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An auxiliary equation technique and exact solutions for a nonlinear Klein–Gordon equation
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Chaos, Solitons and Fractals 41 (2009) 82–90 www.elsevier.com/locate/chaos
An auxiliary equation technique and exact solutions for a nonlinear Klein–Gordon equation q Xiumei Lv a, Shaoyong Lai a,*, YongHong Wu b a
Department of Applied Mathematics, Southwestern University of Finance and Economics, 610074 Chengdu, China b Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia Accepted 9 November 2007
Communicated by Prof. L. Marek-Crnjac
Abstract A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is developed to construct the exact travelling wave solutions to a nonlinear Klein–Gordon equation. Some new solutions including the Jacobi elliptic function solutions, the degenerated soliton-like solutions and the triangle function solutions to the equation are obtained. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Exact solutions of nonlinear partial differential equations are of fundamental and significant importance in applied science because they are widely employed to explain some of the nonlinear phenomena and dynamical processes existed in nature world. In order to obtain the exact solutions for some partial differential equations, scientists have established many mathematical approaches such as the trace method [1–3], the perturbation method [4], the inverse scattering method [5], the Ba¨cklund transformations [6], the Adomian decomposition method [7], the Lie group method [8] and so on. With the development of the symbolic computation system, the direct methods for constructing travelling wave solutions to differential equations become feasible. Sirendaoreji [9–12] used the auxiliary equation method to investigate KdV and mKdV equations, Boussinesq equations, sine-Gordon equations and the nonlinear Klein–Gordon equations, respectively. The tanh method and the extended tanh method are reliable techniques and have been utilized by many authors to seek exact solutions of nonlinear equations (see [13–15,21–23]). The Jacobi elliptic function expansion method is confirmed as a powerful technique to solve some nonlinear differential equations (see [16–19]). For other methods relating to use symbolic computation system to investigate nonlinear dispersive equations, the reader is referred to [24,25] and the references therein. q *
This work is supported by the SWUFE’s third period construction item funds of the 211 project. Corresponding author. E-mail address: [email protected] (S. Lai).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.11.013
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
83
We write a nonlinear Klein–Gordon equation in the form utt a2 uxx þ au bun ¼ 0;
n > 1;
ð1Þ
where a, a, b are real constants. Sirendaoreji [12] took advantage of an auxiliary differential equation method and obtained solitons, kinks and anti-kinks, bell and anti-bell solitary wave solutions, periodic solutions, singular solutions and exponential solutions for Eq. (1). Based on Jacobian elliptic function expansion and the modified tanhfunction method, Li and Pan [18] developed a new algebraic method to investigate Eq. (1) in the case n = 3 and obtained its travelling wave solutions. When n = 3, Zheng and Yue [19] used the modified mapping method to acquire the Jacobian elliptic function solutions for Eq. (1). Using the tanh method and the sine–cosine method, Wazwaz [20] found compactons, solitons, solitary patterns and periodic solutions to Eq. (1) with arbitrary exponent n. Motivated by the desire to extend the works made in [12,18–20], we will further investigate Eq. (1). The auxiliary equation technique which differs from that used in [12] is employed to derive the exact travelling wave solutions of Eq. (1), including Jacobi elliptic function solutions, degenerated soliton-like solutions and triangular function solutions. Many of the solutions obtained are different from those presented in previous works [12,18–20].
2. Brief description of the approach The transformation uðx; tÞ ¼ uðnÞðn ¼ lðx ctÞÞ turns a given nonlinear equation P ðu; ut ; ux ; uxx ; uxt ; utt ; . . .Þ ¼ 0
ð2Þ
into the following nonlinear ordinary different equation Qðu; un ; unn ; unnn ; . . .Þ ¼ 0:
ð3Þ
We seek for the solutions of Eq. (3) in the form uðnÞ ¼
N X
gi zi ðnÞ;
ð4Þ
i¼0
where gi ði ¼ 0; 1; 2; . . . ; N Þ are constants which will be determined later. The parameter N is a positive integer and can be determined by balancing the highest order derivative terms and the highest power nonlinear terms in Eq. (3). The op u highest degree of on p can be calculated by 8 h i op u > < O on ¼ N þ p; p ¼ 0; 1; 2; . . . ; p h pi : ð5Þ > o u : O uq on ¼ qN þ p; q; p ¼ 0; 1; 2; . . . p We assume that zðnÞ represents the solutions of the following auxiliary differential equation 2 dz c3 ¼ c1 þ c2 z2 þ z4 ; 2 dn
ð6Þ
where ci ði ¼ 1; 2; 3Þ are real constants. Substituting Eqs. (4) and (6) into Eq. (3) and equating the coefficients of all powers of zðnÞ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zj ðnÞ c1 þ c2 z2 þ c23 z4 ðj ¼ 0; 1; 2; . . .Þ to zero in the resulting equation, several algebraic equations will be obtained. Then solving these algebraic equations by the symbolic computation system Matlab, and combining Eq. (4) and the auxiliary equation Eq. (6), we can get the exact solutions for Eq. (2).
3. Exact travelling wave solutions To find the traveling solutions for Eq. (1), we use the wave variable n ¼ lðx ctÞ, where c–0 and l–0. The wave variable n carries Eq. (1) into the ordinary differential equation l2 ðc2 a2 Þu00 þ au bun ¼ 0: n1
Setting u
ðnÞ ¼ vðnÞ yields
ð7Þ
84
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
(
1
1 u0 ¼ n1 vn11 v0 ; 1 1 1 1 1 1 vn12 ðv0 Þ2 þ n1 vn11 v00 ; u00 ¼ n1 n1
ð8Þ
which turns Eq. (7) into l2 ðc2 a2 Þ½ð2 nÞðv0 Þ2 þ ðn 1Þvv00 þ ðn 1Þ2 ðav2 bv3 Þ ¼ 0:
ð9Þ
From (5), we have N ¼ 2. Therefore, we choose the ansatz vðnÞ ¼ g0 þ g1 z þ g2 z2 ;
ð10Þ
where zðnÞ may be determined by 2 dz c3 ¼ c1 þ c2 z2 þ z4 ; 2 dn
ð11Þ
which possesses several types of solutions listed in Table 1 (see [26]). In Table 1, functions snðnÞ ¼ snðn; rÞ; cnðnÞ ¼ cnðn; rÞ and dnðnÞ ¼ dnðn; rÞ are Jacobian elliptic functions with modulus r ð0 < r < 1Þ, which have the properties snðnÞ ¼ snðnÞ; cnðnÞ ¼ cnðnÞ; dnðnÞ ¼ dnðnÞ; sn2 ðnÞ þ cn2 ðnÞ ¼ 1; dn2 ðnÞ þ r2 sn2 ðnÞ ¼ 1; ðsnðnÞÞ0 ¼ cnðnÞdnðnÞ; ðcnðnÞÞ0 ¼ snðnÞdnðnÞ; ðdnðnÞÞ0 ¼ r2 snðnÞcnðnÞ. Setting r ! 0 yields snðnÞ ! sinðnÞ; cnðnÞ ! cosðnÞ and dnðnÞ ! 1. When r ! 1, it derives that snðnÞ ! tanhðnÞ; cnðnÞ ! sechðnÞ and dnðnÞ ! sechðnÞ.
Table 1 Solutions of auxiliary equation (11) Number
zðnÞ
c3
c2
c1
1
cnðnÞ snðnÞ; cdðnÞ ¼ dnðnÞ
2r2
ðr2 þ 1Þ
1
2
cnðnÞ
2r2
2r2 1
1 r2
2
r2 1
3
dnðnÞ
2
2r
4
1 ncðnÞ ¼ cnðnÞ
2ð1 r2 Þ
2r2 1
r2
5
1 nsðnÞ ¼ snðnÞ ; dcðnÞ ¼ dnðnÞ cnðnÞ
2
ðr2 þ 1Þ
r2
6
1 ndðnÞ ¼ dnðnÞ
2ðr2 1Þ
2 r2
1
7
csðnÞ ¼
8 9 10 11 12
cnðnÞ snðnÞ
2
2
2r
snðnÞ scðnÞ ¼ cnðnÞ
2ð1 r2 Þ
2 r2
snðnÞ sdðnÞ ¼ dnðnÞ
2r2 ðr2 1Þ
2r2 1
dsðnÞ ¼
dnðnÞ snðnÞ
rcnðnÞ dnðnÞ 1 snðnÞ
cnðnÞ snðnÞ snðnÞ cnðnÞ
2
2r 1
2
1 r2
1 r4 r2
12
r þ1 2
ð1r4
1 2
2r2 þ1
1 4
2
2 r þ1 2
1r2 4
1 2
r2 2 2
r4 4
snðnÞ icnðnÞ;
r2 2
r2 2 2
r2 4
16
rsnðnÞ
1 2
12r2 2
1 4
17
snðnÞ 1dnðnÞ
r2 2
r2 2 2
1 4
18
dnðnÞ 1rsnðnÞ
r2 1 2
r2 þ1 2
r2 1 4
19
cnðnÞ 1snðnÞ
1r2 2
r2 þ1 2
r2 þ1 4
13
1 cnðnÞ
14
1 snðnÞ
dnðnÞ snðnÞ
15
20 21
pffiffiffiffiffiffiffiffidnðnÞ 1r2 snðnÞcnðnÞ snðnÞ idnðnÞ; 1cnðnÞ
snðnÞ dnðnÞcnðnÞ cnðnÞ pffiffiffiffiffiffiffiffi 1r2 dnðnÞ
1r 2
2
2
2 2
ð1r Þ 2 4
r 2
2
1 4
2
1 4
r þ1 2 r 2 2
2 2
Þ
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
Substituting Eqs. (10) and (11) into (9) and letting each coefficient of zi obtain l2 ðc2 a2 Þ 2ð2 nÞg22 c3 þ 3ðn 1Þg22 c3 ðn 1Þ2 bg32 ¼ 0; 2
2
2
2
2nl ðc a Þg1 g2 c3 3ðn 1Þ l2 ðc2 a2 Þ
bg1 g22
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 þ c2 z2 þ c23 z4 ð0 6 i 6 6Þ to be zero, we
¼ 0;
1 2 ¼ 0; ng1 c3 þ 4g22 c2 þ 3ðn 1Þg0 g2 c3 þ ðn 1Þ2 ag22 b g0 g22 þ 2g21 g2 þ g2 2g0 g2 þ g21 2
ð12Þ ð13Þ ð14Þ
l2 ðc2 a2 Þ½ðn 1Þg0 g1 c3 þ 5ðn 1Þg1 g2 c2 þ 4ð2 nÞg1 g2 c2 þ ðn 1Þ2 ðbð4g0 g1 g2 þg1 2g0 g2 þ g21 þ 2ag1 g2 ¼ 0;
ð15Þ
2 l2 ðc2 a2 Þ g21 c2 þ 4ð2 nÞg22 c1 þ ð4n 4Þg0 g2 c2 þ þ2ðn 1Þg22 c1 þ ðn 1Þ bg0 2g0 g2 þ g21 þ2g21 g0 þ g2 g20 þ a 2g0 g2 þ g21 ¼ 0;
ð16Þ
l2 ðc2 a2 Þ½4ð2 nÞg1 g2 c1 þ ðn 1Þg0 g1 c2 þ 2ðn 1Þg1 g2 c1 þ ðn 1Þ2 3bg20 g1 þ 2ag0 g1 ¼ 0;
ð17Þ
l2 ðc2 a2 Þ ð2 nÞg21 c1 þ 2ðn 1Þg0 g2 c1 þ ðn 1Þ2 ag20 bg30 ¼ 0:
ð18Þ
Solving (12)–(18) by the use of Matlab, we find 8 ac < g0 ¼ 0; g1 ¼ 0; g2 ¼ bc32 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : l ¼ ða2 ca 2 Þc ; n ¼ 3; 2 and
85
8 < g0 ¼ aðnþ1Þ ; 2b :
g1 ¼ 0; 2
l2 ¼ 2caðn1Þ 2 2 ; 2 ða c Þ
g2 ¼ aðnþ1Þ 2b
c22 ¼ 2c1 c3 ;
ð19Þ
qffiffiffiffiffi
c3 ; 2c1
ð20Þ
n > 1 is a constant;
where ¼ 1; c1 ; c2 ; c3 and c are constants. 3.1. The Jacobi elliptic function solutions to Eq. (1) in the case n = 3 In order to ensure that the wave number l is a real-valued number, we have to choose constants a; a; c and modulus r (c2 depends on r) to satisfy some restrictions. However, in this section, we allow l to take values in complex number domain. From the expressions of g2 and l in (19) and the solutions listed in Table 1, we derive the following Jacobi elliptic function solutions for Eq. (1): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2ar2 a 2 ; ð21Þ sn ðx ctÞ u1:1 ðx; tÞ ¼ bðr2 þ 1Þ ðc2 a2 Þðr2 þ 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2ar2 a 2 ; ð22Þ u1:2 ðx; tÞ ¼ cd ðx ctÞ ðc2 a2 Þðr2 þ 1Þ bðr2 þ 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2ar2 a 2 cn ðx ctÞ ; ð23Þ u1:3 ðx; tÞ ¼ bð2r2 1Þ ðc2 a2 Þð2r2 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a 2 u1:4 ðx; tÞ ¼ ; ð24Þ dn ðx ctÞ bð2 r2 Þ ðc2 a2 Þð2 r2 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2að1 r2 Þ 2 a ; ð25Þ u1:5 ðx; tÞ ¼ nc ðx ctÞ bð2r2 1Þ ðc2 a2 Þð2r2 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a 2 u1:6 ðx; tÞ ¼ ; ð26Þ ns ðx ctÞ bðr2 þ 1Þ ðc2 a2 Þðr2 þ 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a 2 u1:7 ðx; tÞ ¼ ; ð27Þ dc ðx ctÞ bðr2 þ 1Þ ðc2 a2 Þðr2 þ 1Þ
86
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2aðr2 1Þ 2 a nd ðx ctÞ 2 u1:8 ðx; tÞ ¼ ; bð2 r2 Þ ðc a2 Þð2 r2 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a 2 cs ðx ctÞ u1:9 ðx; tÞ ¼ ; bð2 r2 Þ ðc2 a2 Þð2 r2 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2að1 r2 Þ 2 a sc ðx ctÞ ðaðc2 a2 Þ < 0Þ; u1:10 ðx; tÞ ¼ bð2 r2 Þ ðc2 a2 Þð2 r2 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2ar2 ðr2 1Þ 2 a sd ðx ctÞ ; u1:11 ðx; tÞ ¼ bð2r2 1Þ ðc2 a2 Þð2r2 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a 2 ds ðx ctÞ u1:12 ðx; tÞ ¼ ; bð2r2 1Þ ðc2 a2 Þð2r2 1Þ
ð28Þ ð29Þ ð30Þ ð31Þ ð32Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = a 2a 2a ðx ctÞ þ dn ðx ctÞ u1:13 ðx; tÞ ¼ ; rcn ; :bðr2 þ 1Þ ðc2 a2 Þðr2 þ 1Þ ðc2 a2 Þðr2 þ 1Þ 8 <
ð33Þ
8 <
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = a 2a 2a ns 2 ðx ctÞ þ cs 2 ðx ctÞ u1:14 ðx; tÞ ¼ ; 2 2 2 2 2 : bð1 2r Þ ; ðc a Þð1 2r Þ ðc a Þð1 2r Þ ð34Þ 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = < að1 r2 Þ 2a 2a ðx ctÞ þ sc 2 ðx ctÞ u1:15 ðx; tÞ ¼ ; nc 2 2 2 2 2 2 ; : bðr þ 1Þ ðc a Þðr þ 1Þ ðc a Þðr þ 1Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = a 2a 2a ðx ctÞ þ ds 2 ðx ctÞ u1:16 ðx; tÞ ¼ ; ns 2 2 2 2 2 2 ; : bðr 2Þ ðc a Þðr 2Þ ðc a Þðr 2Þ 8 <
8 <
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = ar2 2a 2a sn ðx ctÞ þ icn ðx ctÞ u1:17 ðx; tÞ ¼ ; : bðr2 2Þ ; ðc2 a2 Þðr2 2Þ ðc2 a2 Þðr2 2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = a 2a 2a ðx ctÞ þ idn ðx ctÞ ; rsn u1:18 ðx; tÞ ¼ 2 2 2 2 2 2 2 ; : bð1 2r Þ ðc a Þð2r 1Þ ðc a Þð2r 1Þ 8 <
912 8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a > > = < ar2 dn2 ðx ctÞ ðc2 a2 Þðr2 2Þ u1:19 ðx; tÞ ¼ h
i q q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2> pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > 2a 2a :bðr 2Þ ðx ctÞ þ cn ðx ctÞ ; 1 r2 sn ðc2 a2 Þðr2 2Þ ðc2 a2 Þðr2 2Þ
ð35Þ
ð36Þ
ð37Þ
ð38Þ
ðaðc2 a2 Þ > 0Þ; ð39Þ
912
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a > = ðx ctÞ 2 2 2 ðc a Þð12r Þ a u1:20 ðx; tÞ ¼ h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 ; 2 > > : bð1 2r Þ 1 þ cn ðc2 a22a ðx ctÞ ; Þð12r2 Þ 8 > <
sn2
912
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a > = ðx ctÞ 2 2 2 ðc a Þðr 2Þ ar u1:21 ðx; tÞ ¼ h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 ; 2 > > : bðr 2Þ 1 þ dn 2 22a 2 ðx ctÞ ; 8 > <
2
ð40Þ
sn2
ðc a Þðr 2Þ
ð41Þ
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
912 8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > = < aðr2 1Þ dn2 ðc2 a22aÞðr2 þ1Þðx ctÞ u1:22 ðx; tÞ ¼ h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 ; 2 > > : bðr þ 1Þ 1 þ rsn ðc2 a22aÞðr2 þ1Þðx ctÞ ; 8 > < að1 r2 Þ u1:23 ðx; tÞ ¼ h 2 > : bðr þ 1Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 912 > = ðc2 a22aÞðr2 þ1Þðx ctÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 ; > 1 þ sn 2 22a 2 ðx ctÞ ;
87
ð42Þ
cn2
ð43Þ
ðc a Þðr þ1Þ
91 8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > > = < að1 r2 Þ2 sn2 ðc2 a22aÞðr2 þ1Þðx ctÞ u1:24 ðx; tÞ ¼ h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 ; 2 > > : bðr þ 1Þ dn ðc2 a22aÞðr2 þ1Þðx ctÞ ; ðc2 a22aÞðr2 þ1Þðx ctÞ þ cn
ð44Þ
8 912
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a > > < ar4 = cn2 ðx ctÞ ðc2 a2 Þðr2 2Þ ; u1:25 ðx; tÞ ¼ h
i q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2> pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > 2a :bðr 2Þ ðx ctÞ ; 1 r2 þ dn ðc2 a2 Þðr2 2Þ
ð45Þ
where 0 < r < 1 and c is an arbitrary constant. Remark 1. Solutions from u1:1 to u1:12 are in full agreement with those presented in Li and Pan’s paper [18], solution u1:19 was found by Zheng [19] and the other solutions are new ones. Remark 2. When r ! 0; u1:1 ; u1:2 ; u1:3 ; u1:11 ; u1:17 ; u1:19 ; u1:21 and u1:25 become zero, u1:4 ; u1:8 ; u1:13 ; u1:18 and u1:22 become constants while other solutions degenerate to triangular solutions of Eq. (1). In particular, as r ! 0; u1:5 ; u1:6 ; u1:9 and u1:10 become rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a a 2 u1:5:0 ðx; tÞ ¼ ðx ctÞ ; > 0; ð46Þ sec b c2 a2 c2 a2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a 2 a a u1:6:0 ðx; tÞ ¼ ðx ctÞ ; > 0; ð47Þ csc b c2 a2 c2 a 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 a a a u1:9:0 ðx; tÞ ¼ cot2 ; < 0; ð48Þ ðx ctÞ 2 b 2ðc a2 Þ c2 a2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 a a a 2 ðx ctÞ 2 ; < 0; ð49Þ u1:10:0 ðx; tÞ ¼ tan b 2ðc a2 Þ c2 a2 respectively. Solutions from (46)–(49) were found in [18] while u1:5:0 and u1:6:0 were obtained by Sirendaoreji [12]. Remark 3. When r ! 1, some of the solutions listed in (21)–(45) become zero or constants, and the other solutions degenerate to soliton-like solutions. Specifically, u1:1 ; u1:3 ; u1:6 ; and u1:9 are turned into rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 a a a ðx ctÞ ; > 0; ð50Þ tanh2 u1:1:1 ðx; tÞ ¼ b 2ðc2 a2 Þ c2 a2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a a 2 2 ðx ctÞ u1:3:1 ðx; tÞ ¼ ; < 0; ð51Þ sech b c a2 c2 a2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 a a a u1:6:1 ðx; tÞ ¼ ðx ctÞ ; > 0; ð52Þ coth2 b 2ðc2 a2 Þ c2 a2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a a 2 ðx ctÞ u1:9:1 ðx; tÞ ¼ csch2 ; < 0; ð53Þ 2 2 b c a c a2 respectively, which are the solutions obtained in [12,18].
88
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
3.2. Soliton and triangular function solutions of Eq. (1) From the expression c22 ¼ 2c1 c3 in the formula of (20) and the conditions listed in Table 1, we find that the modulus r is either 1 or 0. Therefore, the degenerated soliton-like and triangular function solutions for Eq. (1) are expressed by 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ aðn þ 1Þ a 2 ðn 1Þ u2:1 ðx; tÞ ¼ ðaðc2 a2 Þ < 0Þ; ð54Þ ðx ctÞ þ tanh 2 2b 2b 2 c a2 u2:2 ðx; tÞ ¼
u2:3 ðx; tÞ ¼
u2:4 ðx; tÞ ¼
u2:5 ðx; tÞ ¼
u2:6 ðx; tÞ ¼
u2:7 ðx; tÞ ¼
u2:8 ðx; tÞ ¼
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ aðn þ 1Þ ðn 1Þ a ðx ctÞ þ coth2 2 2b 2b 2 c a2
8 > :
2b
8 > :
2b
8 > :
2b
8 > :
2b
8 > :
2b
8 > :
u2:9 ðx; tÞ ¼
2b
1 h i2 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = 1 þ cosh ðn 1Þ ðx ctÞ 2 2 c a aðn þ 1Þ h i þ pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; sinh2 ðn 1Þ c2 a 2 ðx ctÞ 1 h i2 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = 1 þ cosh ðn 1Þ ðx ctÞ 2 2 c a aðn þ 1Þ h i pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; sinh2 ðn 1Þ c2 a 2 ðx ctÞ
u2:12 ðx; tÞ ¼
1 h i 32 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = ðx ctÞ sinh ðn 1Þ 2 2 c a aðn þ 1Þ 4 h i5 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; 1 þ cosh ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ < 0Þ;
ð57Þ
ðaðc2 a2 Þ < 0Þ;
ð58Þ
1 h i 32 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = sinh ðn 1Þ ðx ctÞ 2 2 c a aðn þ 1Þ 4 h i5 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; 1 þ cosh ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ < 0Þ;
ð59Þ
2
1 h i32 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = i þ sinh ðn 1Þ ðx ctÞ 2 2 c a aðn þ 1Þ 4 h i 5 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; cosh ðn 1Þ c2 a2 ðx ctÞ
2
1 h i32 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = ðx ctÞ i þ sinh ðn 1Þ 2 2 c a aðn þ 1Þ 4 h i 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; cosh ðn 1Þ c2 a2 ðx ctÞ
2b
8 >
ð56Þ
2
8 >
> :
ðaðc2 a2 Þ < 0Þ;
ðaðc2 a2 Þ < 0Þ;
ð60Þ
ðaðc2 a2 Þ < 0Þ;
ð61Þ
2
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ aðn þ 1Þ 2 ðn 1Þ a ðx ctÞ þ cot 2b 2b 2 c2 a2
> :
ð55Þ
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ aðn þ 1Þ a 2 ðn 1Þ ðx ctÞ þ tan u2:10 ðx; tÞ ¼ 2b 2b 2 c2 a2
u2:11 ðx; tÞ ¼
ðaðc2 a2 Þ < 0Þ;
ðaðc2 a2 Þ > 0Þ;
ðaðc2 a2 Þ > 0Þ;
1
h i2 9n1 pffiffiffiffiffiffiffiffi ffi > a = ðx ctÞ 1 þ cos ðn 1Þ c2 a2 aðn þ 1Þ h i þ ffi pffiffiffiffiffiffiffiffi 2 > a 2b ; sin ðn 1Þ c2 a2 ðx ctÞ 1
h i2 9n1 ffi pffiffiffiffiffiffiffiffi > a = 1 þ cos ðn 1Þ ðx ctÞ c2 a2 aðn þ 1Þ h i ffi pffiffiffiffiffiffiffiffi 2 > a 2b ; sin ðn 1Þ c2 a2 ðx ctÞ
ð62Þ
ð63Þ
ðaðc2 a2 Þ > 0Þ;
ð64Þ
ðaðc2 a2 Þ > 0Þ;
ð65Þ
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
u2:13 ðx; tÞ ¼
u2:14 ðx; tÞ ¼
u2:15 ðx; tÞ ¼
u2:16 ðx; tÞ ¼
u2:17 ðx; tÞ ¼
u2:18 ðx; tÞ ¼
8 > :
2b
8 > :
2b
8 > > > :
2b
8 > > > :
2b
8 > :
2b
8 > :
2b
89
9n1 1 h i ffi pffiffiffiffiffiffiffiffi 2 > a = sin ðn 1Þ ðx ctÞ c2 a2 aðn þ 1Þ þ h i 2 pffiffiffiffiffiffiffiffi ffi > 2b a ; 1 þ cos ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ > 0Þ;
ð66Þ
9n1 1 h i ffi pffiffiffiffiffiffiffiffi 2 > a = sin ðn 1Þ ðx ctÞ c2 a2 aðn þ 1Þ h i 2 pffiffiffiffiffiffiffiffi ffi > 2b a ; 1 þ cos ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ > 0Þ;
ð67Þ
ðaðc2 a2 Þ > 0Þ;
ð68Þ
ðaðc2 a2 Þ > 0Þ;
ð69Þ
9n1 1 h i ffi pffiffiffiffiffiffiffiffi > a 2 = cos ðn 1Þ ðx ctÞ c2 a2 aðn þ 1Þ þ h i2 pffiffiffiffiffiffiffiffi ffi > 2b a ; 1 þ sin ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ > 0Þ;
ð70Þ
9n1 1 h i pffiffiffiffiffiffiffiffi ffi > a = cos2 ðn 1Þ c2 a 2 ðx ctÞ h i2 ffi pffiffiffiffiffiffiffiffi > a ; 1 þ sin ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ > 0Þ:
ð71Þ
1 i2 9 n1 pffiffiffiffiffiffiffiffi ffi > > a aðn þ 1Þ 1 þ sin ðn 1Þ c2 a2 ðx ctÞ =
þ qffiffiffiffiffiffiffiffi > 2b > ; cos2 ðn 1Þ ðacÞ ðx ctÞ bc
h
1 h i2 9 n1 pffiffiffiffiffiffiffiffi ffi > > a = ðx ctÞ 1 þ sin ðn 1Þ c2 a2 aðn þ 1Þ
qffiffiffiffiffiffiffiffi > 2b > ; ðx ctÞ cos2 ðn 1Þ ðacÞ bc
aðn þ 1Þ 2b
Remark 4. Setting ¼ 1 and using the identities 1 1 ¼ sech2 a; 1 þ coshð2aÞ 2 1 1 ¼ csch2 a; 1 coshð2aÞ 2
1 1 ¼ sech2 a; 1 þ cosð2aÞ 2 1 1 ¼ csc2 a; 1 cosð2aÞ 2
we find that some of the solutions listed in (54)–(71) possess the same form. For example, u2:3 and u2:4 become u2:1 while u2:5 and u2:6 are the same as u2:2 . Remark 5. According to the identities 1 tanh2 ðaÞ ¼ sech2 ðaÞ; 1 coth2 ðaÞ ¼ csch2 ðaÞ and letting ¼ 1, solutions u2:1 and u2:2 become 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ a 2 ðn 1Þ u2:1:0 ðx; tÞ ¼ ðx ctÞ sech 2 2b 2 c a2
u2:2:0 ðx; tÞ ¼
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ ðn 1Þ a ðx ctÞ csch2 2 2b 2 c a2
ðaðc2 a2 Þ < 0Þ;
ðaðc2 a2 Þ < 0Þ:
ð72Þ
ð73Þ
From the expressions 1 þ tan2 ðaÞ ¼ sec2 ðaÞ; 1 þ cot2 ðaÞ ¼ csc2 ðaÞ and choosing ¼ 1, u2:9 and u2:10 are turned into 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ 2 ðn 1Þ a u2:9:0 ðx; tÞ ¼ ðaðc2 a2 Þ > 0Þ; ð74Þ ðx ctÞ csc 2b 2 c2 a2
90
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ a 2 ðn 1Þ ðx ctÞ u2:10:0 ðx; tÞ ¼ sec 2b 2 c2 a2
ðaðc2 a2 Þ > 0Þ:
ð75Þ
Solutions in (72)–(75) were obtained by Sirendaoreji [12] and Wazwaz [20].
4. Conclusion Using a different auxiliary equation presented in Sirendaoreji [12] and the symbolic computation system Matlab, we obtain the Jacobi function solutions, the solitons and the triangular function solutions to nonlinear Klein–Gordon equation (1) in the case n = 3. As seen in the above, the method not only can recover the previously known solutions which were found by tanh and sin–cosine method, but also can provide new and more distinct solutions for Eq. (1). However, for arbitrary exponent n, we only obtain the degenerated soliton-like and triangle function solutions to Eq. (1). It may need other techniques to deal with the Jacobi elliptic function solutions for the nonlinear Klein–Gordon equation (1) with arbitrary exponent n.
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Wadati M. Introduction to solitons, Pramana. J Phys 2001;57(5–6):841–7. Wadati M. The exact solution of the modified Kortweg–de Vries equation. J Phys Soc Jpn 1972;32:1681–7. Wadati M. The modified Kortweg–de Vries equation. J Phys Soc Jpn 1973;34:1289–96. Logan DJ. An introduction to nonlinear partial differential equations. New York: Wiley; 1994. Ablowtiz MJ, Clarkson PA. Nonlinear evolution equations and inverse scattering. Cambridge University Press; 1991. Alhumaizi KI. Comparison of finite difference methods for the numerical simulation of reaction flow. Comput Chem Eng 2004;28:1759–69. Hashim I. Adomian decomposition method for solving BVPs for fourth-order integro-differential equations. J Comput Appl Math 2006;193:658–64. Olver PL. Applications of Lie groups to differential equations. Springer; 1986. Sirendaoreji, Jiong Sun. Auxiliary equation method for solving nonlinear partial differential equations. Phys Lett A 2003;309:387–96. Sirendaoreji. A new auxiliary equation and exact traveling wave solutions of nonlinear equations. Phys Lett A 2006;356:124–30. Sirendaoreji, Sun J. A direct method for solving sine-Gordon type equations. Phys Lett A 2002;298:133–9. Sirendaoreji. Exact traveling wave solutions for four forms of nonlinear Klein–Gordon equations. Phys Lett A 2007;303:440–7. Li W, Xie FD. Using symbolic computation to construct traveling wave solution of nonlinear partial differential equation. Chinese Phys 2004;13:1639–64. Fan E. Using symbolic computation to exactly solve a new coupled mKdV system. Phys Lett A 2002;299:46–8. Abdou MA, Soliman AA. Modified extended tanh-function method and its application on nonlinear physics equations. Phys Lett A 2006;353:487–92. El-Wakil SA, Abdou MA. New solitons and periodic wave solutions for nonlinear evolution equations. Phys Lett A 2006;353:40–7. Wang Q, Chen Y. An extended Jacobi elliptic function rational expansion method and its application to (2 + 1)-dimensional dispersive long wave equation. Phys Lett A 2005;340:411–26. Li P, Pan ZL. A new development on Jacobian elliptic function expansion method. Phys Lett A 2004;332:39–48. Zheng Q, Yue P. New exact solution of nonlinear Klein–Gordon equation. Chinese Phys 2006;15:35–8. Wazwaz AM. Compactons, solitons and periodic solutions for some forms of nonlinear Klein–Gordon equations. Chaos Solitons & Fractals 2006;28:1005–13. Wazwaz AM. The tanh method, solitons and periodic solutions for the Dodd–Bullough–Mikhailov and Tzitzeica–Dodd– Bullough equations. Chaos Solitons & Fractals 2005;25:55–63. Wazwaz AM. Exact solutions for the generalized sine-Gordon and the generalized sinh-Gordon equations. Chaos Solitons & Fractals 2006;28:127–35. Wazwaz AM. Two reliable methods for solving variants of the KdV equations with compact and noncompact structures. Chaos Solitons & Fractals 2006;28:454–62. Wazwaz AM. A variable separated ODE method for solving the triple sine-Gordon and the triple sinh-Gordon equations. Chaos Solitons & Fractals 2007;33:703–10. Zhang S. Symbolic computation and new families of exact non-travelling wave solution of (2 + 1)-dimensional Konopelchenko– Dubrovsky equations. Chaos Solitons & Fractals 2007;31:951–9. Li HM. New exact solutions of nonlinear Gross–Pitaevskii equation with weak bias magnetic and time-dependent laser fields. Chinese Phys 2005;14:251–6.
Chaos, Solitons and Fractals 41 (2009) 82–90 www.elsevier.com/locate/chaos
An auxiliary equation technique and exact solutions for a nonlinear Klein–Gordon equation q Xiumei Lv a, Shaoyong Lai a,*, YongHong Wu b a
Department of Applied Mathematics, Southwestern University of Finance and Economics, 610074 Chengdu, China b Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia Accepted 9 November 2007
Communicated by Prof. L. Marek-Crnjac
Abstract A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is developed to construct the exact travelling wave solutions to a nonlinear Klein–Gordon equation. Some new solutions including the Jacobi elliptic function solutions, the degenerated soliton-like solutions and the triangle function solutions to the equation are obtained. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Exact solutions of nonlinear partial differential equations are of fundamental and significant importance in applied science because they are widely employed to explain some of the nonlinear phenomena and dynamical processes existed in nature world. In order to obtain the exact solutions for some partial differential equations, scientists have established many mathematical approaches such as the trace method [1–3], the perturbation method [4], the inverse scattering method [5], the Ba¨cklund transformations [6], the Adomian decomposition method [7], the Lie group method [8] and so on. With the development of the symbolic computation system, the direct methods for constructing travelling wave solutions to differential equations become feasible. Sirendaoreji [9–12] used the auxiliary equation method to investigate KdV and mKdV equations, Boussinesq equations, sine-Gordon equations and the nonlinear Klein–Gordon equations, respectively. The tanh method and the extended tanh method are reliable techniques and have been utilized by many authors to seek exact solutions of nonlinear equations (see [13–15,21–23]). The Jacobi elliptic function expansion method is confirmed as a powerful technique to solve some nonlinear differential equations (see [16–19]). For other methods relating to use symbolic computation system to investigate nonlinear dispersive equations, the reader is referred to [24,25] and the references therein. q *
This work is supported by the SWUFE’s third period construction item funds of the 211 project. Corresponding author. E-mail address: [email protected] (S. Lai).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.11.013
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
83
We write a nonlinear Klein–Gordon equation in the form utt a2 uxx þ au bun ¼ 0;
n > 1;
ð1Þ
where a, a, b are real constants. Sirendaoreji [12] took advantage of an auxiliary differential equation method and obtained solitons, kinks and anti-kinks, bell and anti-bell solitary wave solutions, periodic solutions, singular solutions and exponential solutions for Eq. (1). Based on Jacobian elliptic function expansion and the modified tanhfunction method, Li and Pan [18] developed a new algebraic method to investigate Eq. (1) in the case n = 3 and obtained its travelling wave solutions. When n = 3, Zheng and Yue [19] used the modified mapping method to acquire the Jacobian elliptic function solutions for Eq. (1). Using the tanh method and the sine–cosine method, Wazwaz [20] found compactons, solitons, solitary patterns and periodic solutions to Eq. (1) with arbitrary exponent n. Motivated by the desire to extend the works made in [12,18–20], we will further investigate Eq. (1). The auxiliary equation technique which differs from that used in [12] is employed to derive the exact travelling wave solutions of Eq. (1), including Jacobi elliptic function solutions, degenerated soliton-like solutions and triangular function solutions. Many of the solutions obtained are different from those presented in previous works [12,18–20].
2. Brief description of the approach The transformation uðx; tÞ ¼ uðnÞðn ¼ lðx ctÞÞ turns a given nonlinear equation P ðu; ut ; ux ; uxx ; uxt ; utt ; . . .Þ ¼ 0
ð2Þ
into the following nonlinear ordinary different equation Qðu; un ; unn ; unnn ; . . .Þ ¼ 0:
ð3Þ
We seek for the solutions of Eq. (3) in the form uðnÞ ¼
N X
gi zi ðnÞ;
ð4Þ
i¼0
where gi ði ¼ 0; 1; 2; . . . ; N Þ are constants which will be determined later. The parameter N is a positive integer and can be determined by balancing the highest order derivative terms and the highest power nonlinear terms in Eq. (3). The op u highest degree of on p can be calculated by 8 h i op u > < O on ¼ N þ p; p ¼ 0; 1; 2; . . . ; p h pi : ð5Þ > o u : O uq on ¼ qN þ p; q; p ¼ 0; 1; 2; . . . p We assume that zðnÞ represents the solutions of the following auxiliary differential equation 2 dz c3 ¼ c1 þ c2 z2 þ z4 ; 2 dn
ð6Þ
where ci ði ¼ 1; 2; 3Þ are real constants. Substituting Eqs. (4) and (6) into Eq. (3) and equating the coefficients of all powers of zðnÞ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zj ðnÞ c1 þ c2 z2 þ c23 z4 ðj ¼ 0; 1; 2; . . .Þ to zero in the resulting equation, several algebraic equations will be obtained. Then solving these algebraic equations by the symbolic computation system Matlab, and combining Eq. (4) and the auxiliary equation Eq. (6), we can get the exact solutions for Eq. (2).
3. Exact travelling wave solutions To find the traveling solutions for Eq. (1), we use the wave variable n ¼ lðx ctÞ, where c–0 and l–0. The wave variable n carries Eq. (1) into the ordinary differential equation l2 ðc2 a2 Þu00 þ au bun ¼ 0: n1
Setting u
ðnÞ ¼ vðnÞ yields
ð7Þ
84
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
(
1
1 u0 ¼ n1 vn11 v0 ; 1 1 1 1 1 1 vn12 ðv0 Þ2 þ n1 vn11 v00 ; u00 ¼ n1 n1
ð8Þ
which turns Eq. (7) into l2 ðc2 a2 Þ½ð2 nÞðv0 Þ2 þ ðn 1Þvv00 þ ðn 1Þ2 ðav2 bv3 Þ ¼ 0:
ð9Þ
From (5), we have N ¼ 2. Therefore, we choose the ansatz vðnÞ ¼ g0 þ g1 z þ g2 z2 ;
ð10Þ
where zðnÞ may be determined by 2 dz c3 ¼ c1 þ c2 z2 þ z4 ; 2 dn
ð11Þ
which possesses several types of solutions listed in Table 1 (see [26]). In Table 1, functions snðnÞ ¼ snðn; rÞ; cnðnÞ ¼ cnðn; rÞ and dnðnÞ ¼ dnðn; rÞ are Jacobian elliptic functions with modulus r ð0 < r < 1Þ, which have the properties snðnÞ ¼ snðnÞ; cnðnÞ ¼ cnðnÞ; dnðnÞ ¼ dnðnÞ; sn2 ðnÞ þ cn2 ðnÞ ¼ 1; dn2 ðnÞ þ r2 sn2 ðnÞ ¼ 1; ðsnðnÞÞ0 ¼ cnðnÞdnðnÞ; ðcnðnÞÞ0 ¼ snðnÞdnðnÞ; ðdnðnÞÞ0 ¼ r2 snðnÞcnðnÞ. Setting r ! 0 yields snðnÞ ! sinðnÞ; cnðnÞ ! cosðnÞ and dnðnÞ ! 1. When r ! 1, it derives that snðnÞ ! tanhðnÞ; cnðnÞ ! sechðnÞ and dnðnÞ ! sechðnÞ.
Table 1 Solutions of auxiliary equation (11) Number
zðnÞ
c3
c2
c1
1
cnðnÞ snðnÞ; cdðnÞ ¼ dnðnÞ
2r2
ðr2 þ 1Þ
1
2
cnðnÞ
2r2
2r2 1
1 r2
2
r2 1
3
dnðnÞ
2
2r
4
1 ncðnÞ ¼ cnðnÞ
2ð1 r2 Þ
2r2 1
r2
5
1 nsðnÞ ¼ snðnÞ ; dcðnÞ ¼ dnðnÞ cnðnÞ
2
ðr2 þ 1Þ
r2
6
1 ndðnÞ ¼ dnðnÞ
2ðr2 1Þ
2 r2
1
7
csðnÞ ¼
8 9 10 11 12
cnðnÞ snðnÞ
2
2
2r
snðnÞ scðnÞ ¼ cnðnÞ
2ð1 r2 Þ
2 r2
snðnÞ sdðnÞ ¼ dnðnÞ
2r2 ðr2 1Þ
2r2 1
dsðnÞ ¼
dnðnÞ snðnÞ
rcnðnÞ dnðnÞ 1 snðnÞ
cnðnÞ snðnÞ snðnÞ cnðnÞ
2
2r 1
2
1 r2
1 r4 r2
12
r þ1 2
ð1r4
1 2
2r2 þ1
1 4
2
2 r þ1 2
1r2 4
1 2
r2 2 2
r4 4
snðnÞ icnðnÞ;
r2 2
r2 2 2
r2 4
16
rsnðnÞ
1 2
12r2 2
1 4
17
snðnÞ 1dnðnÞ
r2 2
r2 2 2
1 4
18
dnðnÞ 1rsnðnÞ
r2 1 2
r2 þ1 2
r2 1 4
19
cnðnÞ 1snðnÞ
1r2 2
r2 þ1 2
r2 þ1 4
13
1 cnðnÞ
14
1 snðnÞ
dnðnÞ snðnÞ
15
20 21
pffiffiffiffiffiffiffiffidnðnÞ 1r2 snðnÞcnðnÞ snðnÞ idnðnÞ; 1cnðnÞ
snðnÞ dnðnÞcnðnÞ cnðnÞ pffiffiffiffiffiffiffiffi 1r2 dnðnÞ
1r 2
2
2
2 2
ð1r Þ 2 4
r 2
2
1 4
2
1 4
r þ1 2 r 2 2
2 2
Þ
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
Substituting Eqs. (10) and (11) into (9) and letting each coefficient of zi obtain l2 ðc2 a2 Þ 2ð2 nÞg22 c3 þ 3ðn 1Þg22 c3 ðn 1Þ2 bg32 ¼ 0; 2
2
2
2
2nl ðc a Þg1 g2 c3 3ðn 1Þ l2 ðc2 a2 Þ
bg1 g22
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 þ c2 z2 þ c23 z4 ð0 6 i 6 6Þ to be zero, we
¼ 0;
1 2 ¼ 0; ng1 c3 þ 4g22 c2 þ 3ðn 1Þg0 g2 c3 þ ðn 1Þ2 ag22 b g0 g22 þ 2g21 g2 þ g2 2g0 g2 þ g21 2
ð12Þ ð13Þ ð14Þ
l2 ðc2 a2 Þ½ðn 1Þg0 g1 c3 þ 5ðn 1Þg1 g2 c2 þ 4ð2 nÞg1 g2 c2 þ ðn 1Þ2 ðbð4g0 g1 g2 þg1 2g0 g2 þ g21 þ 2ag1 g2 ¼ 0;
ð15Þ
2 l2 ðc2 a2 Þ g21 c2 þ 4ð2 nÞg22 c1 þ ð4n 4Þg0 g2 c2 þ þ2ðn 1Þg22 c1 þ ðn 1Þ bg0 2g0 g2 þ g21 þ2g21 g0 þ g2 g20 þ a 2g0 g2 þ g21 ¼ 0;
ð16Þ
l2 ðc2 a2 Þ½4ð2 nÞg1 g2 c1 þ ðn 1Þg0 g1 c2 þ 2ðn 1Þg1 g2 c1 þ ðn 1Þ2 3bg20 g1 þ 2ag0 g1 ¼ 0;
ð17Þ
l2 ðc2 a2 Þ ð2 nÞg21 c1 þ 2ðn 1Þg0 g2 c1 þ ðn 1Þ2 ag20 bg30 ¼ 0:
ð18Þ
Solving (12)–(18) by the use of Matlab, we find 8 ac < g0 ¼ 0; g1 ¼ 0; g2 ¼ bc32 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : l ¼ ða2 ca 2 Þc ; n ¼ 3; 2 and
85
8 < g0 ¼ aðnþ1Þ ; 2b :
g1 ¼ 0; 2
l2 ¼ 2caðn1Þ 2 2 ; 2 ða c Þ
g2 ¼ aðnþ1Þ 2b
c22 ¼ 2c1 c3 ;
ð19Þ
qffiffiffiffiffi
c3 ; 2c1
ð20Þ
n > 1 is a constant;
where ¼ 1; c1 ; c2 ; c3 and c are constants. 3.1. The Jacobi elliptic function solutions to Eq. (1) in the case n = 3 In order to ensure that the wave number l is a real-valued number, we have to choose constants a; a; c and modulus r (c2 depends on r) to satisfy some restrictions. However, in this section, we allow l to take values in complex number domain. From the expressions of g2 and l in (19) and the solutions listed in Table 1, we derive the following Jacobi elliptic function solutions for Eq. (1): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2ar2 a 2 ; ð21Þ sn ðx ctÞ u1:1 ðx; tÞ ¼ bðr2 þ 1Þ ðc2 a2 Þðr2 þ 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2ar2 a 2 ; ð22Þ u1:2 ðx; tÞ ¼ cd ðx ctÞ ðc2 a2 Þðr2 þ 1Þ bðr2 þ 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2ar2 a 2 cn ðx ctÞ ; ð23Þ u1:3 ðx; tÞ ¼ bð2r2 1Þ ðc2 a2 Þð2r2 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a 2 u1:4 ðx; tÞ ¼ ; ð24Þ dn ðx ctÞ bð2 r2 Þ ðc2 a2 Þð2 r2 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2að1 r2 Þ 2 a ; ð25Þ u1:5 ðx; tÞ ¼ nc ðx ctÞ bð2r2 1Þ ðc2 a2 Þð2r2 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a 2 u1:6 ðx; tÞ ¼ ; ð26Þ ns ðx ctÞ bðr2 þ 1Þ ðc2 a2 Þðr2 þ 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a 2 u1:7 ðx; tÞ ¼ ; ð27Þ dc ðx ctÞ bðr2 þ 1Þ ðc2 a2 Þðr2 þ 1Þ
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X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2aðr2 1Þ 2 a nd ðx ctÞ 2 u1:8 ðx; tÞ ¼ ; bð2 r2 Þ ðc a2 Þð2 r2 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a 2 cs ðx ctÞ u1:9 ðx; tÞ ¼ ; bð2 r2 Þ ðc2 a2 Þð2 r2 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2að1 r2 Þ 2 a sc ðx ctÞ ðaðc2 a2 Þ < 0Þ; u1:10 ðx; tÞ ¼ bð2 r2 Þ ðc2 a2 Þð2 r2 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2ar2 ðr2 1Þ 2 a sd ðx ctÞ ; u1:11 ðx; tÞ ¼ bð2r2 1Þ ðc2 a2 Þð2r2 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a 2 ds ðx ctÞ u1:12 ðx; tÞ ¼ ; bð2r2 1Þ ðc2 a2 Þð2r2 1Þ
ð28Þ ð29Þ ð30Þ ð31Þ ð32Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = a 2a 2a ðx ctÞ þ dn ðx ctÞ u1:13 ðx; tÞ ¼ ; rcn ; :bðr2 þ 1Þ ðc2 a2 Þðr2 þ 1Þ ðc2 a2 Þðr2 þ 1Þ 8 <
ð33Þ
8 <
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = a 2a 2a ns 2 ðx ctÞ þ cs 2 ðx ctÞ u1:14 ðx; tÞ ¼ ; 2 2 2 2 2 : bð1 2r Þ ; ðc a Þð1 2r Þ ðc a Þð1 2r Þ ð34Þ 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = < að1 r2 Þ 2a 2a ðx ctÞ þ sc 2 ðx ctÞ u1:15 ðx; tÞ ¼ ; nc 2 2 2 2 2 2 ; : bðr þ 1Þ ðc a Þðr þ 1Þ ðc a Þðr þ 1Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = a 2a 2a ðx ctÞ þ ds 2 ðx ctÞ u1:16 ðx; tÞ ¼ ; ns 2 2 2 2 2 2 ; : bðr 2Þ ðc a Þðr 2Þ ðc a Þðr 2Þ 8 <
8 <
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = ar2 2a 2a sn ðx ctÞ þ icn ðx ctÞ u1:17 ðx; tÞ ¼ ; : bðr2 2Þ ; ðc2 a2 Þðr2 2Þ ðc2 a2 Þðr2 2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! !#2 912 = a 2a 2a ðx ctÞ þ idn ðx ctÞ ; rsn u1:18 ðx; tÞ ¼ 2 2 2 2 2 2 2 ; : bð1 2r Þ ðc a Þð2r 1Þ ðc a Þð2r 1Þ 8 <
912 8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a > > = < ar2 dn2 ðx ctÞ ðc2 a2 Þðr2 2Þ u1:19 ðx; tÞ ¼ h
i q q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2> pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > 2a 2a :bðr 2Þ ðx ctÞ þ cn ðx ctÞ ; 1 r2 sn ðc2 a2 Þðr2 2Þ ðc2 a2 Þðr2 2Þ
ð35Þ
ð36Þ
ð37Þ
ð38Þ
ðaðc2 a2 Þ > 0Þ; ð39Þ
912
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a > = ðx ctÞ 2 2 2 ðc a Þð12r Þ a u1:20 ðx; tÞ ¼ h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 ; 2 > > : bð1 2r Þ 1 þ cn ðc2 a22a ðx ctÞ ; Þð12r2 Þ 8 > <
sn2
912
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a > = ðx ctÞ 2 2 2 ðc a Þðr 2Þ ar u1:21 ðx; tÞ ¼ h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 ; 2 > > : bðr 2Þ 1 þ dn 2 22a 2 ðx ctÞ ; 8 > <
2
ð40Þ
sn2
ðc a Þðr 2Þ
ð41Þ
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
912 8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > = < aðr2 1Þ dn2 ðc2 a22aÞðr2 þ1Þðx ctÞ u1:22 ðx; tÞ ¼ h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 ; 2 > > : bðr þ 1Þ 1 þ rsn ðc2 a22aÞðr2 þ1Þðx ctÞ ; 8 > < að1 r2 Þ u1:23 ðx; tÞ ¼ h 2 > : bðr þ 1Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 912 > = ðc2 a22aÞðr2 þ1Þðx ctÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 ; > 1 þ sn 2 22a 2 ðx ctÞ ;
87
ð42Þ
cn2
ð43Þ
ðc a Þðr þ1Þ
91 8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > > = < að1 r2 Þ2 sn2 ðc2 a22aÞðr2 þ1Þðx ctÞ u1:24 ðx; tÞ ¼ h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 ; 2 > > : bðr þ 1Þ dn ðc2 a22aÞðr2 þ1Þðx ctÞ ; ðc2 a22aÞðr2 þ1Þðx ctÞ þ cn
ð44Þ
8 912
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a > > < ar4 = cn2 ðx ctÞ ðc2 a2 Þðr2 2Þ ; u1:25 ðx; tÞ ¼ h
i q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2> pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > 2a :bðr 2Þ ðx ctÞ ; 1 r2 þ dn ðc2 a2 Þðr2 2Þ
ð45Þ
where 0 < r < 1 and c is an arbitrary constant. Remark 1. Solutions from u1:1 to u1:12 are in full agreement with those presented in Li and Pan’s paper [18], solution u1:19 was found by Zheng [19] and the other solutions are new ones. Remark 2. When r ! 0; u1:1 ; u1:2 ; u1:3 ; u1:11 ; u1:17 ; u1:19 ; u1:21 and u1:25 become zero, u1:4 ; u1:8 ; u1:13 ; u1:18 and u1:22 become constants while other solutions degenerate to triangular solutions of Eq. (1). In particular, as r ! 0; u1:5 ; u1:6 ; u1:9 and u1:10 become rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a a 2 u1:5:0 ðx; tÞ ¼ ðx ctÞ ; > 0; ð46Þ sec b c2 a2 c2 a2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a 2 a a u1:6:0 ðx; tÞ ¼ ðx ctÞ ; > 0; ð47Þ csc b c2 a2 c2 a 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 a a a u1:9:0 ðx; tÞ ¼ cot2 ; < 0; ð48Þ ðx ctÞ 2 b 2ðc a2 Þ c2 a2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 a a a 2 ðx ctÞ 2 ; < 0; ð49Þ u1:10:0 ðx; tÞ ¼ tan b 2ðc a2 Þ c2 a2 respectively. Solutions from (46)–(49) were found in [18] while u1:5:0 and u1:6:0 were obtained by Sirendaoreji [12]. Remark 3. When r ! 1, some of the solutions listed in (21)–(45) become zero or constants, and the other solutions degenerate to soliton-like solutions. Specifically, u1:1 ; u1:3 ; u1:6 ; and u1:9 are turned into rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 a a a ðx ctÞ ; > 0; ð50Þ tanh2 u1:1:1 ðx; tÞ ¼ b 2ðc2 a2 Þ c2 a2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a a 2 2 ðx ctÞ u1:3:1 ðx; tÞ ¼ ; < 0; ð51Þ sech b c a2 c2 a2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 a a a u1:6:1 ðx; tÞ ¼ ðx ctÞ ; > 0; ð52Þ coth2 b 2ðc2 a2 Þ c2 a2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12 2a a a 2 ðx ctÞ u1:9:1 ðx; tÞ ¼ csch2 ; < 0; ð53Þ 2 2 b c a c a2 respectively, which are the solutions obtained in [12,18].
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3.2. Soliton and triangular function solutions of Eq. (1) From the expression c22 ¼ 2c1 c3 in the formula of (20) and the conditions listed in Table 1, we find that the modulus r is either 1 or 0. Therefore, the degenerated soliton-like and triangular function solutions for Eq. (1) are expressed by 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ aðn þ 1Þ a 2 ðn 1Þ u2:1 ðx; tÞ ¼ ðaðc2 a2 Þ < 0Þ; ð54Þ ðx ctÞ þ tanh 2 2b 2b 2 c a2 u2:2 ðx; tÞ ¼
u2:3 ðx; tÞ ¼
u2:4 ðx; tÞ ¼
u2:5 ðx; tÞ ¼
u2:6 ðx; tÞ ¼
u2:7 ðx; tÞ ¼
u2:8 ðx; tÞ ¼
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ aðn þ 1Þ ðn 1Þ a ðx ctÞ þ coth2 2 2b 2b 2 c a2
8 >
2b
8 >
2b
8 >
2b
8 >
2b
8 >
2b
8 >
u2:9 ðx; tÞ ¼
2b
1 h i2 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = 1 þ cosh ðn 1Þ ðx ctÞ 2 2 c a aðn þ 1Þ h i þ pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; sinh2 ðn 1Þ c2 a 2 ðx ctÞ 1 h i2 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = 1 þ cosh ðn 1Þ ðx ctÞ 2 2 c a aðn þ 1Þ h i pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; sinh2 ðn 1Þ c2 a 2 ðx ctÞ
u2:12 ðx; tÞ ¼
1 h i 32 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = ðx ctÞ sinh ðn 1Þ 2 2 c a aðn þ 1Þ 4 h i5 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; 1 þ cosh ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ < 0Þ;
ð57Þ
ðaðc2 a2 Þ < 0Þ;
ð58Þ
1 h i 32 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = sinh ðn 1Þ ðx ctÞ 2 2 c a aðn þ 1Þ 4 h i5 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; 1 þ cosh ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ < 0Þ;
ð59Þ
2
1 h i32 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = i þ sinh ðn 1Þ ðx ctÞ 2 2 c a aðn þ 1Þ 4 h i 5 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; cosh ðn 1Þ c2 a2 ðx ctÞ
2
1 h i32 9n1 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a = ðx ctÞ i þ sinh ðn 1Þ 2 2 c a aðn þ 1Þ 4 h i 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2b ; cosh ðn 1Þ c2 a2 ðx ctÞ
2b
8 >
ð56Þ
2
8 >
> :
ðaðc2 a2 Þ < 0Þ;
ðaðc2 a2 Þ < 0Þ;
ð60Þ
ðaðc2 a2 Þ < 0Þ;
ð61Þ
2
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ aðn þ 1Þ 2 ðn 1Þ a ðx ctÞ þ cot 2b 2b 2 c2 a2
> :
ð55Þ
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ aðn þ 1Þ a 2 ðn 1Þ ðx ctÞ þ tan u2:10 ðx; tÞ ¼ 2b 2b 2 c2 a2
u2:11 ðx; tÞ ¼
ðaðc2 a2 Þ < 0Þ;
ðaðc2 a2 Þ > 0Þ;
ðaðc2 a2 Þ > 0Þ;
1
h i2 9n1 pffiffiffiffiffiffiffiffi ffi > a = ðx ctÞ 1 þ cos ðn 1Þ c2 a2 aðn þ 1Þ h i þ ffi pffiffiffiffiffiffiffiffi 2 > a 2b ; sin ðn 1Þ c2 a2 ðx ctÞ 1
h i2 9n1 ffi pffiffiffiffiffiffiffiffi > a = 1 þ cos ðn 1Þ ðx ctÞ c2 a2 aðn þ 1Þ h i ffi pffiffiffiffiffiffiffiffi 2 > a 2b ; sin ðn 1Þ c2 a2 ðx ctÞ
ð62Þ
ð63Þ
ðaðc2 a2 Þ > 0Þ;
ð64Þ
ðaðc2 a2 Þ > 0Þ;
ð65Þ
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90
u2:13 ðx; tÞ ¼
u2:14 ðx; tÞ ¼
u2:15 ðx; tÞ ¼
u2:16 ðx; tÞ ¼
u2:17 ðx; tÞ ¼
u2:18 ðx; tÞ ¼
8 >
2b
8 >
2b
8 > >
2b
8 > >
2b
8 >
2b
8 >
2b
89
9n1 1 h i ffi pffiffiffiffiffiffiffiffi 2 > a = sin ðn 1Þ ðx ctÞ c2 a2 aðn þ 1Þ þ h i 2 pffiffiffiffiffiffiffiffi ffi > 2b a ; 1 þ cos ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ > 0Þ;
ð66Þ
9n1 1 h i ffi pffiffiffiffiffiffiffiffi 2 > a = sin ðn 1Þ ðx ctÞ c2 a2 aðn þ 1Þ h i 2 pffiffiffiffiffiffiffiffi ffi > 2b a ; 1 þ cos ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ > 0Þ;
ð67Þ
ðaðc2 a2 Þ > 0Þ;
ð68Þ
ðaðc2 a2 Þ > 0Þ;
ð69Þ
9n1 1 h i ffi pffiffiffiffiffiffiffiffi > a 2 = cos ðn 1Þ ðx ctÞ c2 a2 aðn þ 1Þ þ h i2 pffiffiffiffiffiffiffiffi ffi > 2b a ; 1 þ sin ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ > 0Þ;
ð70Þ
9n1 1 h i pffiffiffiffiffiffiffiffi ffi > a = cos2 ðn 1Þ c2 a 2 ðx ctÞ h i2 ffi pffiffiffiffiffiffiffiffi > a ; 1 þ sin ðn 1Þ c2 a 2 ðx ctÞ
ðaðc2 a2 Þ > 0Þ:
ð71Þ
1 i2 9 n1 pffiffiffiffiffiffiffiffi ffi > > a aðn þ 1Þ 1 þ sin ðn 1Þ c2 a2 ðx ctÞ =
þ qffiffiffiffiffiffiffiffi > 2b > ; cos2 ðn 1Þ ðacÞ ðx ctÞ bc
h
1 h i2 9 n1 pffiffiffiffiffiffiffiffi ffi > > a = ðx ctÞ 1 þ sin ðn 1Þ c2 a2 aðn þ 1Þ
qffiffiffiffiffiffiffiffi > 2b > ; ðx ctÞ cos2 ðn 1Þ ðacÞ bc
aðn þ 1Þ 2b
Remark 4. Setting ¼ 1 and using the identities 1 1 ¼ sech2 a; 1 þ coshð2aÞ 2 1 1 ¼ csch2 a; 1 coshð2aÞ 2
1 1 ¼ sech2 a; 1 þ cosð2aÞ 2 1 1 ¼ csc2 a; 1 cosð2aÞ 2
we find that some of the solutions listed in (54)–(71) possess the same form. For example, u2:3 and u2:4 become u2:1 while u2:5 and u2:6 are the same as u2:2 . Remark 5. According to the identities 1 tanh2 ðaÞ ¼ sech2 ðaÞ; 1 coth2 ðaÞ ¼ csch2 ðaÞ and letting ¼ 1, solutions u2:1 and u2:2 become 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ a 2 ðn 1Þ u2:1:0 ðx; tÞ ¼ ðx ctÞ sech 2 2b 2 c a2
u2:2:0 ðx; tÞ ¼
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ ðn 1Þ a ðx ctÞ csch2 2 2b 2 c a2
ðaðc2 a2 Þ < 0Þ;
ðaðc2 a2 Þ < 0Þ:
ð72Þ
ð73Þ
From the expressions 1 þ tan2 ðaÞ ¼ sec2 ðaÞ; 1 þ cot2 ðaÞ ¼ csc2 ðaÞ and choosing ¼ 1, u2:9 and u2:10 are turned into 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ 2 ðn 1Þ a u2:9:0 ðx; tÞ ¼ ðaðc2 a2 Þ > 0Þ; ð74Þ ðx ctÞ csc 2b 2 c2 a2
90
X. Lv et al. / Chaos, Solitons and Fractals 41 (2009) 82–90 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 aðn þ 1Þ a 2 ðn 1Þ ðx ctÞ u2:10:0 ðx; tÞ ¼ sec 2b 2 c2 a2
ðaðc2 a2 Þ > 0Þ:
ð75Þ
Solutions in (72)–(75) were obtained by Sirendaoreji [12] and Wazwaz [20].
4. Conclusion Using a different auxiliary equation presented in Sirendaoreji [12] and the symbolic computation system Matlab, we obtain the Jacobi function solutions, the solitons and the triangular function solutions to nonlinear Klein–Gordon equation (1) in the case n = 3. As seen in the above, the method not only can recover the previously known solutions which were found by tanh and sin–cosine method, but also can provide new and more distinct solutions for Eq. (1). However, for arbitrary exponent n, we only obtain the degenerated soliton-like and triangle function solutions to Eq. (1). It may need other techniques to deal with the Jacobi elliptic function solutions for the nonlinear Klein–Gordon equation (1) with arbitrary exponent n.
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