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Abstract
In this paper, we apply thefirst integral method to generalized ZK-BBM equation and Drinefel’d-SokolovWilson system and one-dimensional modified EW-Burgers equation.
Thefirst integral method is a powerful solution method for obtaining exact solutions of some nonlinear evolution equations. This method wasfirst proposed by Feng [8] in solving Burgers–KdV equation which is based on the ring theory of commutative algebra. This method can be applied to nonintegrable equations as well as to integrable ones.
Key words
Firstintegralmethod;GeneralizedZK-BBMequation; Drinefel’d-Sokolov-Wilsonsystem; One-dimensional modified EW-Burgers equation
1.INTRODUCTION
Nonlinear evolution equations are widely used to describe complex phenomena in various sciences such asfluid physics, condensed matter, biophysics, plasma physics, nonlinear optics, quantumfield theory and particle physics, etc. In recent decades, several powerful methods have been proposed to construct exact solutionsfornonlinearevolutionequations,suchas tanhmethod[1-3],extendedtanhmethod[4,5],multiple exp-functionmethod [6], transformed rational function method [7] and so on.
Inthepioneerwork,Feng[8]introducedthefirst integralmethodforareliabletreatmentofthenonlinear PDEs. Theusefulfirstintegralmethodiswidelyusedbymanysuchasin[9-13]andbythereferencetherein.
Raslan [10] proposed thefirst integral method to solve the Fisher equation. Taghizadeh et al., [11] solved nonlinear Schr¨odinger equation by using thefirst integral method. Tascan et al., [12] used thefirst integral method to obtain the exact solutions of the modified Zakharov–Kuznetsovequation and ZK–MEW equation. Hosseini et al., [13] applied thefirst integral method to obtain the exact solutions of KdV system and Kaup–Boussinesq system and Wu–Zhang system.
The aim of this paper is tofind exact soliton solutions of generalizedZK-BBM equation and Drinefel’dSokolov-Wilsonsystemandone-dimensionalgeneralizedEW-Burgersequation.bythefirst integralmethod.
The paper is arranged as follows. In Section 2, we describe briefly thefirst integral method. In Sections 3 - 5 , we apply this method to generalized ZK-BBM equation and Drinefel’d-Sokolov-Wilsonsystem and one-dimensional modified EW-Burgers equation.
2.FIRST INTEGRAL METHOD
Raslan summarized for usingfirst integral method [10]. Step 1.Consider a general nonlinear PDE in the form
F(u,ux,uy,ut,uxx,uxy,uxt,...) = 0,(1)
Using a wave variable whereξ= k(x + ly?λt), we can rewrite Eq. (1) in the following nonlinear ODE
G(u,u′,u′′,u′′′,...) = 0,(2)
where the prime denotes the derivation with respect toξ.
Step 2.Suppose that the solution of ODE (2) can be written as follows: u(x,y,t) = u(ξ) = f(ξ).(3) Step 3.We introduce a new independent variable X(ξ) = f(ξ),Y =
Step 4.By the qualitative theory of ordinary differential equations [14] , if we canfind the integrals to Eq. (5) under the same conditions, then the general solutions to Eq. (5) can be solved directly. However, in general, it is really difficult for us to realize this even for onefirst integral, because for a given plane autonomous system, there is no systematic theory that can tell us how tofind itsfirst integrals, nor is there a logical way for telling us what thesefirst integrals are. We will apply the Division Theorem to obtain one first integral to Eq.(5) which reduces Eq.(2) to afirst order integrable ordinary differential equation. An exact solution to Eq. (1) is then obtained by solving this equation. Now, let us recall the Division Theorem: Division Theorem.Supposethat P(w,z) and Q(w,z) are polynomialsinC[w,z], and P(w,z) is irreducible in C[w,z]. If Q(w,z)vanishes at all zero points of P(w,z), then there exists a polynomial G(w,z) in C[w,z] such that
Q(w,z) = P(w,z)G(w,z).
3.GENERALIZED ZK-BBM EQUATION
Consider the generalized ZK-BBM equation [15]
ut+ ux+ a(u3)x+ b(uxt+ uyy)x= 0,(6)
where a,b are real constants.
By make the transformation
u(x,y,t) = f(ξ),ξ= k(x + ly?λt),(7)
the generalized ZK-BBM equation becomes(1?λ)f′(ξ) + 3af2(ξ)f′(ξ) + bk2(l2?λ)f′′′(ξ) = 0.(8)
2A1X2,(16) where A0is arbitraryintegrationconstant. Substituting a0(X),a1(X) and g(X)in the last equationin (15) and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain B0= 0,A0=?2a + A21bk2(l2?1)
bk2(l2?λ))X4,(24) where d is arbitrary integrationconstant. Substituting a0(X),a1(X) and g(X), in the last equation in (22) and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it with aid Maple, we obtain B0= 0,A0=?8a + A21bk2(l2?1)
4.DRINEFEL’D-SOKOLOV-WILSON SYSTEM
Consider the Drinefel’d-Sokolov-Wilsonsystem
ut+ pvvx= 0,
vt+ qvxxx+ ruvx+ suxv = 0,
X2(ξ)?A0.(42) Combining (42) with (35), we obtain the exact solution to equation (34) and then the exact solution to Drinefel’d-Sokolov-Wilsonsystem can be written as
5.ONE-DIMENSIONAL MODIFIED EW-BURGERS EQUATION
Let us consider one-dimensional modified EW-Burgers equation [16]
ut+ au2ux?δuxx?μuxxt= 0,(44)
where a,δ,μare real constants.
We use the wave transformation
u(x,t) = f(ξ),ξ= x?ct.(45)
Substituting (45) into (44), we obtain ordinary differential equation:?cf′(ξ) + af2(ξ)f′(ξ)?δf′′(ξ) +μcf′′′(ξ) = 0.(46) Integrating Eq. (46) with respect toξ, then we have
?cf(ξ) +a
Case B:Suppose that m = 2, by equating with the coefficients of Yi(i = 3,2,1,0) of both sides of (12), we have
˙a2(X) = h(X)a2(X),(55)˙a1(X) = (?2δ
6.CONCLUSION
In this paper, thefirst integral method is applied successfully for solving generalized ZK-BBM equation and Drinefel’d-Sokolov-Wilson system and one-dimensional modified EW-Burgers equation. The results show that this method is efficient infinding the exact solutions of nonlinear differential equations.
REFERENCES
[1]Malfliet, W. (1992). Solitary Wave Solutions of Nonlinear Wave Equations. Am. J. Phys, 60(7), 650-654.
[2]Malfliet, W., Hereman, W. (1996). The Tanh Method: I. Exact Solutions of Nonlinear Evolution and Wave Equations. Phys. Scripta, 54, 563-568.
[3]Malfliet, W., Hereman, W. (1996). The Tanh Method: II. Perturbation Technique for Conservative Systems. Phys. Scripta, 54, 569-575.
[4]Ma, W. X., Fuchssteiner, B. (1996). Explicit and Exact Solutions to a Kolmogorov–Petrovskii–Piskunov Equation. Internat. J. Non-Linear Mech 31, 329–338.
[5]Fan, E. (2000). Extended Tanh-function Method and Its Applications to Nonlinear Equations, Phys. Lett. A., 277(4-5), 212-218.
[6]Ma, W.X.Huang,T.W., Zhang,Y.(2010).AMultipleExp-functionMethodforNonlinearDifferential Equations and Its Application. Phys. Scr., 82, 065003.
[7]Ma, W. X., Lee, J.-H. (2009). A Transformed Rational Function Method and Exact Solutions to the(3+1)-dimensional Jimbo-Miwa Equation. Chaos Solitons Fract., 42, 1356-1363.
[8]Feng, Z. S. (2002). The First Integral Method to Study the Burgers-Korteweg-de Vries Equation. J.Phys. A., 35(2), 343-349.
[9]Feng, Z. S., Wang, X. H. (2002) . The First Integral Method to the Two-dimensional Burgers-KdV Equation. Phys. Lett. A., 308, 173-178.
[10] Raslan, K. R. (2008) . The First Integral Method for Solving Some Important Nonlinear Partial Differential Equations. Nonlinear Dynam, 53, 281.
[11] Taghizadeh, N., Mirzazadeh, M., Farahrooz, F. (2011) . Exact Solutions of the Nonlinear Schr¨odinger Equation by the First Integral Method. J. Math. Anal. Appl, 374, 549-553.
[12] Tascan, F., Bekir, A. and Koparan, M. (2009). Travelling Wave Solutions of Nonlinear Evolutions by Using the First Integral Method. Commun. None. Sci. Numer.Simul., 14, 1810-1815.
[13] Hosseini, K., Ansari, R., Gholamin, P. (2012). Exact Solutions of Some Nonlinear Systems of Partial Differential Equations by Using the First Integral Method. J. Math. Anal. Appl., 387, 807–814.
[14] Bourbaki, N. (1972). Commutative Algebra. Addison-Wesley, Paris.
[15] Li, H., Zhang, J. (2009). The Auxiliary Elliptic-like Equation and the Exp-function Method. Indian Academy of Sciences, 72(6), 915-925.
[16] Hamdi, S., Enright, WH., Schiesser, WE. Gottlieb, JJ. (2003). Exact Solutions of the Generalized Equal Width Wave Equation. ICCSA, 2, 725-734.
In this paper, we apply thefirst integral method to generalized ZK-BBM equation and Drinefel’d-SokolovWilson system and one-dimensional modified EW-Burgers equation.
Thefirst integral method is a powerful solution method for obtaining exact solutions of some nonlinear evolution equations. This method wasfirst proposed by Feng [8] in solving Burgers–KdV equation which is based on the ring theory of commutative algebra. This method can be applied to nonintegrable equations as well as to integrable ones.
Key words
Firstintegralmethod;GeneralizedZK-BBMequation; Drinefel’d-Sokolov-Wilsonsystem; One-dimensional modified EW-Burgers equation
1.INTRODUCTION
Nonlinear evolution equations are widely used to describe complex phenomena in various sciences such asfluid physics, condensed matter, biophysics, plasma physics, nonlinear optics, quantumfield theory and particle physics, etc. In recent decades, several powerful methods have been proposed to construct exact solutionsfornonlinearevolutionequations,suchas tanhmethod[1-3],extendedtanhmethod[4,5],multiple exp-functionmethod [6], transformed rational function method [7] and so on.
Inthepioneerwork,Feng[8]introducedthefirst integralmethodforareliabletreatmentofthenonlinear PDEs. Theusefulfirstintegralmethodiswidelyusedbymanysuchasin[9-13]andbythereferencetherein.
Raslan [10] proposed thefirst integral method to solve the Fisher equation. Taghizadeh et al., [11] solved nonlinear Schr¨odinger equation by using thefirst integral method. Tascan et al., [12] used thefirst integral method to obtain the exact solutions of the modified Zakharov–Kuznetsovequation and ZK–MEW equation. Hosseini et al., [13] applied thefirst integral method to obtain the exact solutions of KdV system and Kaup–Boussinesq system and Wu–Zhang system.
The aim of this paper is tofind exact soliton solutions of generalizedZK-BBM equation and Drinefel’dSokolov-Wilsonsystemandone-dimensionalgeneralizedEW-Burgersequation.bythefirst integralmethod.
The paper is arranged as follows. In Section 2, we describe briefly thefirst integral method. In Sections 3 - 5 , we apply this method to generalized ZK-BBM equation and Drinefel’d-Sokolov-Wilsonsystem and one-dimensional modified EW-Burgers equation.
2.FIRST INTEGRAL METHOD
Raslan summarized for usingfirst integral method [10]. Step 1.Consider a general nonlinear PDE in the form
F(u,ux,uy,ut,uxx,uxy,uxt,...) = 0,(1)
Using a wave variable whereξ= k(x + ly?λt), we can rewrite Eq. (1) in the following nonlinear ODE
G(u,u′,u′′,u′′′,...) = 0,(2)
where the prime denotes the derivation with respect toξ.
Step 2.Suppose that the solution of ODE (2) can be written as follows: u(x,y,t) = u(ξ) = f(ξ).(3) Step 3.We introduce a new independent variable X(ξ) = f(ξ),Y =
Step 4.By the qualitative theory of ordinary differential equations [14] , if we canfind the integrals to Eq. (5) under the same conditions, then the general solutions to Eq. (5) can be solved directly. However, in general, it is really difficult for us to realize this even for onefirst integral, because for a given plane autonomous system, there is no systematic theory that can tell us how tofind itsfirst integrals, nor is there a logical way for telling us what thesefirst integrals are. We will apply the Division Theorem to obtain one first integral to Eq.(5) which reduces Eq.(2) to afirst order integrable ordinary differential equation. An exact solution to Eq. (1) is then obtained by solving this equation. Now, let us recall the Division Theorem: Division Theorem.Supposethat P(w,z) and Q(w,z) are polynomialsinC[w,z], and P(w,z) is irreducible in C[w,z]. If Q(w,z)vanishes at all zero points of P(w,z), then there exists a polynomial G(w,z) in C[w,z] such that
Q(w,z) = P(w,z)G(w,z).
3.GENERALIZED ZK-BBM EQUATION
Consider the generalized ZK-BBM equation [15]
ut+ ux+ a(u3)x+ b(uxt+ uyy)x= 0,(6)
where a,b are real constants.
By make the transformation
u(x,y,t) = f(ξ),ξ= k(x + ly?λt),(7)
the generalized ZK-BBM equation becomes(1?λ)f′(ξ) + 3af2(ξ)f′(ξ) + bk2(l2?λ)f′′′(ξ) = 0.(8)
2A1X2,(16) where A0is arbitraryintegrationconstant. Substituting a0(X),a1(X) and g(X)in the last equationin (15) and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain B0= 0,A0=?2a + A21bk2(l2?1)
bk2(l2?λ))X4,(24) where d is arbitrary integrationconstant. Substituting a0(X),a1(X) and g(X), in the last equation in (22) and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it with aid Maple, we obtain B0= 0,A0=?8a + A21bk2(l2?1)
4.DRINEFEL’D-SOKOLOV-WILSON SYSTEM
Consider the Drinefel’d-Sokolov-Wilsonsystem
ut+ pvvx= 0,
vt+ qvxxx+ ruvx+ suxv = 0,
X2(ξ)?A0.(42) Combining (42) with (35), we obtain the exact solution to equation (34) and then the exact solution to Drinefel’d-Sokolov-Wilsonsystem can be written as
5.ONE-DIMENSIONAL MODIFIED EW-BURGERS EQUATION
Let us consider one-dimensional modified EW-Burgers equation [16]
ut+ au2ux?δuxx?μuxxt= 0,(44)
where a,δ,μare real constants.
We use the wave transformation
u(x,t) = f(ξ),ξ= x?ct.(45)
Substituting (45) into (44), we obtain ordinary differential equation:?cf′(ξ) + af2(ξ)f′(ξ)?δf′′(ξ) +μcf′′′(ξ) = 0.(46) Integrating Eq. (46) with respect toξ, then we have
?cf(ξ) +a
Case B:Suppose that m = 2, by equating with the coefficients of Yi(i = 3,2,1,0) of both sides of (12), we have
˙a2(X) = h(X)a2(X),(55)˙a1(X) = (?2δ
6.CONCLUSION
In this paper, thefirst integral method is applied successfully for solving generalized ZK-BBM equation and Drinefel’d-Sokolov-Wilson system and one-dimensional modified EW-Burgers equation. The results show that this method is efficient infinding the exact solutions of nonlinear differential equations.
REFERENCES
[1]Malfliet, W. (1992). Solitary Wave Solutions of Nonlinear Wave Equations. Am. J. Phys, 60(7), 650-654.
[2]Malfliet, W., Hereman, W. (1996). The Tanh Method: I. Exact Solutions of Nonlinear Evolution and Wave Equations. Phys. Scripta, 54, 563-568.
[3]Malfliet, W., Hereman, W. (1996). The Tanh Method: II. Perturbation Technique for Conservative Systems. Phys. Scripta, 54, 569-575.
[4]Ma, W. X., Fuchssteiner, B. (1996). Explicit and Exact Solutions to a Kolmogorov–Petrovskii–Piskunov Equation. Internat. J. Non-Linear Mech 31, 329–338.
[5]Fan, E. (2000). Extended Tanh-function Method and Its Applications to Nonlinear Equations, Phys. Lett. A., 277(4-5), 212-218.
[6]Ma, W.X.Huang,T.W., Zhang,Y.(2010).AMultipleExp-functionMethodforNonlinearDifferential Equations and Its Application. Phys. Scr., 82, 065003.
[7]Ma, W. X., Lee, J.-H. (2009). A Transformed Rational Function Method and Exact Solutions to the(3+1)-dimensional Jimbo-Miwa Equation. Chaos Solitons Fract., 42, 1356-1363.
[8]Feng, Z. S. (2002). The First Integral Method to Study the Burgers-Korteweg-de Vries Equation. J.Phys. A., 35(2), 343-349.
[9]Feng, Z. S., Wang, X. H. (2002) . The First Integral Method to the Two-dimensional Burgers-KdV Equation. Phys. Lett. A., 308, 173-178.
[10] Raslan, K. R. (2008) . The First Integral Method for Solving Some Important Nonlinear Partial Differential Equations. Nonlinear Dynam, 53, 281.
[11] Taghizadeh, N., Mirzazadeh, M., Farahrooz, F. (2011) . Exact Solutions of the Nonlinear Schr¨odinger Equation by the First Integral Method. J. Math. Anal. Appl, 374, 549-553.
[12] Tascan, F., Bekir, A. and Koparan, M. (2009). Travelling Wave Solutions of Nonlinear Evolutions by Using the First Integral Method. Commun. None. Sci. Numer.Simul., 14, 1810-1815.
[13] Hosseini, K., Ansari, R., Gholamin, P. (2012). Exact Solutions of Some Nonlinear Systems of Partial Differential Equations by Using the First Integral Method. J. Math. Anal. Appl., 387, 807–814.
[14] Bourbaki, N. (1972). Commutative Algebra. Addison-Wesley, Paris.
[15] Li, H., Zhang, J. (2009). The Auxiliary Elliptic-like Equation and the Exp-function Method. Indian Academy of Sciences, 72(6), 915-925.
[16] Hamdi, S., Enright, WH., Schiesser, WE. Gottlieb, JJ. (2003). Exact Solutions of the Generalized Equal Width Wave Equation. ICCSA, 2, 725-734.