- Email: [email protected]
New double periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation
Chaos, Solitons and Fractals 20 (2004) 765–769 www.elsevier.com/locate/chaos
New double periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation Chen Huai-Tang a
a,b,*
, Zhang Hong-Qing
a
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China b Department of Mathematics, Linyi Teachers University, Linyi 276005, China Accepted 11 August 2003 Communicated by Prof. M. Wadati
Abstract A new generalized Jacobi elliptic function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new double periodic and multiple soliton solutions are obtained for the generalized (2 + 1)-dimensional Boussinesq equation. This method can be applied to many other equations. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, directly searching for exact solutions of nonlinear partial differential equations (PDEs) has become increasingly attractive partly due to the availability of computer symbolic systems like Maple or Mathematica, which allow us to perform complicated and tedious algebraic calculations as well as help us to find exact solutions of PDEs [1– 5]. A number of methods have been presented, such as inverse scattering theory [6–8], Hirota’s bilinear method [9–11], the truncated Painleve expansion [12], homogeneous balance method [13–17], the hyperbolic tangent function series method [18], the sine–cosine method [19], the Jacobi elliptic function expansion method [20–23] and other methods [24,25]. The purpose of this paper is to present a generalized Jacobi elliptic function method and to solve the generalized (2 + 1)-dimensional Boussinesq equation.
2. Our method based on the elliptic equation The main idea of our method is to take full advantage of the elliptic equation that Jacobi elliptic functions satisfy and use its solutions to obtain new doubly periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation. The desired elliptic equation reads /02 ¼ A þ B/2 þ C/4 ;
ð1Þ
where /0 :¼ d/=dn, n ¼ nðx; y; z; tÞ, and A, B, C are constants.
* Corresponding author. Address: Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China. Tel.: +86-411-4701381. E-mail address: [email protected] (C. Huai-Tang).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.08.006
766
C. Huai-Tang, Z. Hong-Qing / Chaos, Solitons and Fractals 20 (2004) 765–769
8 : 2 m C¼ 4 ( Case 7. If
2
A ¼ C ¼ m4 2
B ¼ m 22 (
Case 8. If
A ¼ C ¼ 14 B ¼ 12m 2
( Case 9. If
dnn pffiffiffiffiffiffiffiffi ; msnn idnn; snn ; pffiffiffiffiffiffiffifficnn , then (1) has solution /ðnÞ ¼ mcnni . 1cnn 1m2 1m2 snndnn
2
A ¼ C ¼ m 41 B¼
( Case 10. If
2
dnn . , then (1) has solution /ðnÞ ¼ snn icnn; ipffiffiffiffiffiffiffiffi 1m2 snncnn
m2
þ1 2
dnn , then (1) has solution /ðnÞ ¼ 1msnn .
2
A ¼ C ¼ 1m 4 , then (1) has solution /ðnÞ ¼ cnn . 2 1snn B ¼ 1þm 2
8 2 Þ2 > < A ¼ ð1m 4 2 Case 11. If B ¼ 1þm , then (1) has solution /ðnÞ ¼ mcnn dnn. > : C ¼ 2 1 4 8 1 > : ð1m2 Þ2 C¼ 4 8 1 > : m4 C¼ 4 Here, snn ¼ snðn; mÞ, were m denote the modulus of the Jacobi elliptic functions. For a given PDE with uðx; y; z; tÞ in four independent variables x, y, z, t H ðu; ut ; ux ; uy ; uz ; uxt ; uyt ; uzt ; utt ; uxx ; . . .Þ ¼ 0:
ð2Þ
C. Huai-Tang, Z. Hong-Qing / Chaos, Solitons and Fractals 20 (2004) 765–769
We assume that the solutions of Eq. (2) can be expressed in the form n X uðx; y; z; tÞ ¼ a0 ðx; y; z; tÞ þ ½ai ðx; y; z; tÞ/i ðnÞ þ bi ðx; y; z; tÞ/i ðnÞ ;
767
ð3Þ
i¼1
where n ¼ nðx; y; z; tÞ. The process is taken as the following steps. Step 1. Determine n in Eq. (3) by balancing the linear term of the highest order with the nonlinear term in Eq. (2). Step 2. Substituting (3) and (1) into Eq. (2) yields a set of equations for a0 ðx; y; z; tÞ; a1 ðx; y; z; tÞ; . . . ; an ðx; y; z; tÞ because all coefficients of /i have to vanish. Step 3. From the relations in step 2, a0 ðx; y; z; tÞ; a1 ðx; y; z; tÞ; . . . ; an ðx; y; z; tÞ can be determined by solving the set of equations. Step 4. Substituting the obtained coefficients a0 ðx; y; z; tÞ; a1 ðx; y; z; tÞ; . . . ; an ðx; y; z; tÞ into (3) and using the solutions of Eq. (1) gives the explicit and exact solutions of Eq. (2). In the following we illustrate the method by considering the generalized (2 + 1)-dimensional Boussinesq equation. 3. The generalized (2 + 1)-dimensional Boussinesq equation The generalized (2 + 1)-dimensional Boussinesq equation is given by utt auxx buyy rðu2 Þxx suxxxx ¼ 0;
ð4Þ
where a, b, r, and s are arbitrary constants with r 6¼ 0. The special cases of Eq. (4) are studied by several authors. If a ¼ b ¼ r ¼ s ¼ 1, then Eq. (4) is studied by Chen [26]. If a ¼ 1, b ¼ 1, r ¼ 3, s ¼ 14, then Eq. (4) is studied by Johnson [27]. If a ¼ c20 , b ¼ 0, r ¼ b, s ¼ a, then Eq. (4) is studied by Liu [21]. In order to obtain exact solutions of Eq. (4), we firstly Balance uxxxx with ðu2 Þxx to get n ¼ 2. Therefore, we choose the following ansatz: u ¼ a0 þ a1 /ðnÞ þ a2 /2 ðnÞ þ b1 =/ðnÞ þ b2 =/2 ðnÞ;
ð5Þ
where n ¼ ax þ by þ xt, and a, b, x are arbitrary nonzero constants. Substituting (5) into (4) along with Eq. (1) and using Maple yields a system of equations w.r.t /i . Setting the coefficients of /i in the obtained system of equations to zero, we can deduce the following set of equations with the respect unknowns a0 , a1 , a2 , b1 , b2 , namely: 24sa1 a4 C 2 24ra1 a2 a2 C ¼ 0; 120sa2 a4 C 2 20ra22 a2 C ¼ 0; 20rb22 a2 A 120sb2 a4 A2 ¼ 0; 24rb1 b2 a2 A 24sb1 a4 A2 ¼ 0; 2aa2 a2 A 2ba2 b2 A 4ra0 b2 a2 C 4ra0 a2 a2 A 2ra21 Aa2 8sa2 a4 BA 8sb2 a4 BC 2ab2 a2 C 2bb2 b2 C þ 2b2 x2 C þ 2a2 x2 A 2rb21 a2 C ¼ 0; 2ba1 b2 C þ 2a1 x2 C 4rb1 a2 a2 C 4ra0 a1 a2 C 2aa1 a2 C 20sa1 a4 BC 18ra1 a2 a2 B ¼ 0; 18rb1 b2 a2 B 2ab1 a2 A 4ra1 b2 a2 A þ 2b1 x2 A 20sb1 a4 BA 4ra0 b1 a2 A 2bb1 b2 A ¼ 0; bb1 b2 B 12rb1 b2 a2 C 2ra0 b1 a2 B 12sb1 a4 CA 2ra1 b2 a2 B sb1 a4 B2 ab1 a2 B þ b1 x2 B ¼ 0 12ra0 b2 a2 A 6rb21 a2 A 6ab2 a2 A þ 6b2 x2 A 16rb22 a2 B 6bb2 b2 A 120sb2 a4 BA ¼ 0; 8ra0 b2 a2 B 16sb2 a4 B2 4ab2 a2 B þ 4b2 x2 B 12rb22 a2 C 4rb21 a2 B 4bb2 b2 B 72sb2 a4 CA ¼ 0; 16ra22 a2 B 6aa2 a2 C 120sa2 a4 BC 6ba2 b2 C 6ra21 a2 C 12ra0 a2 a2 C þ 6a2 x2 C ¼ 0; 4ba2 b2 B 16sa2 a4 B2 4ra21 a2 B 8ra0 a2 a2 B 72sa2 a4 CA 12ra22 a2 A 4aa2 a2 B þ 4a2 x2 B ¼ 0; ba1 b2 B 2ra2 b1 a2 B aa1 a2 B sa1 a4 B2 2ra0 a1 a2 B þ a1 x2 B 12ra1 a2 a2 A 12sa1 a4 CA ¼ 0:
ð6Þ
With the aid of Maple or Mathematica, we find 1: a1 ¼ b1 ¼ a2 ¼ 0;
a0 ¼
aa2 þ bb2 x2 þ 4sa4 B ; 2ra2
b2 ¼
6sa2 A : r
ð7Þ
2: a1 ¼ b1 ¼ b2 ¼ 0;
a0 ¼
aa2 þ bb2 x2 þ 4sa4 B ; 2ra2
a2 ¼
6sa2 C : r
ð8Þ
768
C. Huai-Tang, Z. Hong-Qing / Chaos, Solitons and Fractals 20 (2004) 765–769
3: a1 ¼ b1 ¼ 0;
a0 ¼
aa2 þ bb2 x2 þ 4sa4 B ; 2ra2
b2 ¼
6sa2 A ; r
a2 ¼
6sa2 C : r
ð9Þ
Substituting (7)–(9) into (5) and using the solutions of Eq. (1), we obtain the following double periodic solutions of Eq. (4) u1 ¼
aa2 þ bb2 x2 4sa4 ð1 þ m2 Þ 6sa2 ðe þ gm2 sn4 nÞ ; 2ra2 rsn2 n
ð10Þ
u2 ¼
aa2 þ bb2 x2 4sa4 ð1 þ m2 Þ 6sa2 ðedn4 n þ gm2 cn4 nÞ ; 2ra2 rcn2 ndn2 n
ð11Þ
u3 ¼
aa2 þ bb2 x2 þ 4sa4 ð2m2 1Þ 6sa2 ½eð1 m2 Þ gm2 cn4 n
; 2ra2 rcn2 n
ð12Þ
u4 ¼
aa2 þ bb2 x2 þ 4sa4 ð2 m2 Þ 6sa2 ½eð1 m2 Þ þ gdn4 n
; þ 2ra2 rdn2 n
ð13Þ
u5 ¼
aa2 þ bb2 x2 þ 4sa4 ð2 m2 Þ 6sa2 ½eð1 m2 Þsn4 n þ gcn4 n
; 2ra2 rsn2 ncn2 n
ð14Þ
u6 ¼
aa2 þ bb2 x2 þ 4sa4 ð2m2 1Þ 6sa2 ½edn4 n gm2 ð1 m2 Þsn4 n
; 2ra2 rsn2 ndn2 n
ð15Þ
u7 ¼
aa2 þ bb2 x2 þ 2sa4 ðm2 2Þ 3sa2 ½em2 sn4 n þ gð1 dnnÞ4
; 2ra2 2rsn2 nð1 dnnÞ2
ð16Þ
aa2 þ bb2 x2 þ 2sa4 ð1 2m2 Þ 3sa2 ½esn4 n þ gð1 cnnÞ4
; 2ra2 2rsn2 nð1 cnnÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi aa2 þ bb2 x2 þ 2sa4 ð1 2m2 Þ 3sa2 ½ecn4 n þ gð 1 m2 snn dnnÞ4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ; u9 ¼ 2ra2 2rcn2 nð 1 m2 snn dnnÞ2
u8 ¼
ð17Þ
ð18Þ
u10 ¼
aa2 þ bb2 x2 þ 2sa4 ð1 þ m2 Þ 3sa2 ð1 m2 Þ½edn4 n þ gð1 msnnÞ4
þ ; 2ra2 2rdn2 nð1 msnnÞ2
ð19Þ
u11 ¼
aa2 þ bb2 x2 þ 2sa4 ð1 þ m2 Þ 3sa2 ð1 m2 Þ½ecn4 n þ gð1 snnÞ4
; 2ra2 2rcn2 nð1 snnÞ2
ð20Þ
u12 ¼
aa2 þ bb2 x2 þ 2sa4 ð1 þ m2 Þ 3sa2 ½eð1 m2 Þ2 þ gðmcnn dnnÞ4
þ ; 2ra2 2rðmcnn dnnÞ2
ð21Þ
aa2 þ bb2 x2 þ 2sa4 ð1 þ m2 Þ 3sa2 ½eð1 m2 Þ2 sn4 n þ gðdnn cnnÞ4
; 2ra2 2rsn2 nðdnn cnnÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi aa2 þ bb2 x2 þ 2sa4 ðm2 2Þ 3sa2 ½em4 cn4 n þ gð 1 m2 dnnÞ4
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : u14 ¼ 2ra2 2rcn2 nð 1 m2 dnnÞ2
u13 ¼
ð22Þ
ð23Þ
If m ! 1, then snn ! tanh n; cnn ! sechn; dnn ! sechn. Thus we can obtain the following multiple soliton solutions of Eq. (4) u15 ¼
aa2 þ bb2 x2 8sa4 6a2 sðe þ g tanh4 nÞ ; 2ra2 r tanh2 n
ð24Þ
u16 ¼
aa2 þ bb2 x2 2sa4 3sa2 ½e tanh4 n þ gð1 sechnÞ4
; 2ra2 2r tanh2 nð1 sechnÞ2
ð25Þ
where n ¼ ax þ by þ xt, e and g are arbitrary elements of {0,1}.
C. Huai-Tang, Z. Hong-Qing / Chaos, Solitons and Fractals 20 (2004) 765–769
769
Remark 1. (10)–(23) are only some solutions of Eq. (4) obtained by Cases 1–13. (24) and (25) are only some special cases of (10)–(23). Other solutions are omitted from verbosity. Remark 2. If m ! 0, then snn ! sin n; cnn ! cos n; dnn ! 1. We can obtain triangular periodic solutions of Eq. (4). The triangular periodic solutions of Eq. (4) are omitted.
4. Conclusion and discussion We have presented a generalized Jacobi elliptic function method and used it to solve the generalized (2 + 1)-dimensional Boussinesq equation. In fact, we can obtain not only the double periodic solutions, but also triangular periodic solutions as well as the multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation. This method can be applied to many other nonlinear PDEs.
Acknowledgements This work is supported by the National Key Basic Research Development Project of China (grant no. 1998030600), the National Natural Science Foundation of China (grant no. 10072013).
References [1] Duffy BR, Parkes EJ. Phys Lett A 1996;214:271–2. [2] Parkes EJ, Duffy BR. Phys Lett A 1997;229:217–20. [3] Li ZB. Exact solitary wave solutions of nonlinear evolution equations. In: Gao XS, Wang DM, editors. Mathematics mechanization and application. Academic Press; 2000. [4] Li ZB, Wang ML. J Phys A 1993;26:6027–31. [5] Parkes EJ. J Phys A 1994;27:L497–501. [6] Gardner CS, Greene JM, Kruskal MD, Miura RM. Phys Rev Lett 1967;19:1095–7. [7] Ablowitz MJ, Kaup DJ, Newell AC, Segur H. Stu Appl Math 1974;53:315–429. [8] Reals R, Coifman RR. Commu Pure Appl Math 1984;37:39–90. [9] Hirota R. J Math Phys 1973;14:810. [10] Nozaki K. J Phys Soc Jpn 1987;56:3052. [11] Hu XB, Ma WX. Phys Lett A 2002;293:161–5. [12] Weis J, Tabor M, Garnevale G. J Math Phys 1983;24:522–6. [13] Wang ML. Phys Lett A 1995;199:169–72. [14] Wang ML, Zhou YB, Li ZB. Phys Lett A 1996;216:67–75. [15] Fan EG, Zhang HQ. Phys Lett A 1998;246:403–6. [16] Yang L, Zhu Z, Wang Y. Phys Lett A 1999;260:55–9. [17] Fan EG. Phys Lett A 2000;265:353–7. [18] Blickle T, Lakatos BG, Mihalyk o C, Ulbert Z. Powder Technol 1998;97:100–8. [19] Yan C. Phys Lett A 1996;224:77–84. [20] Fu ZT, Liu SK, Liu SD, Zhao Q. Phys Lett A 2001;290:72–6. [21] Liu SK, Fu ZT, Liu SD, Zhao Q. Phys Lett A 2001;289:69–74. [22] Chen HT, Zhang HQ. Chaos, Solitons & Fractals 2003;15:585–91. [23] Chen HT, Zhang HQ. J Appl Math Comput 2002;10:119–30. [24] Yang L, Liu JB, Yang KQ. Phys Lett A 2001;278:267–70. [25] Xia TC, Zhang HQ, Yan ZY. Chin Phys 2001;10:694–7. [26] Chen Y et al. Phys Lett A 2003;307:107–13. [27] Johnson RS. J Fluid Mech 1996;323:65–78.
New double periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation Chen Huai-Tang a
a,b,*
, Zhang Hong-Qing
a
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China b Department of Mathematics, Linyi Teachers University, Linyi 276005, China Accepted 11 August 2003 Communicated by Prof. M. Wadati
Abstract A new generalized Jacobi elliptic function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new double periodic and multiple soliton solutions are obtained for the generalized (2 + 1)-dimensional Boussinesq equation. This method can be applied to many other equations. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, directly searching for exact solutions of nonlinear partial differential equations (PDEs) has become increasingly attractive partly due to the availability of computer symbolic systems like Maple or Mathematica, which allow us to perform complicated and tedious algebraic calculations as well as help us to find exact solutions of PDEs [1– 5]. A number of methods have been presented, such as inverse scattering theory [6–8], Hirota’s bilinear method [9–11], the truncated Painleve expansion [12], homogeneous balance method [13–17], the hyperbolic tangent function series method [18], the sine–cosine method [19], the Jacobi elliptic function expansion method [20–23] and other methods [24,25]. The purpose of this paper is to present a generalized Jacobi elliptic function method and to solve the generalized (2 + 1)-dimensional Boussinesq equation.
2. Our method based on the elliptic equation The main idea of our method is to take full advantage of the elliptic equation that Jacobi elliptic functions satisfy and use its solutions to obtain new doubly periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation. The desired elliptic equation reads /02 ¼ A þ B/2 þ C/4 ;
ð1Þ
where /0 :¼ d/=dn, n ¼ nðx; y; z; tÞ, and A, B, C are constants.
* Corresponding author. Address: Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China. Tel.: +86-411-4701381. E-mail address: [email protected] (C. Huai-Tang).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.08.006
766
C. Huai-Tang, Z. Hong-Qing / Chaos, Solitons and Fractals 20 (2004) 765–769
8 : 2 m C¼ 4 ( Case 7. If
2
A ¼ C ¼ m4 2
B ¼ m 22 (
Case 8. If
A ¼ C ¼ 14 B ¼ 12m 2
( Case 9. If
dnn pffiffiffiffiffiffiffiffi ; msnn idnn; snn ; pffiffiffiffiffiffiffifficnn , then (1) has solution /ðnÞ ¼ mcnni . 1cnn 1m2 1m2 snndnn
2
A ¼ C ¼ m 41 B¼
( Case 10. If
2
dnn . , then (1) has solution /ðnÞ ¼ snn icnn; ipffiffiffiffiffiffiffiffi 1m2 snncnn
m2
þ1 2
dnn , then (1) has solution /ðnÞ ¼ 1msnn .
2
A ¼ C ¼ 1m 4 , then (1) has solution /ðnÞ ¼ cnn . 2 1snn B ¼ 1þm 2
8 2 Þ2 > < A ¼ ð1m 4 2 Case 11. If B ¼ 1þm , then (1) has solution /ðnÞ ¼ mcnn dnn. > : C ¼ 2 1 4 8 1 > : ð1m2 Þ2 C¼ 4 8 1 > : m4 C¼ 4 Here, snn ¼ snðn; mÞ, were m denote the modulus of the Jacobi elliptic functions. For a given PDE with uðx; y; z; tÞ in four independent variables x, y, z, t H ðu; ut ; ux ; uy ; uz ; uxt ; uyt ; uzt ; utt ; uxx ; . . .Þ ¼ 0:
ð2Þ
C. Huai-Tang, Z. Hong-Qing / Chaos, Solitons and Fractals 20 (2004) 765–769
We assume that the solutions of Eq. (2) can be expressed in the form n X uðx; y; z; tÞ ¼ a0 ðx; y; z; tÞ þ ½ai ðx; y; z; tÞ/i ðnÞ þ bi ðx; y; z; tÞ/i ðnÞ ;
767
ð3Þ
i¼1
where n ¼ nðx; y; z; tÞ. The process is taken as the following steps. Step 1. Determine n in Eq. (3) by balancing the linear term of the highest order with the nonlinear term in Eq. (2). Step 2. Substituting (3) and (1) into Eq. (2) yields a set of equations for a0 ðx; y; z; tÞ; a1 ðx; y; z; tÞ; . . . ; an ðx; y; z; tÞ because all coefficients of /i have to vanish. Step 3. From the relations in step 2, a0 ðx; y; z; tÞ; a1 ðx; y; z; tÞ; . . . ; an ðx; y; z; tÞ can be determined by solving the set of equations. Step 4. Substituting the obtained coefficients a0 ðx; y; z; tÞ; a1 ðx; y; z; tÞ; . . . ; an ðx; y; z; tÞ into (3) and using the solutions of Eq. (1) gives the explicit and exact solutions of Eq. (2). In the following we illustrate the method by considering the generalized (2 + 1)-dimensional Boussinesq equation. 3. The generalized (2 + 1)-dimensional Boussinesq equation The generalized (2 + 1)-dimensional Boussinesq equation is given by utt auxx buyy rðu2 Þxx suxxxx ¼ 0;
ð4Þ
where a, b, r, and s are arbitrary constants with r 6¼ 0. The special cases of Eq. (4) are studied by several authors. If a ¼ b ¼ r ¼ s ¼ 1, then Eq. (4) is studied by Chen [26]. If a ¼ 1, b ¼ 1, r ¼ 3, s ¼ 14, then Eq. (4) is studied by Johnson [27]. If a ¼ c20 , b ¼ 0, r ¼ b, s ¼ a, then Eq. (4) is studied by Liu [21]. In order to obtain exact solutions of Eq. (4), we firstly Balance uxxxx with ðu2 Þxx to get n ¼ 2. Therefore, we choose the following ansatz: u ¼ a0 þ a1 /ðnÞ þ a2 /2 ðnÞ þ b1 =/ðnÞ þ b2 =/2 ðnÞ;
ð5Þ
where n ¼ ax þ by þ xt, and a, b, x are arbitrary nonzero constants. Substituting (5) into (4) along with Eq. (1) and using Maple yields a system of equations w.r.t /i . Setting the coefficients of /i in the obtained system of equations to zero, we can deduce the following set of equations with the respect unknowns a0 , a1 , a2 , b1 , b2 , namely: 24sa1 a4 C 2 24ra1 a2 a2 C ¼ 0; 120sa2 a4 C 2 20ra22 a2 C ¼ 0; 20rb22 a2 A 120sb2 a4 A2 ¼ 0; 24rb1 b2 a2 A 24sb1 a4 A2 ¼ 0; 2aa2 a2 A 2ba2 b2 A 4ra0 b2 a2 C 4ra0 a2 a2 A 2ra21 Aa2 8sa2 a4 BA 8sb2 a4 BC 2ab2 a2 C 2bb2 b2 C þ 2b2 x2 C þ 2a2 x2 A 2rb21 a2 C ¼ 0; 2ba1 b2 C þ 2a1 x2 C 4rb1 a2 a2 C 4ra0 a1 a2 C 2aa1 a2 C 20sa1 a4 BC 18ra1 a2 a2 B ¼ 0; 18rb1 b2 a2 B 2ab1 a2 A 4ra1 b2 a2 A þ 2b1 x2 A 20sb1 a4 BA 4ra0 b1 a2 A 2bb1 b2 A ¼ 0; bb1 b2 B 12rb1 b2 a2 C 2ra0 b1 a2 B 12sb1 a4 CA 2ra1 b2 a2 B sb1 a4 B2 ab1 a2 B þ b1 x2 B ¼ 0 12ra0 b2 a2 A 6rb21 a2 A 6ab2 a2 A þ 6b2 x2 A 16rb22 a2 B 6bb2 b2 A 120sb2 a4 BA ¼ 0; 8ra0 b2 a2 B 16sb2 a4 B2 4ab2 a2 B þ 4b2 x2 B 12rb22 a2 C 4rb21 a2 B 4bb2 b2 B 72sb2 a4 CA ¼ 0; 16ra22 a2 B 6aa2 a2 C 120sa2 a4 BC 6ba2 b2 C 6ra21 a2 C 12ra0 a2 a2 C þ 6a2 x2 C ¼ 0; 4ba2 b2 B 16sa2 a4 B2 4ra21 a2 B 8ra0 a2 a2 B 72sa2 a4 CA 12ra22 a2 A 4aa2 a2 B þ 4a2 x2 B ¼ 0; ba1 b2 B 2ra2 b1 a2 B aa1 a2 B sa1 a4 B2 2ra0 a1 a2 B þ a1 x2 B 12ra1 a2 a2 A 12sa1 a4 CA ¼ 0:
ð6Þ
With the aid of Maple or Mathematica, we find 1: a1 ¼ b1 ¼ a2 ¼ 0;
a0 ¼
aa2 þ bb2 x2 þ 4sa4 B ; 2ra2
b2 ¼
6sa2 A : r
ð7Þ
2: a1 ¼ b1 ¼ b2 ¼ 0;
a0 ¼
aa2 þ bb2 x2 þ 4sa4 B ; 2ra2
a2 ¼
6sa2 C : r
ð8Þ
768
C. Huai-Tang, Z. Hong-Qing / Chaos, Solitons and Fractals 20 (2004) 765–769
3: a1 ¼ b1 ¼ 0;
a0 ¼
aa2 þ bb2 x2 þ 4sa4 B ; 2ra2
b2 ¼
6sa2 A ; r
a2 ¼
6sa2 C : r
ð9Þ
Substituting (7)–(9) into (5) and using the solutions of Eq. (1), we obtain the following double periodic solutions of Eq. (4) u1 ¼
aa2 þ bb2 x2 4sa4 ð1 þ m2 Þ 6sa2 ðe þ gm2 sn4 nÞ ; 2ra2 rsn2 n
ð10Þ
u2 ¼
aa2 þ bb2 x2 4sa4 ð1 þ m2 Þ 6sa2 ðedn4 n þ gm2 cn4 nÞ ; 2ra2 rcn2 ndn2 n
ð11Þ
u3 ¼
aa2 þ bb2 x2 þ 4sa4 ð2m2 1Þ 6sa2 ½eð1 m2 Þ gm2 cn4 n
; 2ra2 rcn2 n
ð12Þ
u4 ¼
aa2 þ bb2 x2 þ 4sa4 ð2 m2 Þ 6sa2 ½eð1 m2 Þ þ gdn4 n
; þ 2ra2 rdn2 n
ð13Þ
u5 ¼
aa2 þ bb2 x2 þ 4sa4 ð2 m2 Þ 6sa2 ½eð1 m2 Þsn4 n þ gcn4 n
; 2ra2 rsn2 ncn2 n
ð14Þ
u6 ¼
aa2 þ bb2 x2 þ 4sa4 ð2m2 1Þ 6sa2 ½edn4 n gm2 ð1 m2 Þsn4 n
; 2ra2 rsn2 ndn2 n
ð15Þ
u7 ¼
aa2 þ bb2 x2 þ 2sa4 ðm2 2Þ 3sa2 ½em2 sn4 n þ gð1 dnnÞ4
; 2ra2 2rsn2 nð1 dnnÞ2
ð16Þ
aa2 þ bb2 x2 þ 2sa4 ð1 2m2 Þ 3sa2 ½esn4 n þ gð1 cnnÞ4
; 2ra2 2rsn2 nð1 cnnÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi aa2 þ bb2 x2 þ 2sa4 ð1 2m2 Þ 3sa2 ½ecn4 n þ gð 1 m2 snn dnnÞ4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ; u9 ¼ 2ra2 2rcn2 nð 1 m2 snn dnnÞ2
u8 ¼
ð17Þ
ð18Þ
u10 ¼
aa2 þ bb2 x2 þ 2sa4 ð1 þ m2 Þ 3sa2 ð1 m2 Þ½edn4 n þ gð1 msnnÞ4
þ ; 2ra2 2rdn2 nð1 msnnÞ2
ð19Þ
u11 ¼
aa2 þ bb2 x2 þ 2sa4 ð1 þ m2 Þ 3sa2 ð1 m2 Þ½ecn4 n þ gð1 snnÞ4
; 2ra2 2rcn2 nð1 snnÞ2
ð20Þ
u12 ¼
aa2 þ bb2 x2 þ 2sa4 ð1 þ m2 Þ 3sa2 ½eð1 m2 Þ2 þ gðmcnn dnnÞ4
þ ; 2ra2 2rðmcnn dnnÞ2
ð21Þ
aa2 þ bb2 x2 þ 2sa4 ð1 þ m2 Þ 3sa2 ½eð1 m2 Þ2 sn4 n þ gðdnn cnnÞ4
; 2ra2 2rsn2 nðdnn cnnÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi aa2 þ bb2 x2 þ 2sa4 ðm2 2Þ 3sa2 ½em4 cn4 n þ gð 1 m2 dnnÞ4
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : u14 ¼ 2ra2 2rcn2 nð 1 m2 dnnÞ2
u13 ¼
ð22Þ
ð23Þ
If m ! 1, then snn ! tanh n; cnn ! sechn; dnn ! sechn. Thus we can obtain the following multiple soliton solutions of Eq. (4) u15 ¼
aa2 þ bb2 x2 8sa4 6a2 sðe þ g tanh4 nÞ ; 2ra2 r tanh2 n
ð24Þ
u16 ¼
aa2 þ bb2 x2 2sa4 3sa2 ½e tanh4 n þ gð1 sechnÞ4
; 2ra2 2r tanh2 nð1 sechnÞ2
ð25Þ
where n ¼ ax þ by þ xt, e and g are arbitrary elements of {0,1}.
C. Huai-Tang, Z. Hong-Qing / Chaos, Solitons and Fractals 20 (2004) 765–769
769
Remark 1. (10)–(23) are only some solutions of Eq. (4) obtained by Cases 1–13. (24) and (25) are only some special cases of (10)–(23). Other solutions are omitted from verbosity. Remark 2. If m ! 0, then snn ! sin n; cnn ! cos n; dnn ! 1. We can obtain triangular periodic solutions of Eq. (4). The triangular periodic solutions of Eq. (4) are omitted.
4. Conclusion and discussion We have presented a generalized Jacobi elliptic function method and used it to solve the generalized (2 + 1)-dimensional Boussinesq equation. In fact, we can obtain not only the double periodic solutions, but also triangular periodic solutions as well as the multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation. This method can be applied to many other nonlinear PDEs.
Acknowledgements This work is supported by the National Key Basic Research Development Project of China (grant no. 1998030600), the National Natural Science Foundation of China (grant no. 10072013).
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