0 and t = 0.
Fig. 5. Hump-shaped curve of solution (3.25) and (3.26) at k = 1, b = 5 and t = 0.
Fig. 6. Loop-shaped curve of solution (3.25) and (3.26) at k = 1, b = 1.5 and t = 0.
1177
ð3:23Þ
ð3:24Þ
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Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
By using (3.10), (3.12) and (3.23), we obtain a parameterized double periodic wave solution of Eq. (1.1) as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
( 4k 2 uðx; tÞ ¼ pþq m þ 1 þ m4 m2 þ 1 3m2 sn2 ðn1 Þ ; ð3:25Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4k 12k x ¼ T þ pþq m 2 þ m4 m2 þ 1 n1 þ pþq Eðsnðn; mÞ; mÞ; pkqffiffiffiffiffiffiffiffiffiffiffiffiffiffi T . where E(sn(n1, m), m) is the Jacobi EllipticE function, n1 ¼ k t 2 bþ4k
m4 m2 þ1
When m ! 1, the double periodic solution (3.25) is degenerated to 8 h i 2 kq > < uðx; tÞ ¼ 12k sech2 k t bþ4k ; 2 T pþq h i kq > 12k : x ¼ T þ pþq tanh k t bþ4k . 2 T
Fig. 7. Hump-shaped curve of solution (3.27) and (3.28) at k = 1, b = 0 and t = 0.
Fig. 8. Cusp-shaped curve of solution (3.27) and (3.28) at k = 1, b = 5 and t = 0.
Fig. 9. Hump-shaped curve of solution (3.27) and (3.28) at k = 1, b = 1.5 and t = 0.
ð3:26Þ
Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
By using (3.10), (3.12) and (3.24), we obtain another double periodic wave solution of Eq. (1.1) as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
( 4k 2 uðx; tÞ ¼ pþq m þ 1 m4 m2 þ 1 3m2 sn2 ðn2 Þ ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4k 12k x ¼ T þ pþq m 2 m4 m2 þ 1 n2 þ pþq Eðsnðn2 ; mÞ; mÞ; pkqffiffiffiffiffiffiffiffiffiffiffiffiffiffi T . where n2 ¼ k t 2 4 2 b4k
1179
ð3:27Þ
m m þ1
When m ! 1, the double periodic solution (3.27) is degenerated to 8 h i kq 4k 2 > < uðx; tÞ ¼ pþq 2 þ 3 sech2 k t b4k ; 2 T h h ii kq > 4k : x ¼ T pþq 2n 3 tanh k t b4k . 2 T
ð3:28Þ
Taking p = 2q, the solitary wave solution (3.26) is reduced to the soliton solution (4.4) and (4.5) in [4], it can be seen that the 1-soliton solution of the mGVE is just a special case of solution (3.26). In [4] it was shown that the solutions (4.4) and (4.5) for different values of parameters b, k may be different types, namely, loops, cusps or humps. So do the solitary wave solutions (3.26) and (3.28) for parameters b, k, p. And also we known that different types of double periodic wave solutions (3.25) and (3.27) are possible for different values of parameters b, k, p, namely, periodic loopshaped wave solutions, periodic cusp-shaped wave solutions or periodic hump-shaped wave solutions. Furthermore, besides parameters b, k, parameter p can independently determine the types of all our obtained solutions. This conclusion can be shown in Figs. 1–3 for fixing k = 1, b = 1. Parameter q mainly works on the periods of solutions. In addition, the periodic loop-shaped wave solutions will be degenerated to loop-like solitary wave solutions. So do the cusp-shaped solutions and hump-shaped solutions. We display this conclusion for fixing p = 2 and m = 0.8 in Figs. 4–6. Finally, the plots of solutions (3.27) and (3.28) are shown in Figs. 7–9 for the same parameter values with those in Figs. 4–6. We can see that there are big difference for the type and shape between solutions (3.25), (3.26) and (3.27), (3.28).
4. Summary In this paper, by the Jacobi elliptic function method, some new exact double periodic wave solutions and solitary wave solutions of a nonlinear evolution equation are constructed. It can be seen that the soliton solutions of the VE equation, the GVE equation and the mGVE equation are just special cases of our obtained new solitary wave solutions. Moreover, it is very interesting that not only our obtained new solitary wave solutions may be loop-shaped, cuspshaped and hump-shaped for different values of parameters, but also our new more general double periodic solutions have this property. Furthermore, apart from parameters b, k, parameter p exerts a great impact on the shape of solutions.
Acknowledgements This work was supported by a National Key Basic Research Project of China (2004CB318000). Liu also would like to express her sincere thanks for the support of the Graduate School of East China Normal University.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Parkes EJ, Vakhnenko VO. Chaos, Solitons & Fractals 2002;13:1819. Vakhnenko VO, Morrison AJ. Chaos, Solitons & Fractals 2003;17:683. Morrison AJ, Parkes EJ. Glasgow Math J 2001;43A:65. Morrison AJ, Parkes EJ. Chaos, Solitons & Fractals 2003;16:13. Ablowitz MJ, Kaup DJ, Newell AC, Segur H. Stud Appl Math 1974;53:249. Hirota R, Satsuma J. J Phys Soc Jpn 1976;40:611. Konno K, Ichikawa YH, Wadati M. J Phys Soc Jpn 1981;50(3):1025. Konno K, Jeffrey A. J Phys Soc Jpn 1983;52:1. Ishimori Y. J Phys Soc Jpn 1981;50:2471. El Naschie MS. Stress, stability and chaos in structural engineering. London: McGraw-Hill; 1990.
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Liu SK, Fu ZT, Liu SD, Zhao Q. Appl Math Mech 2001;22:326. Fan EG. Phys Lett A 2000;277:212. Liu YP, Li ZB. Chin Phys Lett 2002;19(9):1228. Wu WT. Mechanical theorem proving in geometries: basic principles. New York: Springer; 1994. Wang DM. Elimination method. New York: Springer; 2001.
Symbolic computation of exact solutions for a nonlinear evolution equation Liu Yinping *, Li Zhibin, Wang Kuncheng Department of Computer Science, East China Normal University, Shanghai 200062, China Accepted 14 September 2005
Abstract In this paper, by means of the Jacobi elliptic function method, exact double periodic wave solutions and solitary wave solutions of a nonlinear evolution equation are presented. It can be shown that not only the obtained solitary wave solutions have the property of loop-shaped, cusp-shaped and hump-shaped for different values of parameters, but also different types of double periodic wave solutions are possible, namely periodic loop-shaped wave solutions, periodic hump-shaped wave solutions or periodic cusp-shaped wave solutions. Furthermore, periodic loop-shaped wave solutions will be degenerated to loop-shaped solitary wave solutions for the same values of parameters. So do cuspshaped solutions and hump-shaped solutions. All these solutions are new and first reported here. Ó 2006 Elseiver Ltd. All rights reserved.
1. Introduction Consider the nonlinear evolution equation o 1 o o D2 u þ pu2 þ bu þ qDu ¼ 0; D :¼ þ u ; ox 2 ot ox
ð1:1Þ
where p, q and b are arbitrary constants. To construct exact solutions of Eq. (1.1) is all-important. For much research has been conducted on special cases of Eq. (1.1). For example, when p = b = 0 and q = 1, (1.1) is reduced to the well known Vakhnenko equation (VE) [1], which governs the propagation of waves in a relaxing medium. In [2] it was shown that the VE has a multi-loop soliton solution. When p = q = 1 and b an arbitrary non-zero constant, (1.1) is reduced as the GVE (generalization of the VE), in [3] it was shown that GVE has N-soliton solutions. When p = 2q and b an arbitrary non-zero constant, (1.1) is reduced to mGVE (modified generalization of the VE), in [4] it was shown that mGVE not only has loop soliton solution, but also has hump-like and cusp-like soliton solution. Some work for other special cases: the case p = 2q, b = 1 was discussed by Ablowitz et al. [5] and was shown to be integrable by inverse scattering, the case p = q, b = 1 was discussed by Hirota and Satsuma [6] and was shown to be integrable using HirotaÕs bilinear technique, etc.
*
Corresponding author.
0960-0779/$ - see front matter Ó 2006 Elseiver Ltd. All rights reserved. doi:10.1016/j.chaos.2005.09.055
1174
Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
Much of the above work is closely related to the following equation: # " dx uxx uxt þ sgn ¼ 0. ds ð1 þ u2x Þ3=2 xx
ð1:2Þ
This equation shares the similar properties with VE, and its loop soliton solutions have been discussed successively in articles [7–10]. Especially Ishimori [9] found a transformation of the dependent and independent variables to Eq. (1.2) and obtained multiple loop solutions. The methods used in studying VE, GVE and mGVE are similar, although it is only the independent variables that are transformed. Our interesting is to construct exact double periodic wave solutions as well as solitary wave solutions of Eq. (1.1) by the Jacobi elliptic function method [11–13] in this paper. In Section 2, we briefly introduce the Jacobi elliptic function method. In Section 3, we construct exact solutions of Eq. (1.1) by this method, not only new loop-like, hump-like and cusp-like soliton solutions, but also more general double periodic wave solutions, were obtained. It is very interesting that different types of double periodic wave solutions are possible, namely, periodic loop-shaped wave solutions, periodic cusp-shaped wave solutions or periodic hump-shaped wave solutions, for different values of the parameters p, q, b, k. A conclusion is then given in the final Section 4.
2. The Jacobi elliptic function method Consider nonlinear evolution equation H ðu; ut ; ux ; uxx ; . . .Þ ¼ 0;
ð2:3Þ
where u = u(x, t). H is a polynomial about u and its derivatives. We seek its explicit solution in the form u ¼ uðnÞ;
n ¼ kðx ctÞ;
ð2:4Þ
where k and c are the wave number and wave speed, respectively. Using the Jacobi elliptic function method, u(n) can be expressed as a polynomial of different Jacobi elliptic functions. Such as uðnÞ ¼
n X
ai snðnÞi ;
ð2:5Þ
i¼0
P P similarly, uðnÞ ¼ ni¼0 ai cnðnÞi , or uðnÞ ¼ ni¼0 ai dnðnÞi , where sn(n) is the Jacobi elliptic sine function, cn(n) the Jacobi elliptic cosine function and dn(n) the Jacobi elliptic function of the third kind. It is apparent that the highest degree of u(n) is O(u(n)) = n. Since cnðnÞ2 ¼ 1 snðnÞ2 ;
dnðnÞ2 ¼ 1 m2 snðnÞ2 ;
ð2:6Þ
and d snðnÞ ¼ cnðnÞ dnðnÞ; dn
d cnðnÞ ¼ snðnÞ dnðnÞ; dn
d dnðnÞ ¼ m2 snðnÞ cnðnÞ; dn
where m (0 < m < 1) is the modulus, the highest degree of dpu/dnp is taken as p p du qd u ¼ n þ p; O u ¼ ðq þ 1Þn þ p; p ¼ 1; 2; 3; . . . ; q ¼ 0; 1; 2; . . . . O dnp dnp
ð2:7Þ
ð2:8Þ
Thus n can be obtained by balancing the linear term(s) of highest order with the highest-order nonlinear term(s) in Eq. (2.3). Here k, c, a0, . . . , an are parameters to be determined later. Substituting Eq. (2.5) into Eq. (2.3) yields a set of algebraic equations for k, c, a0, . . . , an. WuÕs elimination method[14,15] is a powerful tool for solving nonlinear algebraic equations, and k, c, a0, . . . , an can be obtained. Therefore, the exact double periodic wave solutions are obtained. When m ! 1, the Jacobi elliptic functions degenerate to the hyperbolic functions, i.e., snðnÞ ! tanhðnÞ;
cnðnÞ ! sechðnÞ;
dnðnÞ ! sechðnÞ.
So, besides double periodic wave solutions, solitary wave solutions can also be obtained by this method.
ð2:9Þ
Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
1175
3. Symbolic computation of exact solutions For the nonlinear evolution Eq. (1.1), introducing new independent variables X and T, defined by Z X U ðX 0 ; T Þ dX 0 þ x0 ; t ¼ X ; x¼T þ
ð3:10Þ
1
where u(x, t) = U(X, T) and x0 is a constant [4]. Let Z X W ðX ; T Þ ¼ U ðX 0 ; T Þ dX 0 ;
ð3:11Þ
1
we have W X ðX ; T Þ ¼ U ðX ; T Þ.
ð3:12Þ
From (3.10) it follows that o o o ¼ þu ; oX ot ox
o o ¼w ; oT ox
where w(X, T) = 1 + WT. Use of (3.10) transforms (1.1) into the following equation Z 1 U XXT þ pUU T qU X U T ðX 0 ; T Þ dX 0 þ bU T þ qU X ¼ 0;
ð3:13Þ
ð3:14Þ
X
substituting (3.12) into (3.14) yields W XXXT þ pW X W XT þ qW XX W T þ bW XT þ qW XX ¼ 0.
ð3:15Þ
Now, introducing a new independent variable n ¼ kðX cT Þ;
ð3:16Þ
substituting (3.16) into Eq. (3.15) and integrating once yields k 4 cW 3n 1=2ðp þ qÞk 3 cW 2n þ ðqk 2 bk 2 cÞW n ¼ C;
ð3:17Þ
where C is an integral constant. By taking V(n) = Wn, we obtain k 4 cV 2n 1=2ðp þ qÞk 3 cV ðnÞ2 þ ðqk 3 bk 2 cÞV ðnÞ ¼ C. Suppose that Eq. (3.18) has solutions with the form n X ai snðnÞi ; V ðnÞ ¼
ð3:18Þ
ð3:19Þ
i¼0
By balancing the linear term(s) of highest order with the highest-order nonlinear term in Eq. (3.18) we obtain n = 2. So the ansatz solution of Eq. (3.18) can be rewritten as V ðnÞ ¼ a0 þ a1 snðnÞ þ a2 snðnÞ2 .
ð3:20Þ
Substituting (3.20) into Eq. (3.18) and collecting each power of sn yields algebraic equations about parameters aj (j = 0, 1, 2) and k, c, i.e., k 3 cpa20 2qk 3 a0 þ 2bk 2 ca0 þ k 3 cqa20 þ 4k 4 ca2 ¼ 0; k 4 ca1 m2 k 3 cpa1 a0 þ k 4 ca1 bk 2 ca1 þ qk 3 a1 k 3 cqa1 a0 ¼ 0; 8k 4 ca2 m2 þ 2k 3 qa2 k 3 cqa21 þ 8k 4 ca2 2k 3 cqa2 a0 2k 2 bca2 2k 3 cpa2 a0 þ k 3 cpa21 ¼ 0; 4
2
3
3
2k ca1 m þ k cqa1 a2 þ k cpa1 a2 ¼ 0; k 3 cqa22 þ k 3 cpa22 þ 12k 4 ca2 m2 ¼ 0;
ð3:21Þ
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Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
from which we have (
12km2 ; pþq
a1 0;
a2 ¼
a1 ¼ 0;
12km2 ; a2 ¼ pþq
(
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k m2 þ 1 þ m4 m2 þ 1 ; pþq pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k m2 þ 1 m4 m2 þ 1 a0 ¼ ; pþq
a0 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) kðq 4kc m4 m2 þ 1Þ ; c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) k q þ 4kc m4 m2 þ 1 b¼ . c
b¼
Fig. 1. Hump-shaped curve of solution (3.25) at p = 4, q = 1 and t = 0.
Fig. 2. Loop-shaped curve of solution (3.25) at p = 1, q = 1 and t = 0.
Fig. 3. Cusp-shaped curve of solution (3.25) at p = 2, q = 1 and t = 0.
ð3:22Þ
Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
Substituting (3.22) into the ansatz solution (3.20), we obtain two non-trivial solutions of Eq. (3.18) i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k h 2 V 1 ðn1 Þ ¼ m þ 1 þ m4 m2 þ 1 3m2 snðn1 Þ2 ; p þ q kq p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which n1 ¼ k X T , and bþ4k 2 m4 m2 þ1 h i p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k m2 þ 1 m4 m2 þ 1 3m2 snðn2 Þ2 ; V 2 ðn2 Þ ¼ p þq kq p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi T . where n2 ¼ k X 2 4 2 b4k
m m þ1
Fig. 4. Cusp-shaped curve of solution (3.25) and (3.26) at k = 1, b = 0 and t = 0.
Fig. 5. Hump-shaped curve of solution (3.25) and (3.26) at k = 1, b = 5 and t = 0.
Fig. 6. Loop-shaped curve of solution (3.25) and (3.26) at k = 1, b = 1.5 and t = 0.
1177
ð3:23Þ
ð3:24Þ
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Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
By using (3.10), (3.12) and (3.23), we obtain a parameterized double periodic wave solution of Eq. (1.1) as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
( 4k 2 uðx; tÞ ¼ pþq m þ 1 þ m4 m2 þ 1 3m2 sn2 ðn1 Þ ; ð3:25Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4k 12k x ¼ T þ pþq m 2 þ m4 m2 þ 1 n1 þ pþq Eðsnðn; mÞ; mÞ; pkqffiffiffiffiffiffiffiffiffiffiffiffiffiffi T . where E(sn(n1, m), m) is the Jacobi EllipticE function, n1 ¼ k t 2 bþ4k
m4 m2 þ1
When m ! 1, the double periodic solution (3.25) is degenerated to 8 h i 2 kq > < uðx; tÞ ¼ 12k sech2 k t bþ4k ; 2 T pþq h i kq > 12k : x ¼ T þ pþq tanh k t bþ4k . 2 T
Fig. 7. Hump-shaped curve of solution (3.27) and (3.28) at k = 1, b = 0 and t = 0.
Fig. 8. Cusp-shaped curve of solution (3.27) and (3.28) at k = 1, b = 5 and t = 0.
Fig. 9. Hump-shaped curve of solution (3.27) and (3.28) at k = 1, b = 1.5 and t = 0.
ð3:26Þ
Y. Liu et al. / Chaos, Solitons and Fractals 31 (2007) 1173–1180
By using (3.10), (3.12) and (3.24), we obtain another double periodic wave solution of Eq. (1.1) as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
( 4k 2 uðx; tÞ ¼ pþq m þ 1 m4 m2 þ 1 3m2 sn2 ðn2 Þ ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4k 12k x ¼ T þ pþq m 2 m4 m2 þ 1 n2 þ pþq Eðsnðn2 ; mÞ; mÞ; pkqffiffiffiffiffiffiffiffiffiffiffiffiffiffi T . where n2 ¼ k t 2 4 2 b4k
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ð3:27Þ
m m þ1
When m ! 1, the double periodic solution (3.27) is degenerated to 8 h i kq 4k 2 > < uðx; tÞ ¼ pþq 2 þ 3 sech2 k t b4k ; 2 T h h ii kq > 4k : x ¼ T pþq 2n 3 tanh k t b4k . 2 T
ð3:28Þ
Taking p = 2q, the solitary wave solution (3.26) is reduced to the soliton solution (4.4) and (4.5) in [4], it can be seen that the 1-soliton solution of the mGVE is just a special case of solution (3.26). In [4] it was shown that the solutions (4.4) and (4.5) for different values of parameters b, k may be different types, namely, loops, cusps or humps. So do the solitary wave solutions (3.26) and (3.28) for parameters b, k, p. And also we known that different types of double periodic wave solutions (3.25) and (3.27) are possible for different values of parameters b, k, p, namely, periodic loopshaped wave solutions, periodic cusp-shaped wave solutions or periodic hump-shaped wave solutions. Furthermore, besides parameters b, k, parameter p can independently determine the types of all our obtained solutions. This conclusion can be shown in Figs. 1–3 for fixing k = 1, b = 1. Parameter q mainly works on the periods of solutions. In addition, the periodic loop-shaped wave solutions will be degenerated to loop-like solitary wave solutions. So do the cusp-shaped solutions and hump-shaped solutions. We display this conclusion for fixing p = 2 and m = 0.8 in Figs. 4–6. Finally, the plots of solutions (3.27) and (3.28) are shown in Figs. 7–9 for the same parameter values with those in Figs. 4–6. We can see that there are big difference for the type and shape between solutions (3.25), (3.26) and (3.27), (3.28).
4. Summary In this paper, by the Jacobi elliptic function method, some new exact double periodic wave solutions and solitary wave solutions of a nonlinear evolution equation are constructed. It can be seen that the soliton solutions of the VE equation, the GVE equation and the mGVE equation are just special cases of our obtained new solitary wave solutions. Moreover, it is very interesting that not only our obtained new solitary wave solutions may be loop-shaped, cuspshaped and hump-shaped for different values of parameters, but also our new more general double periodic solutions have this property. Furthermore, apart from parameters b, k, parameter p exerts a great impact on the shape of solutions.
Acknowledgements This work was supported by a National Key Basic Research Project of China (2004CB318000). Liu also would like to express her sincere thanks for the support of the Graduate School of East China Normal University.
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