- Email: [email protected]
New explicit and exact solutions for a system of variant RLW equations
Available online at www.sciencedirect.com
Applied Mathematics and Computation 198 (2008) 715–720 www.elsevier.com/locate/amc
New explicit and exact solutions for a system of variant RLW equations Dahe Feng
a,*
, Jibin Li b, Junliang Lu¨ b, Tianlan He
b
a
b
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, PR China Department of Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, PR China
Abstract In this article, a system of regularized long wave equations are studied. With the aid of the mathematic software Maple and using the direct method, some new exact solutions: compactons, solitons, solitary patterns and periodic solutions are obtained. Ó 2007 Elsevier Inc. All rights reserved. Keywords: R(m, n) equations; Compacton; Soliton; Solitary pattern solution; Periodic solution
1. Introduction The investigation of the exact solutions of nonlinear partial differential equations (PDEs) play an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in fluid dynamics, plasma and elastic media are often modelled by the bell-shaped sech functions and the kink-shaped tanh functions. The exact solution, if available, of those nonlinear PDEs facilitates the verification of numerical solvers and aids in the stability analysis of solutions. In the past several decades, various methods for obtaining exact solutions of nonlinear PDEs have been presented, such as inverse scattering method [1], Darboux transformation method [2,3], Hirota bilinear method [4], Lie group method [5,6], bifurcation method of dynamic systems [7], sine–cosine method [8–13], tanh function method [12–14], Fan-expansion method [15,16], homogenous balance method [17] and so on. Recently, Dye and Parker [18] studied the well-known nonlinear regularized long wave equation (RLW equation) ut þ aux 6uux þ buxxt ¼ 0
*
Corresponding author. E-mail addresses: [email protected] (D. Feng), [email protected] (J. Li), [email protected] (J. Lu¨).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.09.009
ð1Þ
716
D. Feng et al. / Applied Mathematics and Computation 198 (2008) 715–720
by using inverse scattering method. This RLW equation was introduced as an alternative model to the KdV equation to describe small amplitude long wave in shallow water. An increasing interest in studying the equation has been attached many mathematicians to investigate the problem to develop new solutions and to examine the physical behavior of the obtained solutions. More recently, Wazwaz [12] introduced a system of nonlinear variant RLW equations ut þ aux kðun Þx þ bðun Þxxt ¼ 0
ð2Þ
and derived some compact and noncompact exact solutions by using the sine–cosine method and the tanh method. Motivated by the rich mathematical and physical properties of the RLW Eqs. (1) and (2), in this paper we study the following generalized variant RLW equations (R(m, n) equations in short): Rðm; nÞ : ut þ aux kðum Þx þ bðun Þxxt ¼ 0:
ð3Þ
With the aid of Maple, we obtain some new exact solutions such as compactons, solitary pattern solutions, solitons and periodic solutions. 2. Explicit and exact solutions of R(m, n) equations Now we seek the travelling wave solutions of (3). Let uðx; tÞ ¼ uðnÞ;
n ¼ x ct;
ð4Þ
where c is wave speed. Substituting (4) into (3) yields ða cÞun kðum Þn bcðun Þnnn ¼ 0:
ð5Þ
Integrating (5) once, we obtain ða cÞu kum bcðun Þnn g ¼ 0; where g is an integral constant. Below we seek compacton solutions and solitary pattern solutions of (3) using the four ansatzs A cosd ðBnÞ; jBnj 6 p2 ; Ansatz 1: uðx; tÞ ¼ 0; otherwise: ( A sind ðBnÞ; jBnj 6 p; Ansatz 2: uðx; tÞ ¼ 0; otherwise: Ansatz 3: Ansatz 4:
uðx; tÞ ¼ A coshd ðBnÞ: d
uðx; tÞ ¼ A sinh ðBnÞ;
ð6Þ
ð7Þ ð8Þ ð9Þ ð10Þ
where A; B; d are parameters to be determined later. 2.1. Compact and noncompact solutions of ansatz 1 Substituting (7) into (3) gives Aða cÞ cosd ðBnÞ Am k cosmd ðBnÞ þ bcn2 d2 An B2 cosnd ðBnÞ bcndAn B2 ðnd 1Þ cosnd2 ðBnÞ g ¼ 0: ð11Þ Thus we can obtain the three possible cases to be discussed: Case 1. When nd 2 ¼ 0; d ¼ md ¼ nd, we have ( Aða cÞ Am k þ bcn2 d2 An B2 ¼ 0; bcndAn B2 ðnd 1Þ þ g ¼ 0
ð12Þ
D. Feng et al. / Applied Mathematics and Computation 198 (2008) 715–720
717
from which we get m ¼ n ¼ 1;
d ¼ 2;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kþca B¼ ; 4bc
A¼
2g : kþca
ð13Þ
Therefore we derive the compacton solution of R(1, 1) equation qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi ( 2g kþca bc p; n ; jnj 6 cos2 kþca 4bc kþca u1 ðx; tÞ ¼ 0; otherwise:
ð14Þ
Case 2. When nd 2 ¼ d; md ¼ nd; g ¼ 0, we have ( Aða cÞ bcndAn B2 ðnd 1Þ ¼ 0;
ð15Þ
Am k þ bcn2 d2 An B2 ¼ 0 which lead to m ¼ n;
2 ; d¼ n1
n1 B¼ 2n
rffiffiffiffiffi k ; bc
1 2nða cÞ n1 ; A¼ kðn þ 1Þ
g ¼ 0:
ð16Þ
Therefore we get the compacton solution of R(n, n) equations ðn > 1Þ 8h 1 qffiffiffi qffiffiffi in1 < 2nðacÞ 2 n1 k np bc n ; cos ; jnj 6 2n bc n1 k kðnþ1Þ u2 ðx; tÞ ¼ : 0; otherwise:
ð17Þ
Case 3. When d ¼ nd; md ¼ nd 2; g ¼ 0, we have (
Aða cÞ þ bcn2 d2 An B2 ¼ 0;
ð18Þ
Am k þ bcndAn B2 ðnd 1Þ ¼ 0 from which we get n ¼ 1;
2 ; d¼ m1
m1 B¼ 2
rffiffiffiffiffiffiffiffiffiffiffi ac ; bc
1 ða cÞðm þ 1Þ m1 ; A¼ 2k
g ¼ 0:
Thus we obtain the compacton solutions of Rðm; 1Þ equations ðm > 1Þ 8h qffiffiffiffiffiffi 1 pffiffiffiffiffiffi imþ1 < 2k 2 mþ1 ca p bc ; cos ; jnj 6 n ðcaÞðm1Þ 2 bc mþ1 ca u3 ðx; tÞ ¼ : 0; otherwise:
ð19Þ
ð20Þ
Remark (1) We can obtain the periodic solution of Rðn; nÞ equations ðn > 1Þ from Case 2 1 " rffiffiffiffiffi !#nþ1 kðn 1Þ nþ1 k 2 u4 ðx; tÞ ¼ sec ; n 2nða cÞ 2n bc (2) We also can get the periodic solution of Rðm; 1Þ equations ðm > 1Þ from Case 3 1 rffiffiffiffiffiffiffiffiffiffiffi m1 ða cÞðm þ 1Þ ca 2 m1 u5 ðx; tÞ ¼ sec : n 2k 2 bc
ð21Þ
ð22Þ
718
D. Feng et al. / Applied Mathematics and Computation 198 (2008) 715–720
2.2. Compact and noncompact solutions of ansatz 2 Substituting (8) into (3), we get the same three possible cases as the Section 2.1. Hence we obtain the following compact and noncompact solutions of R(m, n) equations: (1) Compacton solutions of R(1, 1) equation qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi ( 2g kþca 4bc p; n ; jnj 6 sin2 kþca 4bc kþca u6 ðx; tÞ ¼ 0; otherwise: (2) Compacton solutions of Rðn; nÞ equations ðn > 1Þ 8h 1 qffiffiffi qffiffiffi in1 < 2nðacÞ 2 n1 k 2np bc n ; sin ; jnj 6 2n bc n1 k kðnþ1Þ u7 ðx; tÞ ¼ : 0; otherwise: (3) Compacton solutions of Rðm; 1Þ equations ðm > 1Þ 8h qffiffiffiffiffiffi 1 ffi imþ1 < 2 mþ1 pffiffiffiffiffi 2k ca 2p bc n ; sin ; jnj 6 ðcaÞðm1Þ 2 bc mþ1 ca u8 ðx; tÞ ¼ : 0; otherwise: (4) Periodic solutions of Rðn; nÞ equations ðn > 1Þ 1 " rffiffiffiffiffi !#nþ1 kðn 1Þ k 2 nþ1 csc : n u9 ðx; tÞ ¼ 2nða cÞ 2n bc (5) Periodic solutions of Rðm; 1Þ equations ðm > 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffi m1 ða cÞðm þ 1Þ 2 m 1 c a u10 ðx; tÞ ¼ csc : n 2k 2 bc
ð23Þ
ð24Þ
ð25Þ
ð26Þ
ð27Þ
2.3. Solitary pattern and soliton solutions of ansatz 3 Substituting (9) into (3), we have Aða cÞcoshd ðBnÞ Am kcoshmd ðBnÞ bcn2 d2 An B2 coshnd ðBnÞ þ bcndAn B2 ðnd 1Þcoshnd2 ðBnÞ g ¼ 0 ð28Þ from which we can get the solitary pattern and soliton solutions of Rðm; nÞ equations by using the similar method as the Section 2.1 to determine the parameters A; B; d. (1) Solitary pattern solutions of Rð1; 1Þ equation rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2g ack 2 u11 ðx; tÞ ¼ cosh n : ack 4bc (2) Solitary pattern solutions of Rðn; nÞ equations ðn > 1Þ 1 " rffiffiffiffiffiffiffi !#n1 2nða cÞ k 2 n1 u12 ðx; tÞ ¼ cosh : n kðn þ 1Þ 2n bc (3) Solitary pattern solutions of Rðm; 1Þ equations ðm > 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffi mþ1 2k ac 2 mþ1 cosh u13 ðx; tÞ ¼ : n ðc aÞðm 1Þ 2 bc
ð29Þ
ð30Þ
ð31Þ
D. Feng et al. / Applied Mathematics and Computation 198 (2008) 715–720
(4) Soliton solutions of Rðn; nÞ equations ðn > 1Þ 1 " rffiffiffiffiffiffiffi !#nþ1 kðn 1Þ k 2 nþ1 : sech n u14 ðx; tÞ ¼ 2nða cÞ 2n bc (5) Soliton solutions of Rðm; 1Þ equations ðm > 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffi m1 ða cÞðm þ 1Þ ac 2 m1 sech u15 ðx; tÞ ¼ : n 2k 2 bc
719
ð32Þ
ð33Þ
2.4. Solitary pattern and soliton solutions of ansatz 4 Substituting (10) into (3), similar to the Section 2.3, we obtain the following results: (1) Solitary pattern solutions of Rð1; 1Þ equation rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2g ack 2 sinh u16 ðx; tÞ ¼ n : ack 4bc (2) Solitary pattern solutions of Rðn; nÞ equations ðn > 1Þ 1 " rffiffiffiffiffiffiffi !#n1 2nða cÞ k 2 n1 sinh : u17 ðx; tÞ ¼ n kðn þ 1Þ 2n bc (3) Solitary pattern solutions of Rðm; 1Þ equations ðm > 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffi mþ1 2k ac 2 mþ1 sinh u18 ðx; tÞ ¼ : n ða cÞðm 1Þ 2 bc (4) Soliton solutions of Rðn; nÞ equations ðn > 1Þ 1 " rffiffiffiffiffiffiffi !#nþ1 kðn 1Þ k 2 nþ1 csch u19 ðx; tÞ ¼ : n 2nðc aÞ 2n bc (5) Soliton solutions of Rðm; 1Þ equations ðm > 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffi m1 ðc aÞðm þ 1Þ ac 2 m1 csch u20 ðx; tÞ ¼ : n 2k 2 bc
ð34Þ
ð35Þ
ð36Þ
ð37Þ
ð38Þ
3. Conclusion With the aid of the mathematical software Maple, we obtain 20 families of solutions of Rðm; nÞ equations which contain compactons (solutions with the absence of infinite wings), solitary pattern solutions having infinite slopes or cusps, solitons and singular periodic wave solutions. Thus the direct method is not only efficient, but also has the merit of being widely applicable. References [1] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, London, 1991. [2] V.B. Matveev, M.A. Salle, Darboux Transformation and Solitons, Springer-Verlag, Berlin, 1991. [3] C.H. Gu, H.S. Hu, Z.X. Zhou, Darboux Transformations in Soliton Theory and its Geometric Applications, Shanghai Sci. Tech. Publ., Shanghai, 1999.
720 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
D. Feng et al. / Applied Mathematics and Computation 198 (2008) 715–720 R. Hirota, J. Satsuma, Soliton solutions of a coupled KdV equation, Phys. Lett. A 85 (1981) 407–408. P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer-Verlag, Berlin, 1989. J.B. Li, M. Li, Bounded travelling wave solutions for the (n + 1)-dimensional sine- and sinh-Gordon equations, Chaos Soliton. Fract. 25 (2005) 1037–1047. L. Tian, J. Yin, New compacton solutions and solitary wave solutions of fully nonlinear generalized Camassa–Holm equations, Chaos Soliton. Fract. 20 (2004) 289–299. A.M. Wazwaz, Solutions of compact and noncompact structures for nonlinear Klein–Gordon-type equation, Appl. Math. Comput. 134 (2003) 487–500. A.M. Wazwaz, A sine–cosine method for handling nonlinear wave equations, Math. Comput. Model. 40 (2004) 499–508. A.M. Wazwaz, Solitons and periodic solutions for the fifth-order KdV equation, Appl. Math. Lett. 19 (2006) 1162–1167. A.M. Wazwaz, Analytic study on nonlinear variant of the RLW and the PHI-four equations, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 314–327. Z.Y. Yan, H.Q. Zhang, New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics, Phys. Lett. A 252 (1999) 291–296. Z.Y. Yan, New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. Lett. A 292 (2001) 100–106. E.G. Fan, Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos Soliton. Fract. 16 (2003) 819–839. S. Zhang, New exact solutions of the KdV–Burgers–Kuramoto equation, Phys. Lett. A 358 (2006) 414–420. M.L. Wang, Exact solutions for a compound KdV–Burgers equation, Phys. Lett. A 213 (1996) 279–287. J.M. Dye, A. Parker, An inverse scattering scheme for the regularized long-wave equation, J. Math. Phys. 41 (2000) 2889–2904.
Applied Mathematics and Computation 198 (2008) 715–720 www.elsevier.com/locate/amc
New explicit and exact solutions for a system of variant RLW equations Dahe Feng
a,*
, Jibin Li b, Junliang Lu¨ b, Tianlan He
b
a
b
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, PR China Department of Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, PR China
Abstract In this article, a system of regularized long wave equations are studied. With the aid of the mathematic software Maple and using the direct method, some new exact solutions: compactons, solitons, solitary patterns and periodic solutions are obtained. Ó 2007 Elsevier Inc. All rights reserved. Keywords: R(m, n) equations; Compacton; Soliton; Solitary pattern solution; Periodic solution
1. Introduction The investigation of the exact solutions of nonlinear partial differential equations (PDEs) play an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in fluid dynamics, plasma and elastic media are often modelled by the bell-shaped sech functions and the kink-shaped tanh functions. The exact solution, if available, of those nonlinear PDEs facilitates the verification of numerical solvers and aids in the stability analysis of solutions. In the past several decades, various methods for obtaining exact solutions of nonlinear PDEs have been presented, such as inverse scattering method [1], Darboux transformation method [2,3], Hirota bilinear method [4], Lie group method [5,6], bifurcation method of dynamic systems [7], sine–cosine method [8–13], tanh function method [12–14], Fan-expansion method [15,16], homogenous balance method [17] and so on. Recently, Dye and Parker [18] studied the well-known nonlinear regularized long wave equation (RLW equation) ut þ aux 6uux þ buxxt ¼ 0
*
Corresponding author. E-mail addresses: [email protected] (D. Feng), [email protected] (J. Li), [email protected] (J. Lu¨).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.09.009
ð1Þ
716
D. Feng et al. / Applied Mathematics and Computation 198 (2008) 715–720
by using inverse scattering method. This RLW equation was introduced as an alternative model to the KdV equation to describe small amplitude long wave in shallow water. An increasing interest in studying the equation has been attached many mathematicians to investigate the problem to develop new solutions and to examine the physical behavior of the obtained solutions. More recently, Wazwaz [12] introduced a system of nonlinear variant RLW equations ut þ aux kðun Þx þ bðun Þxxt ¼ 0
ð2Þ
and derived some compact and noncompact exact solutions by using the sine–cosine method and the tanh method. Motivated by the rich mathematical and physical properties of the RLW Eqs. (1) and (2), in this paper we study the following generalized variant RLW equations (R(m, n) equations in short): Rðm; nÞ : ut þ aux kðum Þx þ bðun Þxxt ¼ 0:
ð3Þ
With the aid of Maple, we obtain some new exact solutions such as compactons, solitary pattern solutions, solitons and periodic solutions. 2. Explicit and exact solutions of R(m, n) equations Now we seek the travelling wave solutions of (3). Let uðx; tÞ ¼ uðnÞ;
n ¼ x ct;
ð4Þ
where c is wave speed. Substituting (4) into (3) yields ða cÞun kðum Þn bcðun Þnnn ¼ 0:
ð5Þ
Integrating (5) once, we obtain ða cÞu kum bcðun Þnn g ¼ 0; where g is an integral constant. Below we seek compacton solutions and solitary pattern solutions of (3) using the four ansatzs A cosd ðBnÞ; jBnj 6 p2 ; Ansatz 1: uðx; tÞ ¼ 0; otherwise: ( A sind ðBnÞ; jBnj 6 p; Ansatz 2: uðx; tÞ ¼ 0; otherwise: Ansatz 3: Ansatz 4:
uðx; tÞ ¼ A coshd ðBnÞ: d
uðx; tÞ ¼ A sinh ðBnÞ;
ð6Þ
ð7Þ ð8Þ ð9Þ ð10Þ
where A; B; d are parameters to be determined later. 2.1. Compact and noncompact solutions of ansatz 1 Substituting (7) into (3) gives Aða cÞ cosd ðBnÞ Am k cosmd ðBnÞ þ bcn2 d2 An B2 cosnd ðBnÞ bcndAn B2 ðnd 1Þ cosnd2 ðBnÞ g ¼ 0: ð11Þ Thus we can obtain the three possible cases to be discussed: Case 1. When nd 2 ¼ 0; d ¼ md ¼ nd, we have ( Aða cÞ Am k þ bcn2 d2 An B2 ¼ 0; bcndAn B2 ðnd 1Þ þ g ¼ 0
ð12Þ
D. Feng et al. / Applied Mathematics and Computation 198 (2008) 715–720
717
from which we get m ¼ n ¼ 1;
d ¼ 2;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kþca B¼ ; 4bc
A¼
2g : kþca
ð13Þ
Therefore we derive the compacton solution of R(1, 1) equation qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi ( 2g kþca bc p; n ; jnj 6 cos2 kþca 4bc kþca u1 ðx; tÞ ¼ 0; otherwise:
ð14Þ
Case 2. When nd 2 ¼ d; md ¼ nd; g ¼ 0, we have ( Aða cÞ bcndAn B2 ðnd 1Þ ¼ 0;
ð15Þ
Am k þ bcn2 d2 An B2 ¼ 0 which lead to m ¼ n;
2 ; d¼ n1
n1 B¼ 2n
rffiffiffiffiffi k ; bc
1 2nða cÞ n1 ; A¼ kðn þ 1Þ
g ¼ 0:
ð16Þ
Therefore we get the compacton solution of R(n, n) equations ðn > 1Þ 8h 1 qffiffiffi qffiffiffi in1 < 2nðacÞ 2 n1 k np bc n ; cos ; jnj 6 2n bc n1 k kðnþ1Þ u2 ðx; tÞ ¼ : 0; otherwise:
ð17Þ
Case 3. When d ¼ nd; md ¼ nd 2; g ¼ 0, we have (
Aða cÞ þ bcn2 d2 An B2 ¼ 0;
ð18Þ
Am k þ bcndAn B2 ðnd 1Þ ¼ 0 from which we get n ¼ 1;
2 ; d¼ m1
m1 B¼ 2
rffiffiffiffiffiffiffiffiffiffiffi ac ; bc
1 ða cÞðm þ 1Þ m1 ; A¼ 2k
g ¼ 0:
Thus we obtain the compacton solutions of Rðm; 1Þ equations ðm > 1Þ 8h qffiffiffiffiffiffi 1 pffiffiffiffiffiffi imþ1 < 2k 2 mþ1 ca p bc ; cos ; jnj 6 n ðcaÞðm1Þ 2 bc mþ1 ca u3 ðx; tÞ ¼ : 0; otherwise:
ð19Þ
ð20Þ
Remark (1) We can obtain the periodic solution of Rðn; nÞ equations ðn > 1Þ from Case 2 1 " rffiffiffiffiffi !#nþ1 kðn 1Þ nþ1 k 2 u4 ðx; tÞ ¼ sec ; n 2nða cÞ 2n bc (2) We also can get the periodic solution of Rðm; 1Þ equations ðm > 1Þ from Case 3 1 rffiffiffiffiffiffiffiffiffiffiffi m1 ða cÞðm þ 1Þ ca 2 m1 u5 ðx; tÞ ¼ sec : n 2k 2 bc
ð21Þ
ð22Þ
718
D. Feng et al. / Applied Mathematics and Computation 198 (2008) 715–720
2.2. Compact and noncompact solutions of ansatz 2 Substituting (8) into (3), we get the same three possible cases as the Section 2.1. Hence we obtain the following compact and noncompact solutions of R(m, n) equations: (1) Compacton solutions of R(1, 1) equation qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi ( 2g kþca 4bc p; n ; jnj 6 sin2 kþca 4bc kþca u6 ðx; tÞ ¼ 0; otherwise: (2) Compacton solutions of Rðn; nÞ equations ðn > 1Þ 8h 1 qffiffiffi qffiffiffi in1 < 2nðacÞ 2 n1 k 2np bc n ; sin ; jnj 6 2n bc n1 k kðnþ1Þ u7 ðx; tÞ ¼ : 0; otherwise: (3) Compacton solutions of Rðm; 1Þ equations ðm > 1Þ 8h qffiffiffiffiffiffi 1 ffi imþ1 < 2 mþ1 pffiffiffiffiffi 2k ca 2p bc n ; sin ; jnj 6 ðcaÞðm1Þ 2 bc mþ1 ca u8 ðx; tÞ ¼ : 0; otherwise: (4) Periodic solutions of Rðn; nÞ equations ðn > 1Þ 1 " rffiffiffiffiffi !#nþ1 kðn 1Þ k 2 nþ1 csc : n u9 ðx; tÞ ¼ 2nða cÞ 2n bc (5) Periodic solutions of Rðm; 1Þ equations ðm > 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffi m1 ða cÞðm þ 1Þ 2 m 1 c a u10 ðx; tÞ ¼ csc : n 2k 2 bc
ð23Þ
ð24Þ
ð25Þ
ð26Þ
ð27Þ
2.3. Solitary pattern and soliton solutions of ansatz 3 Substituting (9) into (3), we have Aða cÞcoshd ðBnÞ Am kcoshmd ðBnÞ bcn2 d2 An B2 coshnd ðBnÞ þ bcndAn B2 ðnd 1Þcoshnd2 ðBnÞ g ¼ 0 ð28Þ from which we can get the solitary pattern and soliton solutions of Rðm; nÞ equations by using the similar method as the Section 2.1 to determine the parameters A; B; d. (1) Solitary pattern solutions of Rð1; 1Þ equation rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2g ack 2 u11 ðx; tÞ ¼ cosh n : ack 4bc (2) Solitary pattern solutions of Rðn; nÞ equations ðn > 1Þ 1 " rffiffiffiffiffiffiffi !#n1 2nða cÞ k 2 n1 u12 ðx; tÞ ¼ cosh : n kðn þ 1Þ 2n bc (3) Solitary pattern solutions of Rðm; 1Þ equations ðm > 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffi mþ1 2k ac 2 mþ1 cosh u13 ðx; tÞ ¼ : n ðc aÞðm 1Þ 2 bc
ð29Þ
ð30Þ
ð31Þ
D. Feng et al. / Applied Mathematics and Computation 198 (2008) 715–720
(4) Soliton solutions of Rðn; nÞ equations ðn > 1Þ 1 " rffiffiffiffiffiffiffi !#nþ1 kðn 1Þ k 2 nþ1 : sech n u14 ðx; tÞ ¼ 2nða cÞ 2n bc (5) Soliton solutions of Rðm; 1Þ equations ðm > 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffi m1 ða cÞðm þ 1Þ ac 2 m1 sech u15 ðx; tÞ ¼ : n 2k 2 bc
719
ð32Þ
ð33Þ
2.4. Solitary pattern and soliton solutions of ansatz 4 Substituting (10) into (3), similar to the Section 2.3, we obtain the following results: (1) Solitary pattern solutions of Rð1; 1Þ equation rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2g ack 2 sinh u16 ðx; tÞ ¼ n : ack 4bc (2) Solitary pattern solutions of Rðn; nÞ equations ðn > 1Þ 1 " rffiffiffiffiffiffiffi !#n1 2nða cÞ k 2 n1 sinh : u17 ðx; tÞ ¼ n kðn þ 1Þ 2n bc (3) Solitary pattern solutions of Rðm; 1Þ equations ðm > 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffi mþ1 2k ac 2 mþ1 sinh u18 ðx; tÞ ¼ : n ða cÞðm 1Þ 2 bc (4) Soliton solutions of Rðn; nÞ equations ðn > 1Þ 1 " rffiffiffiffiffiffiffi !#nþ1 kðn 1Þ k 2 nþ1 csch u19 ðx; tÞ ¼ : n 2nðc aÞ 2n bc (5) Soliton solutions of Rðm; 1Þ equations ðm > 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffi m1 ðc aÞðm þ 1Þ ac 2 m1 csch u20 ðx; tÞ ¼ : n 2k 2 bc
ð34Þ
ð35Þ
ð36Þ
ð37Þ
ð38Þ
3. Conclusion With the aid of the mathematical software Maple, we obtain 20 families of solutions of Rðm; nÞ equations which contain compactons (solutions with the absence of infinite wings), solitary pattern solutions having infinite slopes or cusps, solitons and singular periodic wave solutions. Thus the direct method is not only efficient, but also has the merit of being widely applicable. References [1] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, London, 1991. [2] V.B. Matveev, M.A. Salle, Darboux Transformation and Solitons, Springer-Verlag, Berlin, 1991. [3] C.H. Gu, H.S. Hu, Z.X. Zhou, Darboux Transformations in Soliton Theory and its Geometric Applications, Shanghai Sci. Tech. Publ., Shanghai, 1999.
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