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Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using Sine–Cosine method
Available online at www.sciencedirect.com
Chaos, Solitons and Fractals 37 (2008) 1193–1197 www.elsevier.com/locate/chaos
Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using Sine–Cosine method E. Yusufog˘lu *, A. Bekir, M. Alp Dumlupınar University, Faculty of Art-Science, Department of Mathematics, Ku¨tahya, Turkey Accepted 9 October 2006
Communicated by Ji-Huan He
Abstract In this paper, we establish exact solutions for nonlinear evolution equations. The sine–cosine method is used to construct periodic and solitary wave solutions of the Kawahara and modified Kawahara equations. These solutions may be important of significance for the explanation of some practical physical problems. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The world around us is inherently nonlinear. Nonlinear evolution equations (NEEs) are widely used as models to describe complex physical phenomena in various fields of sciences, especially in fluid mechanics, solid state physics, plasma physics, plasma wave and chemical physics. Particularly, various methods have been utilized to explore different kinds of solutions of physical models described by nonlinear PDEs. One of the basic physical problems for those models is to obtain their traveling wave solutions. Concepts like solitons, peakons, kinks, breathers, cusps and compactons are now being thoroughly investigated in the scientific literature [14–16]. During the past decades, quite a few methods for obtaining explicit traveling and solitary wave solutions of nonlinear evolution equations have proposed a variety of powerfull methods, such as inverse scattering method [1,11], bilinear transformation [5], the tanh–sech method [8,17], extended tanh method [4,13] and homogeneous balance method [3]. The sine–cosine method was developed by Wazwaz [21] and was successfully applied to nonlinear evolution equations [12,17–19], to nonlinear equations systems [20]. By using the solutions of an auxilary ordinary differential equation, a direct algebraic method is described to construct the exact traveling wave solutions for nonlinear evolution equations. By this method the Kawahara and modified Kawahara equations are investigated [9]. *
Corresponding author. E-mail address: [email protected] (E. Yusufog˘lu).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.10.012
E. Yusufog˘lu et al. / Chaos, Solitons and Fractals 37 (2008) 1193–1197
1194
Some doubly periodic (Jacobi elliptic function) solutions of the modified Kawahara equation are presented in closed form [23]. The existence of compacton solutions and solitary patterns solutions for Kawahara type equation is demostrated in [22]. Existence and uniqueness of solutions to nonlinear Kawahara equations are obtained in [10]. The Kawahara equation ut þ uux þ puxxx quxxxxx ¼ 0;
ð1:1Þ
occurs in the theory of magneto-acoustic waves in a plasmas and in theory of shallow water waves with surface tension. Eq. (1.1) was first proposed by Kawahara in 1972, as a model equation describing solitary-wave propagation in media [7]. In the literature this equation is also referred as fifth-order KdV equation or singularly perturbed KdV equation [2]. The modified Kawahara equation ut þ u2 ux þ puxxx þ quxxxxx ¼ 0;
ð1:2Þ
where p, q are nonzero real constants [6]. Solitary waves are wave packets or pulses, which propagate in nonlinear dispersive media. Due to dynamical balance between the nonlinear and dispersive effects these waves retain a stable waveform. A soliton is a very special type of solitary wave, which also keep its waveform after collision with other solitons. The motivation of this paper is to extend the analysis of the sine–cosine method to solve two different types of nonlinear equations, namely, the Kawahara equation and modified Kawahara equation. 2. Sine–cosine method 1. We introduce the wave variable n = x ct into the PDE P ðu; ut ; ux ; utt ; uxx ; uxt ; uxxx ; . . .Þ ¼ 0;
ð2:1Þ
where u(x, t) is traveling wave solution. This enables us to use the following changes: o o o2 o2 o o o2 o2 ¼ c ; 2 ¼ c2 2 ; ¼ ; 2 ¼ 2 ; . . . ot on ot on ox on ox on
ð2:2Þ
One can immediately reduce the nonlinear PDE (2.1) into a nonlinear ODE Qðu; un ; unn ; unnn ; . . .Þ ¼ 0
ð2:3Þ
The ordinary differential Eq. (2.3) is then integrated as long as all terms contain derivatives, where we neglect integration constants. 2. The solutions of many nonlinear equations can be expressed in the form ( k sinb ðlnÞ; jnj 6 pl ; uðx; tÞ ¼ ð2:4Þ 0; otherwise; or in the form uðx; tÞ ¼
k cosb ðlnÞ;
p jnj 6 2l ;
0;
otherwise;
ð2:5Þ
where k, l and b are parameters that will be determined, l and c are the wave number and the wave speed, respectively [21]. We use uðnÞ ¼ k sinb ðlnÞ; un ðnÞ ¼ kn sinnb ðlnÞ; ðun Þn ¼ nlbkn cosðlnÞ sinnb1 ðlnÞ; n
2 2 2 n
nb
ð2:6Þ 2 n
ðu Þnn ¼ n l b k sin ðlnÞ þ nl k bðnb 1Þ sin
nb2
ðlnÞ;
and the derivatives of (2.5) become uðnÞ ¼ k cosb ðlnÞ; un ðnÞ ¼ kn cosnb ðlnÞ; ðun Þn ¼ nlbkn sinðlnÞ cosnb1 ðlnÞ; n
2 2 2 n
nb
2 n
ðu Þnn ¼ n l b k cos ðlnÞ þ nl k bðnb 1Þ cos
ð2:7Þ nb2
ðlnÞ;
E. Yusufog˘lu et al. / Chaos, Solitons and Fractals 37 (2008) 1193–1197
1195
and so on for other derivatives. 3. We substitute Eq. (2.6) or (2.7) into the reduced equation obtained above in (2.3), balance the terms of the cosine functions when (2.7) is used, or balance the terms of the sine functions when (2.6) is used, and solving the resulting system of algebraic equations by using the computerized symbolic calculations. We next collect all terms with same power in cosk(ln) or sink(ln) and set to zero their coefficients to get a system of algebraic equations among the unknowns l, b and k. We obtained all possible value of the parameters l, b and k.
3. The Kawahara equation In this section, we employ the sine–cosine method to the Kawahara (1.1). Using the wave variable n = x ct carries (1.1) into ODE cun þ uun þ punnn qunnnnn ¼ 0:
ð3:1Þ
Integrating (3.1) and using the constants of integration to be zero, we find 1 cu þ u2 þ punn qunnnn ¼ 0: 2
ð3:2Þ
Substituting Eq. (2.4) into (3.2) gives 1 ck sinb ðlnÞ þ k2 sin2b ðlnÞ pl2 b2 k sinb ðlnÞ þ pl2 kbðb 1Þ sinb2 ðlnÞ ql4 b4 k sinb ðlnÞ 2 þ 2ql4 kbðb 1Þðb2 2b þ 2Þ sinb2 ðlnÞ ql4 kbðb 1Þðb 2ðb 3Þ sinb4 ðlnÞ ¼ 0:
ð3:3Þ
Equating the exponents and the coefficients of each pair of the sine functions we find the following system of algebraic equations: ðb 1Þðb 2Þðb 3Þ 6¼ 0; b 4 ¼ 2b; 1 2 k 840qkl4 ¼ 0; 2 20pkl2 þ 1040qkl4 ¼ 0;
ð3:4Þ
ck 16pkl2 256qkl4 ¼ 0: Solving the system (3.4) yields b ¼ 4; sffiffiffiffiffiffiffiffiffiffiffi 1 13p ; l¼ 26 q k¼
ð3:5Þ
35 c: 12
The result (3.5) can be easily obtained if we also use the cosine method (2.5). Cosequently, for pq < 0, the following periodic solutions: sffiffiffiffiffiffiffiffiffiffiffi " sffiffiffiffiffiffiffiffiffiffiffi # 35 1 13p 1 13p uðx; tÞ ¼ ccsc4 ðx ctÞ ; 0 < ðx ctÞ < p; ð3:6Þ 12 26 q 26 q and " sffiffiffiffiffiffiffiffiffiffiffi # 35 1 13p 4 uðx; tÞ ¼ c sec ðx ctÞ ; 12 26 q
sffiffiffiffiffiffiffiffiffiffiffi 1 13p p ðx ctÞ < ; 26 2 q
are readily obtained. However, for pq > 0 , we obtain the solitary wave solutions " sffiffiffiffiffiffiffiffi # 35 13p 4 1 uðx; tÞ ¼ c sec h ðx ctÞ ; 12 26 q
ð3:7Þ
ð3:8Þ
E. Yusufog˘lu et al. / Chaos, Solitons and Fractals 37 (2008) 1193–1197
1196
and " sffiffiffiffiffiffiffiffi # 35 13p 4 1 ðx ctÞ : uðx; tÞ ¼ c csc h 12 26 q
ð3:9Þ
4. The modified Kawahara equation We next consider the modified Kawahara Eq. (1.2). Using the wave variable n = x ct and proceeding as before we find ð1 cÞun þ u2 un þ punnn qunnnnn ¼ 0:
ð4:1Þ
After intergating (4.1) and using the constants of integration to be zero we find 1 ð1 cÞu þ u3 þ punn qunnnn ¼ 0: 3 Using the sine ansatz Eqs. (2.4) into (4.2) gives
ð4:2Þ
1 ð1 cÞk sinb ðlnÞ þ k3 sin3b ðlnÞ pl2 b2 k sinb ðlnÞ þ pl2 kbðb 1Þ sinb2 ðlnÞ ql4 b4 k sinb ðlnÞ 3 þ 2ql4 kbðb 1Þðb2 2b þ 2Þ sinb2 ðlnÞ ql4 kbðb 1Þðb 2ðb 3Þ sinb4 ðlnÞ ¼ 0: This equation holds true if the system of algebraic equations ðb 1Þðb 2Þðb 3Þ 6¼ 0; b 4 ¼ 3b; 1 3 k 120qkl4 ¼ 0; 3 6pkl2 þ 120qkl4 ¼ 0;
ð4:4Þ
ð1 cÞk 4pkl2 16qkl4 ¼ 0; is satisfied. Solving this system yields b ¼ 2; sffiffiffiffiffiffiffiffiffi 1 5p ; l¼ 10 q 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10c 10: k¼ 4
ð4:5Þ
Consequently, for pq < 0, the following periodic solutions sffiffiffiffiffiffiffiffiffi " sffiffiffiffiffiffiffiffiffi # 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 5p 1 5p ðx ctÞ ; 0 < ðx ctÞ < p; uðx; tÞ ¼ 10c 10csc 4 10 q 10 q
ð4:6Þ
and " sffiffiffiffiffiffiffiffiffi # 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 5p uðx; tÞ ¼ ðx ctÞ ; 10c 10 sec 4 10 q
sffiffiffiffiffiffiffiffiffi 1 5p p ðx ctÞ < ; 10 2 q
are readily obtained. However, for pq > 0 , we obtain the solitary wave solutions " sffiffiffiffiffiffiffiffiffi # 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5p 2 1 uðx; tÞ ¼ 10c 10 sec h ðx ctÞ ; 4 10 q
ð4:7Þ
ð4:8Þ
and " sffiffiffiffiffiffiffiffiffi # 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5p 2 1 uðx; tÞ ¼ 10c 10 csc h ðx ctÞ : 4 10 q
ð4:9Þ
E. Yusufog˘lu et al. / Chaos, Solitons and Fractals 37 (2008) 1193–1197
1197
5. Conclusion In this study we have used the sine–cosine method to derive exact solutions with distinct physical structures. Some of these results are in agreement with the results reported by others in the literature, and new results are formally developed in this work. The method can be also applied to other nonlinear partial differential equations and their systems. The results revealed remarkable properties of the shames in that the solutions may come as solitary wave and periodic solutions depending on the method used. The availability of computer systems like Mathematica or Maple facilitates the tedious algebraic calculations.
References [1] Ablowitz MJ, Clarkson PA. Solitons nonlinear evolution equations and inverse scattering transform. Cambridge: Cambridge University Press; 1990. [2] Boyd JP. Weakly non-local solitons for capillary-gravity waves: fifth degree KdV equation. Physica D 1991;48:129–46. [3] Fan E, Zhang H. A note on the homogeneous balance method. Phys Lett A 1998;246:403–6. [4] Fan E. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 2000;277:212. [5] Hirota R. Direct method of finding exact solutions of nonlinear evolution equations. In: Bullough R, Caudrey P, editors. Backlund transformations. Berlin: Springer; 1980. p. 1157–75. [6] Hunter JK, Scheule J. Existence of perturbed solitary wave solutions to a model equation for water waves. Physica D 1988;32:253. [7] Kawahara T. Oscillatory solitary waves in dispersive media. J Phys Soc Jpn 1972;33:260. [8] Malfliet W, Hereman W. The tanh method: I. Exact solutions of nonlinear evolution wave equations. Phys Sprica 1996;54:569–75. [9] Sirendaoreji SJ. New exact travelling wave solutions for the Kawahara and modified Kawahara equations. Chaos, Solitons & Fractals 2004;19:147–50. [10] Shuangping T, Shuangbin C. Existence and uniqueness of solutions to nonlinear Kawahara equations. Chin Ann Math Ser A 2002;23(2):221–8. [11] Vakhnenko VO, J Parkes E, Morrison AJ. A Ba¨cklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos, Solitons & Fractals 2003;17(4):683–92. [12] Yan Z. Constructing exact solutions for two-dimensional nonlinear dispersion Boussinesq equation. II: Solitary pattern solutions. Chaos, Solitons & Fractals 2003;18(4):869–80. [13] Yusufoglu E, Bekir A. A travelling wave solution to the Ostrovsky equation. Appl Math Comput, in press. [14] Wadati M. Introduction to solitons. Pramana: J Phys 2001;57(5-6):841–7. [15] Wadati M. The exact solution of the modified Korteweg-de Vries equation. J Phys Soc Jpn 1972;32:1681–7. [16] Wadati M. The modified Korteweg-de Vries equation. J Phys Soc Jpn 1973;34:1289–96. [17] Wazwaz AM. Two reliable methods for solving variants of the KdV equation with compact and noncompact structures. Chaos, Solitons & Fractals 2006;28(2):454–62. [18] Wazwaz AM, Helal MA. Nonlinear variants of the BBM equation with compact and noncompact physical structures. Chaos, Solitons & Fractals 2005;26(3):767–76. [19] M Wazwaz A. New compactons, solitons and periodic solutions for nonlinear variants of the KdV and the KP equations. Chaos, Solitons & Fractals 2004;22(1):249–60. [20] M Wazwaz A. Exact and explicit travelling wave solutions for the nonlinear Drinfeld–Sokolov system. Commun Nonlin Sci Numer Simul 2006;11:311–25. [21] Wazwaz AM. A sine–cosine method for handling nonlinear wave equations. Math Comput Model 2004;40:499–508. [22] Wazwaz AM. Compacton solutions of the Kawahara-type nonlinear dispersive equation. Appl Math Comput 2003;145:133–50. [23] Zhang D, equations Doubly periodic solutions of the modified Kawahara. Chaos, Solitons & Fractals 2005;25:1155–60.
Chaos, Solitons and Fractals 37 (2008) 1193–1197 www.elsevier.com/locate/chaos
Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using Sine–Cosine method E. Yusufog˘lu *, A. Bekir, M. Alp Dumlupınar University, Faculty of Art-Science, Department of Mathematics, Ku¨tahya, Turkey Accepted 9 October 2006
Communicated by Ji-Huan He
Abstract In this paper, we establish exact solutions for nonlinear evolution equations. The sine–cosine method is used to construct periodic and solitary wave solutions of the Kawahara and modified Kawahara equations. These solutions may be important of significance for the explanation of some practical physical problems. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The world around us is inherently nonlinear. Nonlinear evolution equations (NEEs) are widely used as models to describe complex physical phenomena in various fields of sciences, especially in fluid mechanics, solid state physics, plasma physics, plasma wave and chemical physics. Particularly, various methods have been utilized to explore different kinds of solutions of physical models described by nonlinear PDEs. One of the basic physical problems for those models is to obtain their traveling wave solutions. Concepts like solitons, peakons, kinks, breathers, cusps and compactons are now being thoroughly investigated in the scientific literature [14–16]. During the past decades, quite a few methods for obtaining explicit traveling and solitary wave solutions of nonlinear evolution equations have proposed a variety of powerfull methods, such as inverse scattering method [1,11], bilinear transformation [5], the tanh–sech method [8,17], extended tanh method [4,13] and homogeneous balance method [3]. The sine–cosine method was developed by Wazwaz [21] and was successfully applied to nonlinear evolution equations [12,17–19], to nonlinear equations systems [20]. By using the solutions of an auxilary ordinary differential equation, a direct algebraic method is described to construct the exact traveling wave solutions for nonlinear evolution equations. By this method the Kawahara and modified Kawahara equations are investigated [9]. *
Corresponding author. E-mail address: [email protected] (E. Yusufog˘lu).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.10.012
E. Yusufog˘lu et al. / Chaos, Solitons and Fractals 37 (2008) 1193–1197
1194
Some doubly periodic (Jacobi elliptic function) solutions of the modified Kawahara equation are presented in closed form [23]. The existence of compacton solutions and solitary patterns solutions for Kawahara type equation is demostrated in [22]. Existence and uniqueness of solutions to nonlinear Kawahara equations are obtained in [10]. The Kawahara equation ut þ uux þ puxxx quxxxxx ¼ 0;
ð1:1Þ
occurs in the theory of magneto-acoustic waves in a plasmas and in theory of shallow water waves with surface tension. Eq. (1.1) was first proposed by Kawahara in 1972, as a model equation describing solitary-wave propagation in media [7]. In the literature this equation is also referred as fifth-order KdV equation or singularly perturbed KdV equation [2]. The modified Kawahara equation ut þ u2 ux þ puxxx þ quxxxxx ¼ 0;
ð1:2Þ
where p, q are nonzero real constants [6]. Solitary waves are wave packets or pulses, which propagate in nonlinear dispersive media. Due to dynamical balance between the nonlinear and dispersive effects these waves retain a stable waveform. A soliton is a very special type of solitary wave, which also keep its waveform after collision with other solitons. The motivation of this paper is to extend the analysis of the sine–cosine method to solve two different types of nonlinear equations, namely, the Kawahara equation and modified Kawahara equation. 2. Sine–cosine method 1. We introduce the wave variable n = x ct into the PDE P ðu; ut ; ux ; utt ; uxx ; uxt ; uxxx ; . . .Þ ¼ 0;
ð2:1Þ
where u(x, t) is traveling wave solution. This enables us to use the following changes: o o o2 o2 o o o2 o2 ¼ c ; 2 ¼ c2 2 ; ¼ ; 2 ¼ 2 ; . . . ot on ot on ox on ox on
ð2:2Þ
One can immediately reduce the nonlinear PDE (2.1) into a nonlinear ODE Qðu; un ; unn ; unnn ; . . .Þ ¼ 0
ð2:3Þ
The ordinary differential Eq. (2.3) is then integrated as long as all terms contain derivatives, where we neglect integration constants. 2. The solutions of many nonlinear equations can be expressed in the form ( k sinb ðlnÞ; jnj 6 pl ; uðx; tÞ ¼ ð2:4Þ 0; otherwise; or in the form uðx; tÞ ¼
k cosb ðlnÞ;
p jnj 6 2l ;
0;
otherwise;
ð2:5Þ
where k, l and b are parameters that will be determined, l and c are the wave number and the wave speed, respectively [21]. We use uðnÞ ¼ k sinb ðlnÞ; un ðnÞ ¼ kn sinnb ðlnÞ; ðun Þn ¼ nlbkn cosðlnÞ sinnb1 ðlnÞ; n
2 2 2 n
nb
ð2:6Þ 2 n
ðu Þnn ¼ n l b k sin ðlnÞ þ nl k bðnb 1Þ sin
nb2
ðlnÞ;
and the derivatives of (2.5) become uðnÞ ¼ k cosb ðlnÞ; un ðnÞ ¼ kn cosnb ðlnÞ; ðun Þn ¼ nlbkn sinðlnÞ cosnb1 ðlnÞ; n
2 2 2 n
nb
2 n
ðu Þnn ¼ n l b k cos ðlnÞ þ nl k bðnb 1Þ cos
ð2:7Þ nb2
ðlnÞ;
E. Yusufog˘lu et al. / Chaos, Solitons and Fractals 37 (2008) 1193–1197
1195
and so on for other derivatives. 3. We substitute Eq. (2.6) or (2.7) into the reduced equation obtained above in (2.3), balance the terms of the cosine functions when (2.7) is used, or balance the terms of the sine functions when (2.6) is used, and solving the resulting system of algebraic equations by using the computerized symbolic calculations. We next collect all terms with same power in cosk(ln) or sink(ln) and set to zero their coefficients to get a system of algebraic equations among the unknowns l, b and k. We obtained all possible value of the parameters l, b and k.
3. The Kawahara equation In this section, we employ the sine–cosine method to the Kawahara (1.1). Using the wave variable n = x ct carries (1.1) into ODE cun þ uun þ punnn qunnnnn ¼ 0:
ð3:1Þ
Integrating (3.1) and using the constants of integration to be zero, we find 1 cu þ u2 þ punn qunnnn ¼ 0: 2
ð3:2Þ
Substituting Eq. (2.4) into (3.2) gives 1 ck sinb ðlnÞ þ k2 sin2b ðlnÞ pl2 b2 k sinb ðlnÞ þ pl2 kbðb 1Þ sinb2 ðlnÞ ql4 b4 k sinb ðlnÞ 2 þ 2ql4 kbðb 1Þðb2 2b þ 2Þ sinb2 ðlnÞ ql4 kbðb 1Þðb 2ðb 3Þ sinb4 ðlnÞ ¼ 0:
ð3:3Þ
Equating the exponents and the coefficients of each pair of the sine functions we find the following system of algebraic equations: ðb 1Þðb 2Þðb 3Þ 6¼ 0; b 4 ¼ 2b; 1 2 k 840qkl4 ¼ 0; 2 20pkl2 þ 1040qkl4 ¼ 0;
ð3:4Þ
ck 16pkl2 256qkl4 ¼ 0: Solving the system (3.4) yields b ¼ 4; sffiffiffiffiffiffiffiffiffiffiffi 1 13p ; l¼ 26 q k¼
ð3:5Þ
35 c: 12
The result (3.5) can be easily obtained if we also use the cosine method (2.5). Cosequently, for pq < 0, the following periodic solutions: sffiffiffiffiffiffiffiffiffiffiffi " sffiffiffiffiffiffiffiffiffiffiffi # 35 1 13p 1 13p uðx; tÞ ¼ ccsc4 ðx ctÞ ; 0 < ðx ctÞ < p; ð3:6Þ 12 26 q 26 q and " sffiffiffiffiffiffiffiffiffiffiffi # 35 1 13p 4 uðx; tÞ ¼ c sec ðx ctÞ ; 12 26 q
sffiffiffiffiffiffiffiffiffiffiffi 1 13p p ðx ctÞ < ; 26 2 q
are readily obtained. However, for pq > 0 , we obtain the solitary wave solutions " sffiffiffiffiffiffiffiffi # 35 13p 4 1 uðx; tÞ ¼ c sec h ðx ctÞ ; 12 26 q
ð3:7Þ
ð3:8Þ
E. Yusufog˘lu et al. / Chaos, Solitons and Fractals 37 (2008) 1193–1197
1196
and " sffiffiffiffiffiffiffiffi # 35 13p 4 1 ðx ctÞ : uðx; tÞ ¼ c csc h 12 26 q
ð3:9Þ
4. The modified Kawahara equation We next consider the modified Kawahara Eq. (1.2). Using the wave variable n = x ct and proceeding as before we find ð1 cÞun þ u2 un þ punnn qunnnnn ¼ 0:
ð4:1Þ
After intergating (4.1) and using the constants of integration to be zero we find 1 ð1 cÞu þ u3 þ punn qunnnn ¼ 0: 3 Using the sine ansatz Eqs. (2.4) into (4.2) gives
ð4:2Þ
1 ð1 cÞk sinb ðlnÞ þ k3 sin3b ðlnÞ pl2 b2 k sinb ðlnÞ þ pl2 kbðb 1Þ sinb2 ðlnÞ ql4 b4 k sinb ðlnÞ 3 þ 2ql4 kbðb 1Þðb2 2b þ 2Þ sinb2 ðlnÞ ql4 kbðb 1Þðb 2ðb 3Þ sinb4 ðlnÞ ¼ 0: This equation holds true if the system of algebraic equations ðb 1Þðb 2Þðb 3Þ 6¼ 0; b 4 ¼ 3b; 1 3 k 120qkl4 ¼ 0; 3 6pkl2 þ 120qkl4 ¼ 0;
ð4:4Þ
ð1 cÞk 4pkl2 16qkl4 ¼ 0; is satisfied. Solving this system yields b ¼ 2; sffiffiffiffiffiffiffiffiffi 1 5p ; l¼ 10 q 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10c 10: k¼ 4
ð4:5Þ
Consequently, for pq < 0, the following periodic solutions sffiffiffiffiffiffiffiffiffi " sffiffiffiffiffiffiffiffiffi # 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 5p 1 5p ðx ctÞ ; 0 < ðx ctÞ < p; uðx; tÞ ¼ 10c 10csc 4 10 q 10 q
ð4:6Þ
and " sffiffiffiffiffiffiffiffiffi # 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 5p uðx; tÞ ¼ ðx ctÞ ; 10c 10 sec 4 10 q
sffiffiffiffiffiffiffiffiffi 1 5p p ðx ctÞ < ; 10 2 q
are readily obtained. However, for pq > 0 , we obtain the solitary wave solutions " sffiffiffiffiffiffiffiffiffi # 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5p 2 1 uðx; tÞ ¼ 10c 10 sec h ðx ctÞ ; 4 10 q
ð4:7Þ
ð4:8Þ
and " sffiffiffiffiffiffiffiffiffi # 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5p 2 1 uðx; tÞ ¼ 10c 10 csc h ðx ctÞ : 4 10 q
ð4:9Þ
E. Yusufog˘lu et al. / Chaos, Solitons and Fractals 37 (2008) 1193–1197
1197
5. Conclusion In this study we have used the sine–cosine method to derive exact solutions with distinct physical structures. Some of these results are in agreement with the results reported by others in the literature, and new results are formally developed in this work. The method can be also applied to other nonlinear partial differential equations and their systems. The results revealed remarkable properties of the shames in that the solutions may come as solitary wave and periodic solutions depending on the method used. The availability of computer systems like Mathematica or Maple facilitates the tedious algebraic calculations.
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