Research Article  Open Access
GangWei Wang, TianZhou Xu, "Group Analysis and New Explicit Solutions of Simplified Modified Kawahara Equation with Variable Coefficients", Abstract and Applied Analysis, vol. 2013, Article ID 139160, 8 pages, 2013. https://doi.org/10.1155/2013/139160
Group Analysis and New Explicit Solutions of Simplified Modified Kawahara Equation with Variable Coefficients
Abstract
The simplified modified Kawahara equation with variable coefficients is studied by using Lie symmetry method. Then we obtain the corresponding Lie algebra, optimal system, and the similarity reductions. At last, we also give some new explicit solutions for some special forms of the equations.
1. Introduction
Lie’s classical theory of symmetries of differential equations is an inspiring source for various generalizations aiming to find the ways for obtaining explicit solutions. Lie’s theory provides a standard method [1–5] for finding the Lie point symmetry group of a nonlinear system. And above all, Lie’s method of infinitesimal transformation groups which essentially reduces the number of independent variables in partial differential equation (PDE) and reduces the order of ordinary differential equation (ODE) has been widely used in equations of mathematical physics. Lie method is an effective and the simplest method among group theoretic techniques and a large number of equations [6–11] are solved with the aid of this method.
In this paper, by using the Lie symmetry group method, we will consider the following simplified modified Kawahara equation:
Here in (1) the first term represents the evolution term while the second term represents the nonlinear term. The third term represents the linear damping [6, 7, 12] while the fourth term is the dispersion term. The time dependent coefficients of damping and dispersion are, respectively, and which are arbitrary smooth functions of the variable . If and , (1) becomes the standard simplified modified Kawahara equation (see [13, 14] and references therein).
These KdV types of equations have been derived to model many physical phenomena, such as gravitycapillary waves on a shallow layer and magnetosound propagation in plasmas, (see [14] and references therein). Many studies have been conducted with these types of equations [6–8, 12–18]. In [14] similarity solutions for some classes of (1) were considered. Abundant solitons solutions are obtained by using the tanh method in [17]. The paper [18] is mainly concerned with the local wellposedness of the initialvalue problems for the Kawahara and the modified Kawahara equations in Sobolev spaces.
Our aim in the present work is to perform the variable coefficients version of the simplified modified Kawahara equation with the help of Lie’s method. Then we get symmetry reductions and groupinvariant solutions.
2. Lie Group Classification
2.1. Lie Symmetry Analysis of (1)
In this section, we will perform Lie group method for (1).
If (1) is invariant under a oneparameter Lie group of point transformations with infinitesimal generator then the invariant condition reads as where Here, denotes the total derivative operator and is defined by and .
Solving (4) with the help of (5) we obtain
The structure of the determining equations (10) and (11) may give the selection of the following three forms for the coefficient .
Case 1 (). In this case, solving (7)–(11) we get
and (8) becomes
where are constants. The analysis of (13) leads to the following three possibilities for .(1.1) is arbitrary.
For this case, we obtain the vector field (1.2) is a constant.
Equation (1) admits a threedimensional Lie algebra spanned by (1.3), are constant.
We obtain the corresponding two Lie point symmetry generators
Case 2 ( is a nonzero constant as ). Similarly, in this case, solving (7)–(11) for the infinitesimals, we obtain and (8) becomes where are constants. The analysis of (18) gives rise to the following four possibilities for .(2.1) is arbitrary.
For this case, we obtain the vector field (2.2) is a constant.
We have the corresponding twodimensional Lie algebra (2.3), is a constant.
Equation (1) admits a threedimensional Lie algebra spanned by (2.4) is a constant.
Substituting into (18), one can get where ??. We obtain the corresponding twodimensional Lie algebra
Case 3 (). In this case, solving (7)–(11), we get
and (8) becomes
where are constants. Similarly, the analysis of (25) gives rise to the following three possibilities for .(3.1) is arbitrary.
For this case, we obtain the vector field (3.2) is a constant.
We obtain the corresponding twodimensional Lie algebra (3.3), is a constant.
The Lie algebra is extended by the symmetry generators
2.2. Optimal System of OneDimensional Lie Algebras
First of all, we briefly review the main definitions [1] which will be used in the following sections.
Definition 1 (see [1]). Let ?? be a Lie group. An optimal system of sparameter subgroups is a list of conjugacy inequivalent sparameter subgroups with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of sparameter subalgebras forms an optimal system if every sparameter subalgebra of is equivalent to a unique member of the list under some element of the adjoint representation: , .
To obtain the optimal system, we apply the formula [1] where is a real constant. Here is the commutator for the Lie algebra given by
The commutator table of the Lie point symmetries of (1.2) and the adjoint representations of the symmetry group of (1.2) on its Lie algebra are given in Tables 1 and 2, respectively. In the same way, the commutator table and the adjoint representations of , , and are given in Tables 3, 4, 5, 6, 7, and 8, respectively. We give in Table 9 optimal system of subalgebras for (1.2),??,??,??and??, respectively.








 
Here , and is a nonzero constant. 
Remark 2. For simplicity, we will refer to the equation corresponding to the case (1.3) as (1.3), and so on.
Remark 3. For brevity we only consider the optimal system for the equation concerned in case (1.2) in detail and the rest will be listed in Table 9 as they can be derived in a similar manner.
3. Symmetry Reductions and Exact GroupInvariant Solutions
In this section, we will use Table 9 to obtain symmetry reductions and exact groupinvariant solutions for??,??,??,??and??.
3.1. Symmetry Reductions and Exact Solutions to (1.2)
3.1.1.
For the generator , the groupinvariant solution is , where ?? is the groupinvariant, the substitution of this solution into (1.2) gives the trivial solution , and is a constant.
3.1.2.
For the linear combination , we have where is the groupinvariant. Substituting (31) into (1.2), we reduce it to the following ODE:
3.1.3.
For the generator , we have where is the groupinvariant. Substituting (33) into (1.2), we reduce it to the following ODE: where??.
3.2. Symmetry Reductions and Exact Solutions to
3.2.1.
For the generator , we get that the groupinvariant solution of is , where is an arbitrary constant.
3.2.2.
For the generator , we have where ?. Substituting (35) into , one can get where?.
3.2.3.
(i) For , we have where ??. Substituting (37) into yields
(ii) For , we get
3.3. Symmetry Reductions and Exact Solutions to
3.3.1.
For the generator , we get that the groupinvariant solution of is where is an arbitrary constant.
3.3.2.
For this case, we have then substituting (41) into gives rise to
3.4. Symmetry Reductions and Exact Solutions to
3.4.1.
For the generator , we get that the groupinvariant solution of is where is an arbitrary constant.
3.4.2.
For the generator , we have where . Substituting (44) into , one can get where?.
3.4.3.
(i) For , we have where . Substituting (46) into yields
In order to search for other explicit solutions, by using the Jacobi elliptic function expansion method [19]. By virtue of the technique of solution we introduce the ansatz Substituting (48) into (47), one can get
Thus, we obtain Jacobi elliptic function solutions of as follows: In particular, when , we can obtain hyperbolic function solutions When , we can obtain trigonometric function solutions where .
(ii) For , we get
Remark 4. It is not difficult to find out that the reduced ODEs may be classified into four classes:
4. The Explicit Power Series Solutions
In this section, we will consider the explicit analytic solutions of some special forms of reduced equations by using the power series method.
Now, we seek a solution of (45) in a power series of the following form:
Substituting (55) into (45), we get
Now from (56), comparing coefficients, for ?, one can get
Generally, for , we obtain
From (57) and (58), we can obtain all the coefficients of the power series (55). For arbitrary chosen constant numbers , , , , and , the other terms also can be determined successively from (57) and (58) in a unique way. In addition, it is easy to prove the convergence of the power series (55) with the coefficients given by (57) and (58) [20, 21]. The details are omitted here. In this connection, this power series solution is an explicit analytic solution.
So, the power series solution of (45) can be written as follows:
Thus, the exact power series solution of is where are arbitrary constants, and the other coefficients can be determined successively from (57) and (58).
Of course, in physical applications, it will be convenient to write the solution of (45) in the approximate form
Remark 5. The exact solution of the rest of equations can be derived in a similar manner. We have details omitted here.
Remark 6. It is easy to see that the reduced equations (54) are all higherorder nonlinear ODEs or with nonconstant coefficients. If we obtain a oneparameter symmetry group of an ODE, then we could reduce the order of the equation by one. However, we find out that such reduced ODEs are more complicated than the original equation. In general, we cannot obtain the exact explicit solutions for higherorder nonlinear ODEs or with nonconstant coefficients by using the elementary functions and integrals. However, the power series can be used to solve them. In view of this, we can find that the power series method [14, 15, 20–23] is an effective tool of solving such ODEs. Moreover, from our model, we could find that these power series solutions are important for computations in numerical analysis and physical applications. And above all, these power series play an important role in the investigation of physical phenomena and other natural phenomena.
Remark 7. Indeed, for all the rest of the cases presented in Section 2, it is possible to get optimal systems and symmetry reductions, but for brevity we have omitted them here.
5. Conclusions
We have performed Lie symmetry analysis for the simplified modified Kawahara equation with variable coefficients. Then the similarity reductions and exact solutions are obtained based on the optimal system of the onedimensional Lie algebras, for some special forms of the equations. Moreover, the power series solution of the reduced equation are given simultaneously. These are new solutions for the simplified modified Kawahara equation with variable coefficients. The symmetry analysis based on the Lie group method is a very powerful method and is worthy of being studyied further.
Acknowledgments
The project is supported by the National Natural Science Foundation of China (NNSFC) (Grant no. 11171022). The authors express their sincere thanks to the referees for their careful review of this paper and their useful suggestions.
References
 P. J. Olver, Application of Lie Group to Differential Equation, Springer, New York, NY, USA, 1986.
 L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, NY, USA, 1982. View at: MathSciNet
 S. Lie, “On integration of a class of linear partial differential equations by means of definite integrals,” Archive for Mathematical Logic, vol. 6, no. 3, pp. 328–368, 1881. View at: Google Scholar
 G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. View at: MathSciNet
 N. H. Ibragimov, Ed., CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1–3, CRC Press, Boca Raton, Fla, USA, 1994.
 A. G. Johnpillai and C. M. Khalique, “Group analysis of KdV equation with time dependent coefficients,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3761–3771, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. G. Johnpillai and C. M. Khalique, “Lie group classification and invariant solutions of mKdV equation with timedependent coefficients,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1207–1215, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 G. W. Wang, X. Q. Liu, and Y. Y. Zhang, “Lie symmetry analysis to the time fractional generalized fifthorder KdV equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 2321–2326, 2013. View at: Google Scholar
 G. W. Wang, X. Q. Liu, and Y. Y. Zhang, “Symmetry reduction, exact solutions and conservation laws of a new fifthorder nonlinear integrable equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 2313–2320, 2013. View at: Google Scholar
 H. Wang and Y.H. Tian, “NonLie symmetry groups and new exact solutions of a ($2+1$)dimensional generalized BroerKaup system,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 3933–3940, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 S. Kumar, K. Singh, and R. K. Gupta, “Painlevé analysis, Lie symmetries and exact solutions for ($2+1$)dimensional variable coefficients BroerKaup equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1529–1541, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 A. Biswas, “Solitary wave solution for the generalized KdV equation with timedependent damping and dispersion,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 910, pp. 3503–3506, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 I. L. Freire and J. C. S. Sampaio, “Conservation laws for Kawahara equations,” Matematica Aplicada E Computacional, 17 a 21 de setembro de 2012Aguas de Lindola/SP. View at: Google Scholar
 H. Liu, J. Li, and L. Liu, “Lie symmetry analysis, optimal systems and exact solutions to the fifthorder KdV types of equations,” Journal of Mathematical Analysis and Applications, vol. 368, no. 2, pp. 551–558, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. Kaur and R. K. Gupta, “Kawahara equation and modified Kawahara equation with time dependent coefficients: symmetry analysis and generalized ${G}^{\prime}G$expansion method,” Mathematical Methods in the Applied Sciences, vol. 36, no. 5, pp. 584–600, 2013. View at: Publisher Site  Google Scholar
 O. Vaneeva, “Lie symmetries and exact solutions of variable coefficient mKdV equations: An equivalence based approach,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 611–618, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A.M. Wazwaz, “Abundant solitons solutions for several forms of the fifthorder KdV equation by using the tanh method,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 283–300, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 W. Chen, J. Li, C. Miao, and J. Wu, “Low regularity solutions of two fifthorder KDV type equations,” Journal d'Analyse Mathematique, vol. 107, no. 1, pp. 221–238, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. Zhang, “Extended Jacobi elliptic function expansion method and its applications,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 5, pp. 627–635, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. Liu and J. Li, “Lie symmetry analysis and exact solutions for the short pulse equation,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 56, pp. 2126–2133, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. Liu, J. Li, and Q. Zhang, “Lie symmetry analysis and exact explicit solutions for general Burgers' equation,” Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 1–9, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 N. H. Asmar, Partial Differential Equations with Fourier Series and Boundary Value Problems, China Machine Press, Beijing, China, 2nd edition, 2005.
 H. Liu, J. Li, and L. Liu, “Group classifications, symmetry reductions and exact solutions to the nonlinear elastic rod equations,” Advances in Applied Clifford Algebras, vol. 22, no. 1, pp. 107–122, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
Copyright
Copyright © 2013 GangWei Wang and TianZhou Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.