- Email: [email protected]
The homogeneous balance method and its application to the Benjamin–Bona–Mahoney (BBM) equation
Applied Mathematics and Computation 217 (2010) 1385–1390
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The homogeneous balance method and its application to the Benjamin–Bona–Mahoney (BBM) equation A.S. Abdel Rady *, E.S. Osman, Mohammed Khalfallah * Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt
a r t i c l e
i n f o
Keywords: Homogeneous balance method Traveling wave solutions Soliton solutions Benjamin–Bona–Mahoney equation Riccati equation
a b s t r a c t We make use of the homogeneous balance method and symbolic computation to construct new exact traveling wave solutions for the Benjamin–Bona–Mahoney (BBM) equation. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations. Ó 2009 Elsevier Inc. All rights reserved.
Many phenomena in physics and other fields are described by nonlinear evolution equations. When we want to understand the physical mechanism of phenomena in nature, described by nonlinear evolution equations, exact traveling wave solutions have to be explored. The investigation of the exact traveling wave solutions of nonlinear evolutions equations plays an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in fluid dynamics, elastic media, optical fibers, etc. In recent years, the homogeneous balance (HB) method has been widely applied to derive the nonlinear transformation and exact solutions (especially the solitary wave solutions) [1–5] and auto Bäcklund transformations [5,6,13] as well as the similarity reductions [5,13] of nonlinear partial differential equations (PDEs) in mathematical physics. Wang et al. [7–9], Khalfallah [10–12] applied the (HB) method to obtain the new exact traveling wave solutions of a given nonlinear partial differential equations. Fan [13] showed that there is a close connection among the HB method, Wiess, Tabor, Carnevale (WTC) method and Clarkson, Kruskal(CK) method. As the mathematical models of complex physical phenomena, nonlinear evolution equations are involved in many fields from physics to biology, chemistry, engineer, plasma physics, optical fibers and solid state physics etc. Many methods were developed for finding the exact traveling wave solutions of nonlinear evolutions equations, such as Hirota’s method, Backlund and Darboux transformation, Painlevé expansions, Homogeneous balance method, Jacobi elliptic function, Extended tanh-function method, F-expansion method and extended F-expansion method. In this paper, we use the HB method to solve the Riccati equation /0 ¼ a/2 þ b/ þ c and the reduced nonlinear ordinary differential equation for the BBM equation, respectively. It makes the HB method use more extensively. For the regularized long wave or BBM equation [14]
ut þ ux þ uux uxxt ¼ 0;
ð1Þ
which was derived as a model for the unidirectional propagation of long-crested, surface water waves. It arises in other contexts as well, and is generally understood as an alternative to the Korteweg–de Vries equation.
* Corresponding authors. E-mail addresses: [email protected] (A.S. Abdel Rady), [email protected] (M. Khalfallah). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.05.027
1386
A.S. Abdel Rady et al. / Applied Mathematics and Computation 217 (2010) 1385–1390
Long wave in shallow water is a subject of broad interests and has a long colorful history. Physically, it has a rich variety of phenomenological manifestation, especially the existence of wave permanent in form and robust in maintaining their entities through mutual interaction and collision as well as the remarkable property of exhibiting recurrences of initial data when circumstances should prevail. These characteristics are due to the intimate interplay between the roles of nonlinearity and dispersion. Let us consider the traveling wave solutions
uðx; tÞ ¼ uðfÞ;
f ¼ kx þ lt þ d;
ð2Þ
where k; l and d are constants. Substituting (2) into (1), then (1) is reduced to the following nonlinear ordinary differential equation 2
000
0
0
0
k lu ku kuu lu ¼ 0:
ð3Þ
We now seek the solutions of Eq. (3) in the form
u¼
m X
qi /i ;
ð4Þ
i¼0
where qi are constants to be determined later and / satisfy the Riccati equation
/0 ¼ a/2 þ b/ þ c;
ð5Þ
where a; b and c are constants and / satisfy Eq. (5). It is easy to show that m ¼ 2 if balancing u000 with uu0 . Therefore use the ansatz
u ¼ q0 þ q1 / þ q2 /2 ;
ð6Þ i
Substituting Eq. (6) into Eq. (3) along with (5) and collecting all terms with the same power in / ði ¼ 0; 1; 2; 3; 4; 5Þ yields a set of algebraic system for q0 ; q1 ; q2 ; k and l, namely
2 2 cl q1 b þ 6cq2 b þ 2acq1 k þ cð2q0 q1 þ q1 Þk þ clq1 ¼ 0; 2 2 bl q1 b þ 6cq2 b þ 2acq1 k 2cð3abkl q1 Þq1 k þ bð2q0 q1 þ q1 Þk þ blq1 2c 2 k 4 b þ 2ac kl 2q0 1 l q2 ¼ 0; 2 2 al q1 b þ 6cq2 b þ 2acq1 k 2bð3abkl q1 Þq1 k þ að2q0 q1 þ q1 Þk 6c 2 klq1 a2 þ 5bklq2 a q1 q2 k þ alq1 2b k 4 b þ 2ac kl 2q0 1 l q2 ¼ 0; 2 2akð3abkl q1 Þq1 2a k 4 b þ 2ac kl 2q0 1 l q2 4ck 2 6a kl q2 q2 6bk klq1 a2 þ 5bklq2 a q1 q2 ¼ 0; 2 4bk 6a kl q2 q2 6ak klq1 a2 þ 5bklq2 a q1 q2 ¼ 0; 2 4ak 6a kl q2 q2 ¼ 0:
ð7Þ
For which, with the aid of Mathematica, we find 2
q0 ¼
2
2
b lk þ 8aclk k l ; 2k
q1 ¼ 6abkl;
q2 ¼ 6a2 kl:
ð8Þ
It is to be noted that the Riccati equation (5) can be solved using the homogeneous balance method as follows i 0 2 Case: I. Let / ¼ Rm i¼0 bi tanh f. Balancing / with / leads to
/ ¼ b0 þ b1 tanh f:
ð9Þ
Substituting Eq. (9) into (5), we have the following solution of Eq. (5)
/¼
1 ðb þ 2 tanh fÞ; 2a
2
ac ¼
b 1: 4
ð10Þ
Substituting Eqs. (8) and (9) into (6) and (2), we have the following new traveling wave solution of BBM Eq. (1) 2
uðx; tÞ ¼
2 2
8ack l 2b k l k l 2 þ 6kl tanh ðkx þ kt þ dÞ; 2k
This is a bell-shaped solution. i Similarly, let / ¼ Rm i¼0 bi coth f, then we obtain the following new traveling wave soliton solutions of BBM Eq. (1)
ð11Þ
A.S. Abdel Rady et al. / Applied Mathematics and Computation 217 (2010) 1385–1390 2
uðx; tÞ ¼
1387
2 2
8ack l 2b k l k l 2 þ 6kl coth ðkx þ kt þ dÞ: 2k
ð12Þ
Case: II. From [15], when a ¼ 1, b ¼ 0, the Riccati equation (5) has the following solutions
pffiffiffiffiffiffi 8 pffiffiffiffiffiffi > < c tanh cf ; c < 0; c ¼ 0; / ¼ 1f ; > pffiffiffi : pffiffiffi c tan cf : c > 0:
ð13Þ
It is seen that the tanh function in (4) is only a special function in (13), so we conjecture that Eq. (1) may admit other types of traveling wave solutions in Eq. (13) in addition to the tanh-type one. Moreover, we hope to construct them in a unified way. For this purpose, we shall use the Riccati equation (5) once again to generate an associated algebraic system, but not use one of the functions in (13). Another advantage of the Riccati equation (5) is that the sign of c can be used to exactly judge the type of the traveling wave solution for Eq. (1). For example, if c < 0, we are sure that Eq. (1) admits tanh-type and coth-type traveling wave solutions. Especially Eq. (1) will possess three types of traveling wave solutions if c is an arbitrary constant. In this way, we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods. From (6), (8) and (13), we have the following new traveling wave solutions of BBM Eq. (1), which contain traveling wave solutions, periodic wave solutions and rational solutions as follows When c < 0, we have
uðx; tÞ ¼
2 2 2 pffiffiffiffiffiffi pffiffiffiffiffiffi b lk þ 8aclk k l 2 pffiffiffiffiffiffi cðkx þ kt þ dÞ : 6abkl c tanh cðkx þ kt þ dÞ 6a2 klc tanh 2k
ð14Þ
This is a linear combinations of kink wave and bell-shaped wave solutions which is a new solution for (1). When c ¼ 0, we have 2
uðx; tÞ ¼
2
2
b lk þ 8aclk k l 6abkl 6a2 kl þ ; 2k ðkx þ kt þ dÞ ðkx þ kt þ dÞ2
ð15Þ
When c > 0, we have
uðx; tÞ ¼
2 2 2 pffiffiffi pffiffiffi pffiffiffi b lk þ 8aclk k l þ 6abkl c tan cðkx þ kt þ dÞ þ 6a2 klc tan2 cðkx þ kt þ dÞ : 2k
ð16Þ
Which contain a periodic-like solutions. Case: III. We suppose that the Riccati equation (5) has the following solutions of the form
/ ¼ A0 þ
m X ðAi f i þ Bi f i1 gÞ;
ð17Þ
i¼1
with
f ¼
1 ; cosh f þ r
g¼
sinh f ; cosh f þ r
which satisfy
f 0 ðfÞ ¼ f ðfÞgðfÞ;
g 0 ðfÞ ¼ 1 g 2 ðfÞ rf ðfÞ;
g 2 ðfÞ ¼ 1 2rf ðfÞ þ ðr2 1Þf 2 ðfÞ: Balancing /0 with /2 leads to
/ ¼ A0 þ A1 f þ B1 g:
ð18Þ i
j
Substituting Eq. (18) into (5), collecting the coefficient of the same power f ðfÞg ðfÞ ði ¼ 0; 1; 2; j ¼ 0; 1Þ and setting each of the obtained coefficients to zero yield the following set of algebra equations
aA20 þ aB21 þ bA0 þ c ¼ 0; 2aA0 A1 2arB21 rB1 þ bA1 ¼ 0; aA21 þ aðr2 1ÞB21 þ ðr 2 1ÞB1 ¼ 0; 2aA0 B1 þ bB1 ¼ 0; 2aA1 B1 þ A1 ¼ 0;
ð19Þ
which have solutions
b A0 ¼ ; 2a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr 2 1Þ A1 ¼ ; 4a2
B1 ¼
1 ; 2a
2
c¼
b 1 : 4a
ð20Þ
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A.S. Abdel Rady et al. / Applied Mathematics and Computation 217 (2010) 1385–1390
From Eqs. (18)–(20), we have
/¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! sinh f ðr 2 1Þ 1 bþ 2a cosh f þ r
ð21Þ
From Eqs. (6), (8) and (21), we obtain the new wave solutions of BBM (1)
uðx; tÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 2 2 2 sinhðkx þ kt þ dÞ ðr 2 1Þ 8ack l 2b k l k l : þ 6kl 2k 2ðcoshðkx þ kt þ dÞ þ rÞ
ð22Þ
Case: IV. We take / in the Riccati equation (5) being of the form
/ ¼ ep1 f qðzÞ þ p4 ðfÞ;
ð23Þ
where
z ¼ ep2 f þ p3 ; where p1 ; p2 and p3 are constants to be determined. 2 2 P 1 þb Substituting (23) into (5) we find that when c ¼ 4a , we have
/¼
p1 ep1 f p b : þ 1 2a aðep1 f þ p3 Þ
ð24Þ
If p3 ¼ 1 in (24), we have
/¼
p1 1 b p1 f : tanh 2 2a 2a
ð25Þ
If p3 ¼ 1 in (24), we have
/¼
p1 1 b p1 f : coth 2 2a 2a
ð26Þ
From (6), (8) and (24), we obtain the following new wave solutions of BBM Eq. (1)
uðx; tÞ ¼
2 2 2 8ack l þ b k l k l p b p1 ep1 f p1 þ b p1 ep1 f ; þ 6akl 1 2k 2a 2 aðep1 f þ p3 Þ ep1 f þ p3
ð27Þ
where f ¼ kx þ kt þ d. When p3 ¼ 1, we have from (25)
uðx; tÞ ¼
2 2 2 2 8ack l 2b k l k l 3klp1 2 1 þ p1 ðkx þ kt þ dÞ : tanh 2k 2 2
ð28Þ
When p3 ¼ 1, we have from (29)
uðx; tÞ ¼
2 2 2 2 8ack l 2b k l k l 3klp1 2 1 þ p1 ðkx þ kt þ dÞ : coth 2k 2 2
ð29Þ
Clearly, (11), (12) is the special case of (28), (29) with p1 ¼ 2. Case: V. We suppose that the Riccati equation (5) have the following solutions of the form:
/ ¼ A0 þ
m X
i1
sinh
ðAi sinh x þ Bi cosh xÞ;
i¼1
where dx=df ¼ sinh x or dx=df ¼ cosh x. It is easy to find that m ¼ 1 by balancing /0 and /2 . So we choose
/ ¼ A0 þ A1 sinh x þ B1 cosh x;
ð30Þ i
j
when dx=df ¼ sinh x, we substitute (30) and dx=df ¼ sinh x, into (5) and set the coefficient of sinh xcosh xði ¼ 0; 1; 2; j ¼ 0; 1Þ to zero. A set of algebraic equations is obtained as follows:
aA20 þ aB21 þ bA0 þ c ¼ 0; 2aA0 A1 þ bA1 ¼ 0; aA21 þ aB21 ¼ B1 2aA0 B1 þ bB1 ¼ 0; 2aA1 B1 þ A1 ¼ 0;
ð31Þ
A.S. Abdel Rady et al. / Applied Mathematics and Computation 217 (2010) 1385–1390
1389
for which, we have the following solutions:
A0 ¼
b ; 2a
A1 ¼ 0;
B1 ¼
1 ; a
ð32Þ
2
where c ¼ b 4a4, and
A0 ¼
b ; 2a
rffiffiffiffiffiffi 1 ; 2a
A1 ¼
B1 ¼
1 ; 2a
ð33Þ
2
where c ¼ b 4a1. To dx=df ¼ sinh x, we have
sinh x ¼ csch f;
cosh x ¼ coth f:
ð34Þ
From (33)-(34), we obtain
/¼ where c ¼
b þ 2 coth f ; 2a
b2 4 , 4a
/¼
ð35Þ
and
b csch f þ coth f ; 2a
ð36Þ
2
where c ¼ b 4a1. Clearly (35) is the special case of (26) with p1 ¼ 2. From (5), (6), (8), (35) and (36), we get the new traveling wave solutions of Eq. (1) in the following form 2
uðx; tÞ ¼
2 2
8ack l 2b k l k l 2 þ 6kl coth ðkx þ kt þ dÞ: 2k
ð37Þ
2
where c ¼ b 4a4, and 2
uðx; tÞ ¼
2 2
8ack l 2b k l k l 3kl þ ðcothðkx þ kt þ dÞ cschðkx þ kt þ dÞÞ2 ; 2k 2
ð38Þ
2
where c ¼ b 4a1. Clearly (37) is the special case of (29) with p1 ¼ 2. Similarly, when dx=df ¼ cosh x, we obtain the following exact traveling wave (periodic-like) solutions of BBM equation (1) 2
uðx; tÞ ¼
2 2
8ack l 2b k l k l þ 6klcot2 ðkx þ kt þ dÞ: 2k
ð39Þ
2
where c ¼ b 4a4, and 2
uðx; tÞ ¼
2 2
8ack l 2b k l k l 3kl þ ðcotðkx þ kt þ dÞ cscðkx þ kt þ dÞÞ2 : 2k 2
ð40Þ
HB method provides us a solutions polynomials in two elementary bell-shaped and kink-shaped functions, this covers the large majority of physically interesting solitary waves Eqs. (11), (28), in addition we also obtain a periodic-like wave solutions Eqs. (16), (39), (40) which play an important interesting in physics. Remark. The majority of solutions for (5) in [16], contain solitary wave, periodic wave, rational solutions, ... but in this paper we study the solutions of (5) which contains kink-type, bell-shaped, periodic-like, rational solutions which are well-known solutions and also new solutions for instance in case III, V are obtained, i.e. we recovered by HB method the well-known solutions and adding some new solutions. In summary we have used the extended homogeneous balance method to obtain many traveling wave solutions of BBM equation. We now summarize the key steps as follows: Step 1: For a given nonlinear evolution equation
Fðu; ut ; ux ; uxt ; utt ; . . .Þ ¼ 0;
ð41Þ
we consider its traveling wave solutions uðx; tÞ ¼ uðfÞ; f ¼ kx þ kt þ d then Eq. (41) is reduced to an nonlinear ordinary differential equation
Q ðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0; d . df
ð42Þ
where a prime denotes Step 2: For a given ansatz equation (for example, the ansatz equation is /0 ¼ a/2 þ b/ þ c in this paper), the form of u is decided and the homogeneous balance method is used on Eq. (42) to find the coefficients of u.
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A.S. Abdel Rady et al. / Applied Mathematics and Computation 217 (2010) 1385–1390
Step 3: The homogeneous balance method is used to solve the ansatz equation. Step 4: Finally, the traveling wave solutions of Eq. (41) are obtained by combining step 2 and 3. From the above procedure, it is easy to find that the homogeneous balance method is more effective and simple than other methods and a lot of solutions can be obtained in the same time. This method can be also applied to other nonlinear evolution equations. In Eq. (5), if we set b ¼ 0, a ¼ l, c ¼ l HB method is reduced to tanh method [17] but HB method is more general than tanh method, extended tanh method [18] because this method is readily applicable to a large variety of nonlinear PDEs in contrast to the tanh method. Some merits are available for HB method. First, all the nonlinear PDEs which can be solved by tanh-function method can be solved easily by HB method.and we have more multiple soliton solutions and triangular periodic solution (including rational solutions). Second, we used only the special solutions of Eq. (1), we can obtain more traveling wave solutions. Third, not only HB method contains the hyperbolic tangent expansion method, but also it is a computerizable method, which allow us to perform complicated and tedious differential calculation on computer. Therefore, the HB method is a generalized tanh-function method for many nonlinear PDEs. References [1] E.G. Fan, H.Q. Zhang, New exact solutions to a system of coupled KdV equations, Phys. Lett. A 245 (1998) 389–392. [2] Anjan Biswas, Swapan Konar, Soliton perturbation theory for the generalized Benjamin–Bona–Mahoney equation, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 703–706. [3] Mariana Antonova, Anjan Biswas, Adiabatic parameter dynamics of perturbed solitary waves, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 734– 748. [4] E.G. Fan, H.Q. Zhang, A note on the homogeneous balance method, Phys. Lett. A 246 (1998) 403–406. [5] E.G. Fan, Auto-Bäcklund transformation and similarity reductions for general variable coefficient KdV equations, Phys. Lett. A 294 (2002) 26–30. [6] M.L. Wang, Y.M. Wang, A new Bäcklund transformation and multi-soliton solutions to the KdV equation with general variable coefficients, Phys. Lett. A 287 (2001) 211–216. [7] M.L. Wang, Solitary wave solution for variant Boussinesq equations, Phys. Lett. A 199 (1995) 169–172. [8] M.L. Wang, Application of homogeneous balance method to exact solutions of nonlinear equation in mathematical physics, Phys. Lett. A 216 (1996) 67– 75. [9] M.L. Wang, Y.B. Zhou, Z.B. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–75. [10] Mohammed Khalfallah, New exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1169–1175. [11] Mohammed Khalfallah, Exact traveling wave solutions of the Boussinesq–Burgers equation, Math. Comput. Model. 49 (2009) 666–671. [12] A.S. Abdel Rady, A.H. Khater, E.S. Osman, Mohammed Khalfallah, New periodic wave and soliton solutions for system of coupled Korteweg–de Vries equations, Appl. Math. Comput. 207 (2009) 406–414. [13] E.G. Fan, Two new applications of the homogeneous balance method, Phys. Lett. A 265 (2000) 353–357. [14] Houde Han, Zhenli Xu, Numerical solitons of generalized Korteweg–de Vries equations, Appl. Math. Comput. 186 (2007) 483–489. [15] X.Q. Zhao, D.B. Tang, A new note on a homogeneous balance method, Phys. Lett. A 297 (2002) 59–67. [16] Shun-dong Zhu, The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2 + 1)-dimensional BoitiLeonPempinelle equation, Chaos Soliton & Fractals 37 (2008) 1335–1342. [17] E.G. Fan, Soliton solutions for a generalized Hirota Satsuma coupled KdV equation and a coupled MKdV equation, Phys. Lett. A 282 (2001) 18–22. [18] C.L. Bai, H. Zhuo, Generalized extended tanh-function method and its application, Chaos Soliton & Fractals 27 (2006) 1026–1035.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The homogeneous balance method and its application to the Benjamin–Bona–Mahoney (BBM) equation A.S. Abdel Rady *, E.S. Osman, Mohammed Khalfallah * Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt
a r t i c l e
i n f o
Keywords: Homogeneous balance method Traveling wave solutions Soliton solutions Benjamin–Bona–Mahoney equation Riccati equation
a b s t r a c t We make use of the homogeneous balance method and symbolic computation to construct new exact traveling wave solutions for the Benjamin–Bona–Mahoney (BBM) equation. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations. Ó 2009 Elsevier Inc. All rights reserved.
Many phenomena in physics and other fields are described by nonlinear evolution equations. When we want to understand the physical mechanism of phenomena in nature, described by nonlinear evolution equations, exact traveling wave solutions have to be explored. The investigation of the exact traveling wave solutions of nonlinear evolutions equations plays an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in fluid dynamics, elastic media, optical fibers, etc. In recent years, the homogeneous balance (HB) method has been widely applied to derive the nonlinear transformation and exact solutions (especially the solitary wave solutions) [1–5] and auto Bäcklund transformations [5,6,13] as well as the similarity reductions [5,13] of nonlinear partial differential equations (PDEs) in mathematical physics. Wang et al. [7–9], Khalfallah [10–12] applied the (HB) method to obtain the new exact traveling wave solutions of a given nonlinear partial differential equations. Fan [13] showed that there is a close connection among the HB method, Wiess, Tabor, Carnevale (WTC) method and Clarkson, Kruskal(CK) method. As the mathematical models of complex physical phenomena, nonlinear evolution equations are involved in many fields from physics to biology, chemistry, engineer, plasma physics, optical fibers and solid state physics etc. Many methods were developed for finding the exact traveling wave solutions of nonlinear evolutions equations, such as Hirota’s method, Backlund and Darboux transformation, Painlevé expansions, Homogeneous balance method, Jacobi elliptic function, Extended tanh-function method, F-expansion method and extended F-expansion method. In this paper, we use the HB method to solve the Riccati equation /0 ¼ a/2 þ b/ þ c and the reduced nonlinear ordinary differential equation for the BBM equation, respectively. It makes the HB method use more extensively. For the regularized long wave or BBM equation [14]
ut þ ux þ uux uxxt ¼ 0;
ð1Þ
which was derived as a model for the unidirectional propagation of long-crested, surface water waves. It arises in other contexts as well, and is generally understood as an alternative to the Korteweg–de Vries equation.
* Corresponding authors. E-mail addresses: [email protected] (A.S. Abdel Rady), [email protected] (M. Khalfallah). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.05.027
1386
A.S. Abdel Rady et al. / Applied Mathematics and Computation 217 (2010) 1385–1390
Long wave in shallow water is a subject of broad interests and has a long colorful history. Physically, it has a rich variety of phenomenological manifestation, especially the existence of wave permanent in form and robust in maintaining their entities through mutual interaction and collision as well as the remarkable property of exhibiting recurrences of initial data when circumstances should prevail. These characteristics are due to the intimate interplay between the roles of nonlinearity and dispersion. Let us consider the traveling wave solutions
uðx; tÞ ¼ uðfÞ;
f ¼ kx þ lt þ d;
ð2Þ
where k; l and d are constants. Substituting (2) into (1), then (1) is reduced to the following nonlinear ordinary differential equation 2
000
0
0
0
k lu ku kuu lu ¼ 0:
ð3Þ
We now seek the solutions of Eq. (3) in the form
u¼
m X
qi /i ;
ð4Þ
i¼0
where qi are constants to be determined later and / satisfy the Riccati equation
/0 ¼ a/2 þ b/ þ c;
ð5Þ
where a; b and c are constants and / satisfy Eq. (5). It is easy to show that m ¼ 2 if balancing u000 with uu0 . Therefore use the ansatz
u ¼ q0 þ q1 / þ q2 /2 ;
ð6Þ i
Substituting Eq. (6) into Eq. (3) along with (5) and collecting all terms with the same power in / ði ¼ 0; 1; 2; 3; 4; 5Þ yields a set of algebraic system for q0 ; q1 ; q2 ; k and l, namely
2 2 cl q1 b þ 6cq2 b þ 2acq1 k þ cð2q0 q1 þ q1 Þk þ clq1 ¼ 0; 2 2 bl q1 b þ 6cq2 b þ 2acq1 k 2cð3abkl q1 Þq1 k þ bð2q0 q1 þ q1 Þk þ blq1 2c 2 k 4 b þ 2ac kl 2q0 1 l q2 ¼ 0; 2 2 al q1 b þ 6cq2 b þ 2acq1 k 2bð3abkl q1 Þq1 k þ að2q0 q1 þ q1 Þk 6c 2 klq1 a2 þ 5bklq2 a q1 q2 k þ alq1 2b k 4 b þ 2ac kl 2q0 1 l q2 ¼ 0; 2 2akð3abkl q1 Þq1 2a k 4 b þ 2ac kl 2q0 1 l q2 4ck 2 6a kl q2 q2 6bk klq1 a2 þ 5bklq2 a q1 q2 ¼ 0; 2 4bk 6a kl q2 q2 6ak klq1 a2 þ 5bklq2 a q1 q2 ¼ 0; 2 4ak 6a kl q2 q2 ¼ 0:
ð7Þ
For which, with the aid of Mathematica, we find 2
q0 ¼
2
2
b lk þ 8aclk k l ; 2k
q1 ¼ 6abkl;
q2 ¼ 6a2 kl:
ð8Þ
It is to be noted that the Riccati equation (5) can be solved using the homogeneous balance method as follows i 0 2 Case: I. Let / ¼ Rm i¼0 bi tanh f. Balancing / with / leads to
/ ¼ b0 þ b1 tanh f:
ð9Þ
Substituting Eq. (9) into (5), we have the following solution of Eq. (5)
/¼
1 ðb þ 2 tanh fÞ; 2a
2
ac ¼
b 1: 4
ð10Þ
Substituting Eqs. (8) and (9) into (6) and (2), we have the following new traveling wave solution of BBM Eq. (1) 2
uðx; tÞ ¼
2 2
8ack l 2b k l k l 2 þ 6kl tanh ðkx þ kt þ dÞ; 2k
This is a bell-shaped solution. i Similarly, let / ¼ Rm i¼0 bi coth f, then we obtain the following new traveling wave soliton solutions of BBM Eq. (1)
ð11Þ
A.S. Abdel Rady et al. / Applied Mathematics and Computation 217 (2010) 1385–1390 2
uðx; tÞ ¼
1387
2 2
8ack l 2b k l k l 2 þ 6kl coth ðkx þ kt þ dÞ: 2k
ð12Þ
Case: II. From [15], when a ¼ 1, b ¼ 0, the Riccati equation (5) has the following solutions
pffiffiffiffiffiffi 8 pffiffiffiffiffiffi > < c tanh cf ; c < 0; c ¼ 0; / ¼ 1f ; > pffiffiffi : pffiffiffi c tan cf : c > 0:
ð13Þ
It is seen that the tanh function in (4) is only a special function in (13), so we conjecture that Eq. (1) may admit other types of traveling wave solutions in Eq. (13) in addition to the tanh-type one. Moreover, we hope to construct them in a unified way. For this purpose, we shall use the Riccati equation (5) once again to generate an associated algebraic system, but not use one of the functions in (13). Another advantage of the Riccati equation (5) is that the sign of c can be used to exactly judge the type of the traveling wave solution for Eq. (1). For example, if c < 0, we are sure that Eq. (1) admits tanh-type and coth-type traveling wave solutions. Especially Eq. (1) will possess three types of traveling wave solutions if c is an arbitrary constant. In this way, we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods. From (6), (8) and (13), we have the following new traveling wave solutions of BBM Eq. (1), which contain traveling wave solutions, periodic wave solutions and rational solutions as follows When c < 0, we have
uðx; tÞ ¼
2 2 2 pffiffiffiffiffiffi pffiffiffiffiffiffi b lk þ 8aclk k l 2 pffiffiffiffiffiffi cðkx þ kt þ dÞ : 6abkl c tanh cðkx þ kt þ dÞ 6a2 klc tanh 2k
ð14Þ
This is a linear combinations of kink wave and bell-shaped wave solutions which is a new solution for (1). When c ¼ 0, we have 2
uðx; tÞ ¼
2
2
b lk þ 8aclk k l 6abkl 6a2 kl þ ; 2k ðkx þ kt þ dÞ ðkx þ kt þ dÞ2
ð15Þ
When c > 0, we have
uðx; tÞ ¼
2 2 2 pffiffiffi pffiffiffi pffiffiffi b lk þ 8aclk k l þ 6abkl c tan cðkx þ kt þ dÞ þ 6a2 klc tan2 cðkx þ kt þ dÞ : 2k
ð16Þ
Which contain a periodic-like solutions. Case: III. We suppose that the Riccati equation (5) has the following solutions of the form
/ ¼ A0 þ
m X ðAi f i þ Bi f i1 gÞ;
ð17Þ
i¼1
with
f ¼
1 ; cosh f þ r
g¼
sinh f ; cosh f þ r
which satisfy
f 0 ðfÞ ¼ f ðfÞgðfÞ;
g 0 ðfÞ ¼ 1 g 2 ðfÞ rf ðfÞ;
g 2 ðfÞ ¼ 1 2rf ðfÞ þ ðr2 1Þf 2 ðfÞ: Balancing /0 with /2 leads to
/ ¼ A0 þ A1 f þ B1 g:
ð18Þ i
j
Substituting Eq. (18) into (5), collecting the coefficient of the same power f ðfÞg ðfÞ ði ¼ 0; 1; 2; j ¼ 0; 1Þ and setting each of the obtained coefficients to zero yield the following set of algebra equations
aA20 þ aB21 þ bA0 þ c ¼ 0; 2aA0 A1 2arB21 rB1 þ bA1 ¼ 0; aA21 þ aðr2 1ÞB21 þ ðr 2 1ÞB1 ¼ 0; 2aA0 B1 þ bB1 ¼ 0; 2aA1 B1 þ A1 ¼ 0;
ð19Þ
which have solutions
b A0 ¼ ; 2a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr 2 1Þ A1 ¼ ; 4a2
B1 ¼
1 ; 2a
2
c¼
b 1 : 4a
ð20Þ
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From Eqs. (18)–(20), we have
/¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! sinh f ðr 2 1Þ 1 bþ 2a cosh f þ r
ð21Þ
From Eqs. (6), (8) and (21), we obtain the new wave solutions of BBM (1)
uðx; tÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 2 2 2 sinhðkx þ kt þ dÞ ðr 2 1Þ 8ack l 2b k l k l : þ 6kl 2k 2ðcoshðkx þ kt þ dÞ þ rÞ
ð22Þ
Case: IV. We take / in the Riccati equation (5) being of the form
/ ¼ ep1 f qðzÞ þ p4 ðfÞ;
ð23Þ
where
z ¼ ep2 f þ p3 ; where p1 ; p2 and p3 are constants to be determined. 2 2 P 1 þb Substituting (23) into (5) we find that when c ¼ 4a , we have
/¼
p1 ep1 f p b : þ 1 2a aðep1 f þ p3 Þ
ð24Þ
If p3 ¼ 1 in (24), we have
/¼
p1 1 b p1 f : tanh 2 2a 2a
ð25Þ
If p3 ¼ 1 in (24), we have
/¼
p1 1 b p1 f : coth 2 2a 2a
ð26Þ
From (6), (8) and (24), we obtain the following new wave solutions of BBM Eq. (1)
uðx; tÞ ¼
2 2 2 8ack l þ b k l k l p b p1 ep1 f p1 þ b p1 ep1 f ; þ 6akl 1 2k 2a 2 aðep1 f þ p3 Þ ep1 f þ p3
ð27Þ
where f ¼ kx þ kt þ d. When p3 ¼ 1, we have from (25)
uðx; tÞ ¼
2 2 2 2 8ack l 2b k l k l 3klp1 2 1 þ p1 ðkx þ kt þ dÞ : tanh 2k 2 2
ð28Þ
When p3 ¼ 1, we have from (29)
uðx; tÞ ¼
2 2 2 2 8ack l 2b k l k l 3klp1 2 1 þ p1 ðkx þ kt þ dÞ : coth 2k 2 2
ð29Þ
Clearly, (11), (12) is the special case of (28), (29) with p1 ¼ 2. Case: V. We suppose that the Riccati equation (5) have the following solutions of the form:
/ ¼ A0 þ
m X
i1
sinh
ðAi sinh x þ Bi cosh xÞ;
i¼1
where dx=df ¼ sinh x or dx=df ¼ cosh x. It is easy to find that m ¼ 1 by balancing /0 and /2 . So we choose
/ ¼ A0 þ A1 sinh x þ B1 cosh x;
ð30Þ i
j
when dx=df ¼ sinh x, we substitute (30) and dx=df ¼ sinh x, into (5) and set the coefficient of sinh xcosh xði ¼ 0; 1; 2; j ¼ 0; 1Þ to zero. A set of algebraic equations is obtained as follows:
aA20 þ aB21 þ bA0 þ c ¼ 0; 2aA0 A1 þ bA1 ¼ 0; aA21 þ aB21 ¼ B1 2aA0 B1 þ bB1 ¼ 0; 2aA1 B1 þ A1 ¼ 0;
ð31Þ
A.S. Abdel Rady et al. / Applied Mathematics and Computation 217 (2010) 1385–1390
1389
for which, we have the following solutions:
A0 ¼
b ; 2a
A1 ¼ 0;
B1 ¼
1 ; a
ð32Þ
2
where c ¼ b 4a4, and
A0 ¼
b ; 2a
rffiffiffiffiffiffi 1 ; 2a
A1 ¼
B1 ¼
1 ; 2a
ð33Þ
2
where c ¼ b 4a1. To dx=df ¼ sinh x, we have
sinh x ¼ csch f;
cosh x ¼ coth f:
ð34Þ
From (33)-(34), we obtain
/¼ where c ¼
b þ 2 coth f ; 2a
b2 4 , 4a
/¼
ð35Þ
and
b csch f þ coth f ; 2a
ð36Þ
2
where c ¼ b 4a1. Clearly (35) is the special case of (26) with p1 ¼ 2. From (5), (6), (8), (35) and (36), we get the new traveling wave solutions of Eq. (1) in the following form 2
uðx; tÞ ¼
2 2
8ack l 2b k l k l 2 þ 6kl coth ðkx þ kt þ dÞ: 2k
ð37Þ
2
where c ¼ b 4a4, and 2
uðx; tÞ ¼
2 2
8ack l 2b k l k l 3kl þ ðcothðkx þ kt þ dÞ cschðkx þ kt þ dÞÞ2 ; 2k 2
ð38Þ
2
where c ¼ b 4a1. Clearly (37) is the special case of (29) with p1 ¼ 2. Similarly, when dx=df ¼ cosh x, we obtain the following exact traveling wave (periodic-like) solutions of BBM equation (1) 2
uðx; tÞ ¼
2 2
8ack l 2b k l k l þ 6klcot2 ðkx þ kt þ dÞ: 2k
ð39Þ
2
where c ¼ b 4a4, and 2
uðx; tÞ ¼
2 2
8ack l 2b k l k l 3kl þ ðcotðkx þ kt þ dÞ cscðkx þ kt þ dÞÞ2 : 2k 2
ð40Þ
HB method provides us a solutions polynomials in two elementary bell-shaped and kink-shaped functions, this covers the large majority of physically interesting solitary waves Eqs. (11), (28), in addition we also obtain a periodic-like wave solutions Eqs. (16), (39), (40) which play an important interesting in physics. Remark. The majority of solutions for (5) in [16], contain solitary wave, periodic wave, rational solutions, ... but in this paper we study the solutions of (5) which contains kink-type, bell-shaped, periodic-like, rational solutions which are well-known solutions and also new solutions for instance in case III, V are obtained, i.e. we recovered by HB method the well-known solutions and adding some new solutions. In summary we have used the extended homogeneous balance method to obtain many traveling wave solutions of BBM equation. We now summarize the key steps as follows: Step 1: For a given nonlinear evolution equation
Fðu; ut ; ux ; uxt ; utt ; . . .Þ ¼ 0;
ð41Þ
we consider its traveling wave solutions uðx; tÞ ¼ uðfÞ; f ¼ kx þ kt þ d then Eq. (41) is reduced to an nonlinear ordinary differential equation
Q ðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0; d . df
ð42Þ
where a prime denotes Step 2: For a given ansatz equation (for example, the ansatz equation is /0 ¼ a/2 þ b/ þ c in this paper), the form of u is decided and the homogeneous balance method is used on Eq. (42) to find the coefficients of u.
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Step 3: The homogeneous balance method is used to solve the ansatz equation. Step 4: Finally, the traveling wave solutions of Eq. (41) are obtained by combining step 2 and 3. From the above procedure, it is easy to find that the homogeneous balance method is more effective and simple than other methods and a lot of solutions can be obtained in the same time. This method can be also applied to other nonlinear evolution equations. In Eq. (5), if we set b ¼ 0, a ¼ l, c ¼ l HB method is reduced to tanh method [17] but HB method is more general than tanh method, extended tanh method [18] because this method is readily applicable to a large variety of nonlinear PDEs in contrast to the tanh method. Some merits are available for HB method. First, all the nonlinear PDEs which can be solved by tanh-function method can be solved easily by HB method.and we have more multiple soliton solutions and triangular periodic solution (including rational solutions). Second, we used only the special solutions of Eq. (1), we can obtain more traveling wave solutions. Third, not only HB method contains the hyperbolic tangent expansion method, but also it is a computerizable method, which allow us to perform complicated and tedious differential calculation on computer. Therefore, the HB method is a generalized tanh-function method for many nonlinear PDEs. References [1] E.G. Fan, H.Q. Zhang, New exact solutions to a system of coupled KdV equations, Phys. Lett. A 245 (1998) 389–392. [2] Anjan Biswas, Swapan Konar, Soliton perturbation theory for the generalized Benjamin–Bona–Mahoney equation, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 703–706. [3] Mariana Antonova, Anjan Biswas, Adiabatic parameter dynamics of perturbed solitary waves, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 734– 748. [4] E.G. Fan, H.Q. Zhang, A note on the homogeneous balance method, Phys. Lett. A 246 (1998) 403–406. [5] E.G. Fan, Auto-Bäcklund transformation and similarity reductions for general variable coefficient KdV equations, Phys. Lett. A 294 (2002) 26–30. [6] M.L. Wang, Y.M. Wang, A new Bäcklund transformation and multi-soliton solutions to the KdV equation with general variable coefficients, Phys. Lett. A 287 (2001) 211–216. [7] M.L. Wang, Solitary wave solution for variant Boussinesq equations, Phys. Lett. A 199 (1995) 169–172. [8] M.L. Wang, Application of homogeneous balance method to exact solutions of nonlinear equation in mathematical physics, Phys. Lett. A 216 (1996) 67– 75. [9] M.L. Wang, Y.B. Zhou, Z.B. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–75. [10] Mohammed Khalfallah, New exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1169–1175. [11] Mohammed Khalfallah, Exact traveling wave solutions of the Boussinesq–Burgers equation, Math. Comput. Model. 49 (2009) 666–671. [12] A.S. Abdel Rady, A.H. Khater, E.S. Osman, Mohammed Khalfallah, New periodic wave and soliton solutions for system of coupled Korteweg–de Vries equations, Appl. Math. Comput. 207 (2009) 406–414. [13] E.G. Fan, Two new applications of the homogeneous balance method, Phys. Lett. A 265 (2000) 353–357. [14] Houde Han, Zhenli Xu, Numerical solitons of generalized Korteweg–de Vries equations, Appl. Math. Comput. 186 (2007) 483–489. [15] X.Q. Zhao, D.B. Tang, A new note on a homogeneous balance method, Phys. Lett. A 297 (2002) 59–67. [16] Shun-dong Zhu, The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2 + 1)-dimensional BoitiLeonPempinelle equation, Chaos Soliton & Fractals 37 (2008) 1335–1342. [17] E.G. Fan, Soliton solutions for a generalized Hirota Satsuma coupled KdV equation and a coupled MKdV equation, Phys. Lett. A 282 (2001) 18–22. [18] C.L. Bai, H. Zhuo, Generalized extended tanh-function method and its application, Chaos Soliton & Fractals 27 (2006) 1026–1035.