- Open Access
Lie symmetry analysis of the time-variable coefficient B-BBM equation
Advances in Difference Equations volume 2012, Article number: 212 (2012)
We perform Lie group classification of a time-variable coefficient combined Burgers and Benjamin-Bona-Mahony equations (B-BBM equation). The direct analysis of the determining equations is employed to specify the forms of these time-dependent coefficients also known as arbitrary parameters. It is established that these model parameters have time-dependent functional forms of linear, power and exponential type.
The Benjamin-Bona-Mahony equation, also known as the regularized-long-wave equation ,
boasts a wide range of applications in the unidirectional propagation of weakly long dispersive waves in inviscid fluids. Likewise, the Burgers equation 
in its different forms has been studied extensively as a model of shock-waves in applications ranging from traffic flow to turbulence. The diverse approaches, both analytical and numerical, have been employed in the extensive studies of these equations in their various forms. Many of these investigations involve assuming the forms of the arbitrary parameters which appear in the mathematical models. However, the Lie group classification is a systematic approach through which the parameters assume their forms naturally. This is the essence of Lie group analysis of differential equations [3–5]. The direct method of group classification [6, 7] is used to analyze the classifying relations, i.e., the equations which contain the arbitrary model parameters.
We study the one-dimensional modified B-BBM equation with power law nonlinearity and time-dependent coefficients given by 
where , and are nonzero arbitrary elements and q is a positive constant. We deduce the following from Eq. (3): If and , then we obtain the variable coefficient Burgers equation
The outline of this work is as follows. The determining equations for the arbitrary elements (classifying relations) are generated in Section 2. In Section 3, the arbitrary elements are specified via the direct method of group classification. The results of group classification are utilized for symmetry reductions solutions in Section 4. Finally, we summarize our findings in Section 5.
2 Classifying equations and principal Lie algebra
The manual task of generating determining equations is tedious, but nowadays the Lie’s algorithm is implemented using a lot of various computer software packages for symbolic computation. We use the Mathematica software package YaLie  to generate and simplify the determining equations of the underlying Eq. (3) for maximal symmetry Lie algebra.
We look for the symmetry generator of Eq. (3) given by
The coefficients of symmetry generator (6), namely , and η, satisfy the determining equations
where the subscripts denote partial differentiation with respect to the indicated variables.
Upon solving (16), the coefficients of symmetry generator (12) become , and . Therefore, we obtain a one-dimensional principal Lie algebra which is spanned by the operator .
3 Lie group classification
This section deals with specifying the forms of the arbitrary parameters through the direct analysis of the classifying relations with the aim of extending the principal symmetry Lie algebra.
We analyze the classifying relations (13)-(15) for the cases: , , . In addition, we must have in order to extend the principal Lie algebra.
Case 1. .
The classifying equation (13) can be written as
In solving (17), we consider the subcases: and .
Subcase 1.1. Suppose that , then from Eq. (17) we obtain
where and are arbitrary constants.
where , and , are nonzero arbitrary constants. The coefficients of symmetry generator (12) become
It is established already that is the admitted generator, thus, we let without loss of generality. Therefore, the extension of the principal Lie algebra is given by the operator
Now set , then we have
The principal Lie algebra is extended by the operator
The condition: implies that and (this result is discarded).
If , then from (18b) we obtain . Thus, for , we have
where . The operator which extends the principal Lie algebra is given by
Now, for , we obtain and . Therefore, the extra operator is given by .
Subcase 1.2. Next assume that , then solving Eq. (17) we obtain
where and are arbitrary constants. Suppose that , then and take the forms
where and , are arbitrary constants. The corresponding additional operator is given by
Now let , then we obtain
Therefore, the principal symmetry Lie algebra is extended by the operator
We obtain for (this result is excluded).
Now, if , then from (17b) we get . Thus, we obtain the following forms for and :
where provided . The additional operator to the principal Lie algebra is given by
The condition implies that and which corresponds to a constant coefficient case. The principal Lie algebra is extended by the operator .
Case 2. Next we consider the case . Proceeding as in the previous case, here we base our analysis on the classifying equation (14). As a result, we obtain the following classification results.
Subcase 2.1. For some constant , we have and thus,
where , and are nonzero arbitrary constants.
Subcase 2.2. : .
where , and are nonzero arbitrary constants.
Case 3. Lastly, we consider . For a given constant k, the analysis of the classifying equation (15) leads to , where . The corresponding forms of and together with the operator, which extends the principal Lie algebra, read
where and are arbitrary constants. The last set of classification results is a particular case of 2.1.1 for some choice of arbitrary constants. It can also be shown by similar calculations that the remaining subcases are duplication of some subcases in Cases 1 and 2.
4 Symmetry reductions
Since the maximal symmetry Lie algebra is two-dimensional, the derivation of invariant solutions amounts to finding solutions invariant under the linear combination where c is an arbitrary constant. For illustration, we consider Subcase 2.2. Firstly, we look at the Subcase 2.2.1. Thus, for , we have the linear combination
The solution of the characteristic equation for (54) is given by the similarity solution
where is an arbitrary function of the similarity variable
Upon substitution of the corresponding arbitrary parameters and the solution (55) into Eq. (3), we obtain the ordinary differential equation
where a ‘prime’ denotes derivative with respect to z.
If , then the similarity variable takes the form , but the similarity solution and the reduced equation remain the same.
Secondly, we consider the Subcase 2.2.2. Proceeding likewise for , the invariant solution assumes the form
where is the similarity variable and is an arbitrary function. The reduced equation is of the form
Next, let , then the similarity variable is given by , and thus the similarity solution takes the form
where F is an arbitrary function of its argument. The corresponding reduced equation is given by
Remark It is noted that the reduced ordinary differential equations are still highly nonlinear to solve exactly.
The direct analysis of the classifying relations was employed to obtain the functional forms of the time-dependent arbitrary parameters for the B-BBM equation. These time-dependent functional forms are of linear, power and exponential type. The maximal symmetry Lie algebra of dimension two is obtained for each set of classification results. Consequently, the symmetries which span the symmetry Lie algebra are utilized to perform symmetry reductions.
This paper is dedicated to Prof. Ravi P. Agarwal on his 65th birth anniversary.
Benjamin TB, Bona JL, Mahoney JJ: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. 1972, 272: 47–78. 10.1098/rsta.1972.0032
Burgers JM: The Nonlinear Diffusion Equation. Reidel Publishing, Dordrecht; 1974.
Bluman GW, Kumei S: Symmetries and Differential Equations. Springer, New York; 1989.
Olver PJ: Applications of Lie Groups to Differential Equations. Springer, New York; 1986.
Ovsiannikov LV: Group Analysis of Differential Equations. Academic, New York; 1982.
Johnpillai AG, Khalique CM: Group analysis of KdV equation with time dependent coefficients. Appl. Math. Comput. 2010, 216: 3761–3771. 10.1016/j.amc.2010.05.043
Molati M, Ramollo MP: Symmetry classification of the Gardner equation with time-dependent coefficients arising in stratified fluids. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 1542–1548. 10.1016/j.cnsns.2011.09.002
Zhao Y, Zhang WG: Qualitative analysis and solutions of bounded travelling wave for B-BBM equation. Acta Math. Appl. Sin. 2010, 26: 415–426. 10.1007/s10255-010-0007-0
Sophocleous C: Transformation properties of a variable coefficient Burgers equation. Chaos Solitons Fractals 2004, 20: 1047–1057. 10.1016/j.chaos.2003.09.024
Morrison PJ, Meiss JD, Cary JR: Scattering of regularized-long-wave solitary waves. Physica D, Nonlinear Phenom. 1984, 11: 324–336. 10.1016/0167-2789(84)90014-9
Díaz, JM: Short guide to YaLie. Yet another Lie Mathematica package for Lie symmetries. http://library.wolfram.com/infocenter/MathSource/4231/
CMK would like to thank the Organizing Committee of ‘International Conference on Applied Analysis and Algebra’, (ICAAA 2012) for their kind hospitality during the conference. MM thanks the North-West University, Mafikeng campus for the post-doctoral fellowship.
The authors declare that they have no competing interests.
MM and CMK worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
About this article
Cite this article
Molati, M., Khalique, C.M. Lie symmetry analysis of the time-variable coefficient B-BBM equation. Adv Differ Equ 2012, 212 (2012). https://doi.org/10.1186/1687-1847-2012-212
- B-BBM equation
- group classification
- classifying relations