New Analytical Solutions for Two Equations by a Proposed Sub
Transcript Of New Analytical Solutions for Two Equations by a Proposed Sub
2012 International Conference on Computer Technology and Science (ICCTS 2012) IPCSIT vol. 47 (2012) © (2012) IACSIT Press, Singapore DOI: 10.7763/IPCSIT.2012.V47.67
New Analytical Solutions for Two Equations by a Proposed Sub-ODE Method
Bin Zheng+
School of Science, Shandong University of Technology, Zhangzhou Road 12, Zibo, Shandong, China, 255049
Abstract. In this paper, we derive exact traveling wave solutions of mBBM equation and 2D-Burgers
equation by a proposed Bernoulli sub-ODE method. The method appears to be efficient in seeking exact solutions of nonlinear equations. Keywords: Bernoulli sub-ODE method, traveling wave solutions, exact solution, evolution equation, mBBM equation, 2D-Burgers equation
1. Introduction
Recently searching for exact traveling wave solutions of nonlinear equations has gained more and more popularity, and many effective methods have been presented so far. Some of these approaches are the homogeneous balance method [1,2], the hyperbolic tangent expansion method [3,4], the trial function method [5], the tanh-method [6-8], the nonlinear transform method [9], the inverse scattering transform [10], the Backlund transform [11,12], the Hirotas bilinear method [13,14], the generalized Riccati equation [15,16], the theta function method [17-19], the sine-Ccosine method [20], the Jacobi elliptic function expansion [21,22], the complex hyperbolic function method [23-25], and so on.
In this paper, we propose a Bernoulli sub-ODE method to construct exact traveling wave solutionns for NLEES, and try to apply the method to obtain the traveling wave solutions for the mBBM equation and the 2D-Burgers equation.
2. Description of the Bernoulli Sub-ODE method
In this section we present the solutions of the following ODE:
G '+ λG = μG2 ,
(2.1)
where λ ≠ 0,G = G(ξ )
When μ ≠ 0 , Eq. (2.1) is the type of Bernoulli equation, and we can obtain the solution as
G= 1 ,
(2.2)
μ + deλξ
λ
where d is an arbitrary constant.
Suppose that a nonlinear equation, say in two or three independent variables x, y and t , is given by
P(u, ut,ux , uy , utt , uxt , uyt , uxx , uyy ......) = 0
(2.3)
where u = u(x, y, t) is an unknown function, P is a polynomial in u = u(x, y, t) and its various partial derivatives,
+ Corresponding author. Tel.: + 86-13561602410 E-mail address: [email protected]
360
in which the highest order derivatives and nonlinear terms are involved. By using the solutions of Eq. (2.1), we can construct a serials of exact solutions of nonlinear equations:.
Step 1.We suppose that
u(x, y,t) = u(ξ ),ξ = ξ (x, y,t)
(2.4)
the traveling wave variable (2.4) permits us reducing Eq. (2.3) to an ODE for u = u(ξ )
P(u,u ',u '',......) = 0
(2.5)
Step 2. Suppose that the solution of (2.5) can be expressed by a polynomial in G as follows:
u(ξ
)
=
αmGm
+
α Gm−1 m−1
+
......
(2.6)
where G = G(ξ ) satisfies Eq. (2.1), and αm ,αm−1... are constants to be determined later, αm ≠ 0 . The
positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (2.5).
Step 3. Substituting (2.6) into (2.5) and using (2.1), collecting all terms with the same order of G together, the left-hand side of Eq. (2.5) is converted into another polynomial in G . Equating each coefficient of this
polynomial to zero, yields a set of algebraic equations for αm ,αm−1,...λ, μ .
Step 4. Solving the algebraic equations system in Step 3, and by using the solutions of Eq. (2.1), we can
construct the traveling wave solutions of the nonlinear evolution equation (2.5).
In the subsequent sections we will illustrate the proposed method in detail by applying it to mBBM equation and the 2D-Burgers equation.
3. Application Of Bernoulli Sub-ODE Method For mBBM Equation
In this section, we will consider the following mBBM equation:
ut + ux + u2ux + uxxt = 0
(3.1)
In order to obtain the traveling wave solutions of Eq.(3.1), we suppose that
u(x,t) = u(ξ ),ξ = x − ct
(3.2)
where c is a constant that to be determined later. By using (3.2), (3.1) is converted into an ODE
(1− c)u '+ u2u '− cu ''' = 0 (3.3)
Integrating (3.3) once it follows: (1− c)u + u3 − cu '' = g 3 (3.4)
where g is the integration constant. Suppose that the solution of (3.4) can be expressed by a polynomial in G as follows:
∑ m
u(ξ ) = aiGi
i=0
(3.5)
where ai are constants, and G . satisfies Eq. (2.2). Balancing the order of u '' and u3 in Eq. (3.3), we have
m + 2 = 3m ⇒ m = 1 . So
u(ξ ) = a0 + a1G
(3.6)
where a1, a0 are constants to be determined later.
Substituting (3.6) into (3.4) and collecting all the terms with the same power of G together and equating each coefficient to zero, yields a set of simultaneous algebraic equations as follows:
G0 : a0 − ca0 − g + 13 a03 = 0 G1 : a1 + a02a1 − ca1λ 2 − ca1 = 0
361
G2 : 3ca1μλ + a0a12 = 0
G3 : −2ca μ 2 + 1 a 3 = 0
1
31
Solving the algebraic equations above, yields:
Case 1:
a = 2 − 3 μ,a = − − 3 λ,c = − 2 , g = 0
1
λ2 −2 0
λ2 −2
λ2 −2
Substituting (3.7) into (3.6), we have
u (ξ ) = − − 3 λ + 2 − 3 μG , ξ = x + 2 t
1
λ2 −2
λ2 −2
λ2 −2
Combining with Eq. (2.2) and(3.8), we can obtain the traveling wave solutions of (3.1) as follows:
u (x,t) = − − 3 λ + 2 − 3 μ
1
1
λ2 −2
λ 2 − 2 μ + deλ (x+ λ22−2t)
λ
where d is an arbitrary constant.
Case 2:
a = −2 − 3 μ, a = − 3 λ, c = − 2 , g = 0
1
λ2 −2 0
λ2 −2
λ2 −2
Substituting (3.7) into (3.6), we have
u2 (ξ ) = − λ 23− 2λ − 2 − λ23− 2 μG , ξ = x + λ 22− 2 t
Combining with Eq. (2.2) and(3.8), we can obtain the traveling wave solutions of (3.1) as follows:
u (x,t) = − 3 λ − 2 − 3 μ(
1
)
2
λ2 −2
λ 2 − 2 μ + deλ (x+ λ22−2t)
λ
where d is an arbitrary constant.
Remark : Our results (3.9) and (3.12) are new exact traveling wave solutions for Eq. (3.1).
4. Application Of Bernoulli Sub-ODE Method For 2d Burgers Equation
In this section, we will consider the following 2D Burgers equation:
ut − 2uux − uxx − uyy − 2vuy = 0
vt − 2uvx − vxx − vyy − 2vvy = 0
In order to obtain the traveling wave solutions of Eq.(4.1), we suppose that
u(x,t) = u(ξ ),ξ = kx + ω y + st
where k,ω, s are constants that to be determined later.
By using (4.3), (4.1) and (4.2) is converted into ODEs
su '− 2kuu '− (k 2 + ω2 )u ''− 2ωvu ' = 0 sv '− 2kuv '− (k 2 + ω2 )v ''− 2ωvv ' = 0
Suppose that the solution of (4.4) and (4.5) can be expressed by a polynomial in G as follows:
∑ m
u(ξ ) = aiGi
i=0
(3.7) (3.8) (3.9)
(3.10) (3.11) (3.12)
(4.1) (4.2) (4.3) (4.4) (4.5) (4.6)
362
∑ n
v(ξ ) = biGi
i=0
(4.7)
where ai , bi are constants, G . satisfies Eq. (2.2). Balancing the order of uu ' and vu ' in Eq. (4.6), the order of uv ' and v '' in Eq. (4.7), we have 2m +1 = m + n +1, m + n +1 = n + 2 ⇒ m = n = 1. So
u(ξ ) = a0 + a1G
(4.8)
v(ξ ) = b0 + b1G
(4.9)
where a1, a0 , b1, b0 are constants to be determined later.
Substituting (4.8), (4.9) into (4.4) , (4.5), and collecting all the terms with the same power of G together
and equating each coefficient to zero, yields a set of simultaneous algebraic equations as follows:
For Eq. (4.4):
G1 : sa1μ − a1ω 2μ 2 − 2ka1a0μ − 2ωa1b0μ − a1k 2μ 2 = 0 G2 : 2ka1a0λ − sa1λ + 3a1k 2μλ − 2ka12μ + 2ωa1b0λ +3a1ω2μλ − 2ωa1b1μ = 0 G 3 : −2a1ω 2λ 2 − 2a1k 2λ 2 + 2a12kλ + 2ωa1b1λ = 0 For Eq. (4.5):
G1 : sb1μ − b1(k 2 + ω2 )μ 2 − 2kb1a0μ − 2ωb1b0μ = 0 G2 : 2kb1a0λ − sb1λ + 3b1(k 2 + ω2 )μλ − 2kb1a1μ +2ωb1b0λ − 2ωb12μ = 0 G 3 : −2b1(k 2 + ω 2 )λ 2 + 2kb1a1λ + 2ωb12λ = 0
Solving the algebraic equations above, yields:
a = ω2λ + k 2λ − ωb1 , a = a ,b = b ,b = b , ω = ω, k = k, s = ω2μ + 2ka0 + 2ωb0 + k 2μ
1
k
0
00
01 1
Substituting (4.10) into (4.8) and (4.9), we have
u(ξ ) = a0 + ω2λ + kk2λ − ωb1 G
(4.10) (4.11)
v(ξ ) = b0 + b1G
(4.12)
where k,ω, a0 ,b0 ,b1 are arbitrary constants, and k,ω,b1 ≠ 0 , ξ = kx + ω y + (ω2μ + 2ka0 + 2ωb0 + k 2μ)t . Combining with Eq. (2.2), we can obtain the traveling wave solutions of (4.1) and (4.2) as follows:
u(x, y,t) = a + ω2λ + k2λ −ωb1 ( 1 )
0
k
μ + deλξ
λ
v(x, y,t) = b + b ( 1 )
0 1 μ + deλξ
λ
where ξ = kx + ω y + (ω2μ + 2ka0 + 2ωb0 + k 2μ)t
Remark: Our result (4.13) and (4.14) are new exact traveling wave solutions for Eqs. (3.1)-(3.2).
(4.13) (4.14)
5. Conclusions
We have seen that some new traveling wave solution of mBBM equation is successfully found by using the Bernoulli sub-ODE method. One can see the method is concise and effective. Also this method can be used to many other nonlinear problems.
6. References
[1] M. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A 199 (1995) 169-172.
363
[2] E.M.E. Zayed, H.A. Zedan, K.A. Gepreel, On the solitary wave solutions for nonlinear Hirota-Satsuma coupled KdV equations, Chaos, Solitons and Fractals 22 (2004) 285-303.
[3] L. Yang, J. Liu, K. Yang, Exact solutions of nonlinear PDE nonlinear transformations and reduction of nonlinear PDE to a quadrature, Phys. Lett. A 278 (2001) 267-270.
[4] E.M.E. Zayed, H.A. Zedan, K.A. Gepreel, Group analysis. and modified tanh-function to find the invariant solutions and soliton solution for nonlinear Euler equations, Int. J. Nonlinear Sci. Numer. Simul. 5 (2004) 221-234
[5] M. Inc, D.J. Evans, On traveling wave solutions of some nonlinear evolution equations, Int. J. Comput. Math. 81 (2004) 191-202
[6] M.A. Abdou, The extended tanh-method and its applications for solving nonlinear physical models, Appl. Math. Comput. 190 (2007) 988-996.
[7] E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212-218.
[8] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992) 650-654. [9] J.L. Hu, A new method of exact traveling wave solution for coupled nonlinear differential equations, Phys. Lett. A
322 (2004) 211-216. [10] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform,
Cambridge University Press, Cambridge, 1991. [11] M.R. Miura, Backlund Transformation, Springer-Verlag, Berlin, 1978. [12] C. Rogers, W.F. Shadwick, Backlund Transformations, Academic Press, New York, 1982. [13] R. Hirota, Exact envelope soliton solutions of a nonlinear wave equation, J. Math. Phys. 14 (1973) 805-810. [14] R. Hirota, J. Satsuma, Soliton solution of a coupled KdV equation, Phys. Lett. A 85 (1981) 407-408. [15] Z.Y. Yan, H.Q. Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup
equation in shallow water, Phys. Lett. A 285 (2001) 355-362. [16] A.V. Porubov, Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid
layer, Phys. Lett. A 221 (1996) 391-394 [17] E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000)
212-218. [18] E.G. Fan, Multiple traveling wave solutions of nonlinear evolution equations using a unifiex algebraic method, J.
Phys. A, Math. Gen. 35 (2002) 6853-6872. [19] Z.Y. Yan, H.Q. Zhang, New explicit and exact traveling wave solutions for a system of variant Boussinesq
equations in mathematical physics, Phys. Lett. A 252 (1999) 291-296. [20] S.K. Liu, Z.T. Fu, S.D. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of
nonlinear wave equations, Phys. Lett. A 289 (2001) 69-74. [21] Z. Yan, Abundant families of Jacobi elliptic functions of the (2+1)-dimensional integrable Davey-Stawartson-type
equation via a new method, Chaos, Solitons and Fractals 18 (2003) 299-309. [22] C. Bai, H. Zhao, Complex hyperbolic-function method and its applications to nonlinear equations, Phys. Lett. A 355
(2006) 22-30.. [23] E.M.E. Zayed, A.M. Abourabia, K.A. Gepreel, M.M. Horbaty, On the rational solitary wave solutions for the
nonlinear Hirota-Satsuma coupled KdV system, Appl. Anal. 85 (2006) 751- 768. [24] K.W. Chow, A class of exact periodic solutions of nonlinear envelope equation, J. Math. Phys. 36 (1995)
4125-4137. [25] M.L.Wang, Y.B. Zhou, The periodic wave equations for the Klein-Gordon-Schordinger equations, Phys. Lett. A
318 (2003) 84-92.
364
New Analytical Solutions for Two Equations by a Proposed Sub-ODE Method
Bin Zheng+
School of Science, Shandong University of Technology, Zhangzhou Road 12, Zibo, Shandong, China, 255049
Abstract. In this paper, we derive exact traveling wave solutions of mBBM equation and 2D-Burgers
equation by a proposed Bernoulli sub-ODE method. The method appears to be efficient in seeking exact solutions of nonlinear equations. Keywords: Bernoulli sub-ODE method, traveling wave solutions, exact solution, evolution equation, mBBM equation, 2D-Burgers equation
1. Introduction
Recently searching for exact traveling wave solutions of nonlinear equations has gained more and more popularity, and many effective methods have been presented so far. Some of these approaches are the homogeneous balance method [1,2], the hyperbolic tangent expansion method [3,4], the trial function method [5], the tanh-method [6-8], the nonlinear transform method [9], the inverse scattering transform [10], the Backlund transform [11,12], the Hirotas bilinear method [13,14], the generalized Riccati equation [15,16], the theta function method [17-19], the sine-Ccosine method [20], the Jacobi elliptic function expansion [21,22], the complex hyperbolic function method [23-25], and so on.
In this paper, we propose a Bernoulli sub-ODE method to construct exact traveling wave solutionns for NLEES, and try to apply the method to obtain the traveling wave solutions for the mBBM equation and the 2D-Burgers equation.
2. Description of the Bernoulli Sub-ODE method
In this section we present the solutions of the following ODE:
G '+ λG = μG2 ,
(2.1)
where λ ≠ 0,G = G(ξ )
When μ ≠ 0 , Eq. (2.1) is the type of Bernoulli equation, and we can obtain the solution as
G= 1 ,
(2.2)
μ + deλξ
λ
where d is an arbitrary constant.
Suppose that a nonlinear equation, say in two or three independent variables x, y and t , is given by
P(u, ut,ux , uy , utt , uxt , uyt , uxx , uyy ......) = 0
(2.3)
where u = u(x, y, t) is an unknown function, P is a polynomial in u = u(x, y, t) and its various partial derivatives,
+ Corresponding author. Tel.: + 86-13561602410 E-mail address: [email protected]
360
in which the highest order derivatives and nonlinear terms are involved. By using the solutions of Eq. (2.1), we can construct a serials of exact solutions of nonlinear equations:.
Step 1.We suppose that
u(x, y,t) = u(ξ ),ξ = ξ (x, y,t)
(2.4)
the traveling wave variable (2.4) permits us reducing Eq. (2.3) to an ODE for u = u(ξ )
P(u,u ',u '',......) = 0
(2.5)
Step 2. Suppose that the solution of (2.5) can be expressed by a polynomial in G as follows:
u(ξ
)
=
αmGm
+
α Gm−1 m−1
+
......
(2.6)
where G = G(ξ ) satisfies Eq. (2.1), and αm ,αm−1... are constants to be determined later, αm ≠ 0 . The
positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (2.5).
Step 3. Substituting (2.6) into (2.5) and using (2.1), collecting all terms with the same order of G together, the left-hand side of Eq. (2.5) is converted into another polynomial in G . Equating each coefficient of this
polynomial to zero, yields a set of algebraic equations for αm ,αm−1,...λ, μ .
Step 4. Solving the algebraic equations system in Step 3, and by using the solutions of Eq. (2.1), we can
construct the traveling wave solutions of the nonlinear evolution equation (2.5).
In the subsequent sections we will illustrate the proposed method in detail by applying it to mBBM equation and the 2D-Burgers equation.
3. Application Of Bernoulli Sub-ODE Method For mBBM Equation
In this section, we will consider the following mBBM equation:
ut + ux + u2ux + uxxt = 0
(3.1)
In order to obtain the traveling wave solutions of Eq.(3.1), we suppose that
u(x,t) = u(ξ ),ξ = x − ct
(3.2)
where c is a constant that to be determined later. By using (3.2), (3.1) is converted into an ODE
(1− c)u '+ u2u '− cu ''' = 0 (3.3)
Integrating (3.3) once it follows: (1− c)u + u3 − cu '' = g 3 (3.4)
where g is the integration constant. Suppose that the solution of (3.4) can be expressed by a polynomial in G as follows:
∑ m
u(ξ ) = aiGi
i=0
(3.5)
where ai are constants, and G . satisfies Eq. (2.2). Balancing the order of u '' and u3 in Eq. (3.3), we have
m + 2 = 3m ⇒ m = 1 . So
u(ξ ) = a0 + a1G
(3.6)
where a1, a0 are constants to be determined later.
Substituting (3.6) into (3.4) and collecting all the terms with the same power of G together and equating each coefficient to zero, yields a set of simultaneous algebraic equations as follows:
G0 : a0 − ca0 − g + 13 a03 = 0 G1 : a1 + a02a1 − ca1λ 2 − ca1 = 0
361
G2 : 3ca1μλ + a0a12 = 0
G3 : −2ca μ 2 + 1 a 3 = 0
1
31
Solving the algebraic equations above, yields:
Case 1:
a = 2 − 3 μ,a = − − 3 λ,c = − 2 , g = 0
1
λ2 −2 0
λ2 −2
λ2 −2
Substituting (3.7) into (3.6), we have
u (ξ ) = − − 3 λ + 2 − 3 μG , ξ = x + 2 t
1
λ2 −2
λ2 −2
λ2 −2
Combining with Eq. (2.2) and(3.8), we can obtain the traveling wave solutions of (3.1) as follows:
u (x,t) = − − 3 λ + 2 − 3 μ
1
1
λ2 −2
λ 2 − 2 μ + deλ (x+ λ22−2t)
λ
where d is an arbitrary constant.
Case 2:
a = −2 − 3 μ, a = − 3 λ, c = − 2 , g = 0
1
λ2 −2 0
λ2 −2
λ2 −2
Substituting (3.7) into (3.6), we have
u2 (ξ ) = − λ 23− 2λ − 2 − λ23− 2 μG , ξ = x + λ 22− 2 t
Combining with Eq. (2.2) and(3.8), we can obtain the traveling wave solutions of (3.1) as follows:
u (x,t) = − 3 λ − 2 − 3 μ(
1
)
2
λ2 −2
λ 2 − 2 μ + deλ (x+ λ22−2t)
λ
where d is an arbitrary constant.
Remark : Our results (3.9) and (3.12) are new exact traveling wave solutions for Eq. (3.1).
4. Application Of Bernoulli Sub-ODE Method For 2d Burgers Equation
In this section, we will consider the following 2D Burgers equation:
ut − 2uux − uxx − uyy − 2vuy = 0
vt − 2uvx − vxx − vyy − 2vvy = 0
In order to obtain the traveling wave solutions of Eq.(4.1), we suppose that
u(x,t) = u(ξ ),ξ = kx + ω y + st
where k,ω, s are constants that to be determined later.
By using (4.3), (4.1) and (4.2) is converted into ODEs
su '− 2kuu '− (k 2 + ω2 )u ''− 2ωvu ' = 0 sv '− 2kuv '− (k 2 + ω2 )v ''− 2ωvv ' = 0
Suppose that the solution of (4.4) and (4.5) can be expressed by a polynomial in G as follows:
∑ m
u(ξ ) = aiGi
i=0
(3.7) (3.8) (3.9)
(3.10) (3.11) (3.12)
(4.1) (4.2) (4.3) (4.4) (4.5) (4.6)
362
∑ n
v(ξ ) = biGi
i=0
(4.7)
where ai , bi are constants, G . satisfies Eq. (2.2). Balancing the order of uu ' and vu ' in Eq. (4.6), the order of uv ' and v '' in Eq. (4.7), we have 2m +1 = m + n +1, m + n +1 = n + 2 ⇒ m = n = 1. So
u(ξ ) = a0 + a1G
(4.8)
v(ξ ) = b0 + b1G
(4.9)
where a1, a0 , b1, b0 are constants to be determined later.
Substituting (4.8), (4.9) into (4.4) , (4.5), and collecting all the terms with the same power of G together
and equating each coefficient to zero, yields a set of simultaneous algebraic equations as follows:
For Eq. (4.4):
G1 : sa1μ − a1ω 2μ 2 − 2ka1a0μ − 2ωa1b0μ − a1k 2μ 2 = 0 G2 : 2ka1a0λ − sa1λ + 3a1k 2μλ − 2ka12μ + 2ωa1b0λ +3a1ω2μλ − 2ωa1b1μ = 0 G 3 : −2a1ω 2λ 2 − 2a1k 2λ 2 + 2a12kλ + 2ωa1b1λ = 0 For Eq. (4.5):
G1 : sb1μ − b1(k 2 + ω2 )μ 2 − 2kb1a0μ − 2ωb1b0μ = 0 G2 : 2kb1a0λ − sb1λ + 3b1(k 2 + ω2 )μλ − 2kb1a1μ +2ωb1b0λ − 2ωb12μ = 0 G 3 : −2b1(k 2 + ω 2 )λ 2 + 2kb1a1λ + 2ωb12λ = 0
Solving the algebraic equations above, yields:
a = ω2λ + k 2λ − ωb1 , a = a ,b = b ,b = b , ω = ω, k = k, s = ω2μ + 2ka0 + 2ωb0 + k 2μ
1
k
0
00
01 1
Substituting (4.10) into (4.8) and (4.9), we have
u(ξ ) = a0 + ω2λ + kk2λ − ωb1 G
(4.10) (4.11)
v(ξ ) = b0 + b1G
(4.12)
where k,ω, a0 ,b0 ,b1 are arbitrary constants, and k,ω,b1 ≠ 0 , ξ = kx + ω y + (ω2μ + 2ka0 + 2ωb0 + k 2μ)t . Combining with Eq. (2.2), we can obtain the traveling wave solutions of (4.1) and (4.2) as follows:
u(x, y,t) = a + ω2λ + k2λ −ωb1 ( 1 )
0
k
μ + deλξ
λ
v(x, y,t) = b + b ( 1 )
0 1 μ + deλξ
λ
where ξ = kx + ω y + (ω2μ + 2ka0 + 2ωb0 + k 2μ)t
Remark: Our result (4.13) and (4.14) are new exact traveling wave solutions for Eqs. (3.1)-(3.2).
(4.13) (4.14)
5. Conclusions
We have seen that some new traveling wave solution of mBBM equation is successfully found by using the Bernoulli sub-ODE method. One can see the method is concise and effective. Also this method can be used to many other nonlinear problems.
6. References
[1] M. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A 199 (1995) 169-172.
363
[2] E.M.E. Zayed, H.A. Zedan, K.A. Gepreel, On the solitary wave solutions for nonlinear Hirota-Satsuma coupled KdV equations, Chaos, Solitons and Fractals 22 (2004) 285-303.
[3] L. Yang, J. Liu, K. Yang, Exact solutions of nonlinear PDE nonlinear transformations and reduction of nonlinear PDE to a quadrature, Phys. Lett. A 278 (2001) 267-270.
[4] E.M.E. Zayed, H.A. Zedan, K.A. Gepreel, Group analysis. and modified tanh-function to find the invariant solutions and soliton solution for nonlinear Euler equations, Int. J. Nonlinear Sci. Numer. Simul. 5 (2004) 221-234
[5] M. Inc, D.J. Evans, On traveling wave solutions of some nonlinear evolution equations, Int. J. Comput. Math. 81 (2004) 191-202
[6] M.A. Abdou, The extended tanh-method and its applications for solving nonlinear physical models, Appl. Math. Comput. 190 (2007) 988-996.
[7] E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212-218.
[8] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992) 650-654. [9] J.L. Hu, A new method of exact traveling wave solution for coupled nonlinear differential equations, Phys. Lett. A
322 (2004) 211-216. [10] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform,
Cambridge University Press, Cambridge, 1991. [11] M.R. Miura, Backlund Transformation, Springer-Verlag, Berlin, 1978. [12] C. Rogers, W.F. Shadwick, Backlund Transformations, Academic Press, New York, 1982. [13] R. Hirota, Exact envelope soliton solutions of a nonlinear wave equation, J. Math. Phys. 14 (1973) 805-810. [14] R. Hirota, J. Satsuma, Soliton solution of a coupled KdV equation, Phys. Lett. A 85 (1981) 407-408. [15] Z.Y. Yan, H.Q. Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup
equation in shallow water, Phys. Lett. A 285 (2001) 355-362. [16] A.V. Porubov, Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid
layer, Phys. Lett. A 221 (1996) 391-394 [17] E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000)
212-218. [18] E.G. Fan, Multiple traveling wave solutions of nonlinear evolution equations using a unifiex algebraic method, J.
Phys. A, Math. Gen. 35 (2002) 6853-6872. [19] Z.Y. Yan, H.Q. Zhang, New explicit and exact traveling wave solutions for a system of variant Boussinesq
equations in mathematical physics, Phys. Lett. A 252 (1999) 291-296. [20] S.K. Liu, Z.T. Fu, S.D. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of
nonlinear wave equations, Phys. Lett. A 289 (2001) 69-74. [21] Z. Yan, Abundant families of Jacobi elliptic functions of the (2+1)-dimensional integrable Davey-Stawartson-type
equation via a new method, Chaos, Solitons and Fractals 18 (2003) 299-309. [22] C. Bai, H. Zhao, Complex hyperbolic-function method and its applications to nonlinear equations, Phys. Lett. A 355
(2006) 22-30.. [23] E.M.E. Zayed, A.M. Abourabia, K.A. Gepreel, M.M. Horbaty, On the rational solitary wave solutions for the
nonlinear Hirota-Satsuma coupled KdV system, Appl. Anal. 85 (2006) 751- 768. [24] K.W. Chow, A class of exact periodic solutions of nonlinear envelope equation, J. Math. Phys. 36 (1995)
4125-4137. [25] M.L.Wang, Y.B. Zhou, The periodic wave equations for the Klein-Gordon-Schordinger equations, Phys. Lett. A
318 (2003) 84-92.
364