Analytical and Numerical Approaches for Complicated Nonlinear EquationsView this Special Issue
Research Article | Open Access
Conservation Laws for a Variable Coefficient Variant Boussinesq System
We construct the conservation laws for a variable coefficient variant Boussinesq system, which is a third-order system of two partial differential equations. This system does not have a Lagrangian and so we transform it to a system of fourth-order, which admits a Lagrangian. Noether’s approach is then utilized to obtain the conservation laws. Lastly, the conservation laws are presented in terms of the original variables. Infinite numbers of both local and nonlocal conserved quantities are derived for the underlying system.
The first type of variant Boussinesq equations [1, 2] is given byand was introduced as a model for water waves . Wang in his paper  obtained the solitary wave solutions of ((1a) and (1b)) by using homogeneous balance method. The periodic wave solutions of ((1a) and (1b)) were derived in  by using ansatz method and the multisolitary wave solutions were obtained in  using the homogeneous balance method. Xu et al.  obtained traveling wave solutions of ((1a) and (1b)). Conservation laws for ((1a) and (1b)) were derived in .
Conservation laws play a vital role in the solution process of differential equations (DEs) because they describe physical properties that remain constant throughout the various processes that occur in the physical world. Thus it is very important to compute conservation laws for differential equations. One can see from the various published papers (see, e.g., [9–11]) that conservation laws have been used in studying the existence, uniqueness, and stability of solutions of nonlinear partial differential equations. They have also been applied in the development and use of numerical methods (see, e.g., [12, 13]). Most importantly, conserved vectors associated with Lie point symmetries have been used to derive exact solutions of some partial differential equations [14–16].
In this paper, we study the variable coefficient variant Boussinesq system: which generalizes the system ((1a) and (1b)). In ((2a) and (2b)), , , and are arbitrary functions of , with describing the different diffusion strength, representing the field of a horizontal velocity, and representing the amplitude describing the deviation from the equilibrium position of the liquid.
The paper is organized as follows. In Section 2 we briefly give the preliminaries concerning the Noether symmetry approach. Section 3 obtains the conservation laws for the system ((2a) and (2b)). Finally, in Section 4 concluding remarks are presented.
Here we present some salient features of Noether operators concerning the system of two partial differential equations. These results will be utilized in Section 3. The reader is referred to [8, 17–19] for further details.
Consider the vector field which has the second-order prolongation where with The Euler-Lagrange operators are defined by Consider a system of two partial differential equations of two independent variables, and , namely,which has a second-order Lagrangian ; that is, ((8a) and (8b)) are equivalent to the Euler-Lagrange equations:
We recall the following theorem.
Theorem 2 (Noether ). If , as given in (3), is a Noether point symmetry generator corresponding to a Lagrangian of (8a) and (8b), then the vector with components, is a conserved vector for ((8a) and (8b)) associated with the operator , where and are the Lie characteristics functions.
Consider the variable coefficient variant Boussinesq system ((2a) and (2b)); namely, Here we note that the system ((2a) and (2b)) does not admit a Lagrangian. Nevertheless, we can transform the system ((2a) and (2b)) into a variational form by setting and . Thus, the system ((2a) and (2b)), with this transformation, becomes a fourth-order system, namely and has a second-order Lagrangian given by Substituting the value of from (14) to (10) and splitting with respect to the derivatives of and yield the linear overdetermined system of PDEs; namely After some tedious and lengthy calculations, the above system yields The analysis of (17), (18), and (19) prompts the following two cases.
Case 1. , , and are arbitrary but not of the form contained in Case 2.
In this case we obtain four Noether point symmetries. These are given below together with their corresponding gauge functions: Invoking Theorem 2, the four nontrivial conserved vectors associated with these four Noether point symmetries are, respectively, From the above we observe that the conserved vector (24)-(25) is a local conserved vector. In (30)-(31) one can see that the nonlocal part within the parenthesis gives the trivial part of the conserved vector and therefore can be set to zero. Thus, the conserved vector (30)-(31) is a local conserved vector. It is also interesting to notice that the conserved vectors (26)-(27) and (28)-(29) for , , , and yield the local conserved vectors:
Remark 3. We note that for arbitrary values of and infinitely many nonlocal conservation laws exist for the variable coefficient variant Boussinesq system.
Case 2. , , and , where , , and are constants.
This case gives us five Noether point symmetries, namely, , , and , given by the generators (20)–(22) and , given by The application of Theorem 2, due to Noether, gives the five nontrivial conserved vectors: respectively, corresponding to the above five Noether point symmetries. We note that in this case we obtain an extra Noether operator and hence an extra conserved vector, which is given by (40)-(41).
Remark 4. When , we recover the results obtained in .
4. Concluding Remarks
In this paper we studied the variable coefficient variant Boussinesq system ((2a) and (2b)). This system does not have a Lagrangian. Therefore we converted it to a fourth-order system ((13a) and (13b)) which has a Lagrangian. Thereafter, we utilized the Noether’s theorem to construct the conservation laws of system ((13a) and (13b)). Finally, by reverting back to our original variables and we constructed the conservation laws for the third-order variable coefficient variant Boussinesq system. The conservation laws obtained consisted of infinite number of local and nonlocal conserved vectors.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Ben Muatjetjeja would like to thank the Faculty Research Committee of FAST, North-West University, Mafikeng Campus, South Africa, for its continuing support.
- J. Boussinesq, “Thorie de l'intumescence liquide, applele onde solitaire ou de translation, se propageant dans un canal rectangulaire,” Comptes Rendus de L'Academie des Sciences, vol. 72, pp. 755–759, 1871.
- R. L. Sachs, “On the integrable variant of the Boussinesq system: Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy,” Physica D, vol. 30, no. 1-2, pp. 1–27, 1988.
- E. Fan and Y. C. Hon, “A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves,” Chaos, Solitons & Fractals, vol. 15, no. 3, pp. 559–566, 2003.
- M. Wang, “Solitary wave solution variant Boussinesq equations travelling wave solutions for two variant Boussinesq equations in shallow water waves,” Chaos, Solitons & Fractals, vol. 15, pp. 559–566, 2003.
- Z. Fu, S. Liu, and S. Liu, “New transformations and new approach to find exact solutions to nonlinear equations,” Physics Letters A, vol. 299, no. 5-6, pp. 507–512, 2002.
- J. F. Zhang, “Multi-solitary wave solutions for variant Boussinesq equations and Kupershmidt equations,” Applied Mathematics and Mechanics, vol. 21, no. 2, pp. 171–175, 2000.
- G.-Q. Xu, Z.-B. Li, and Y.-P. Liu, “Exact solutions to a large class of nonlinear evolution equations,” The Chinese Journal of Physics, vol. 41, no. 3, pp. 232–241, 2003.
- R. Naz, F. M. Mahomed, and T. Hayat, “Conservation laws for third-order variant Boussinesq system,” Applied Mathematics Letters, vol. 23, no. 8, pp. 883–886, 2010.
- P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Communications on Pure and Applied Mathematics, vol. 21, no. 5, pp. 467–490, 1968.
- T. B. Benjamin, “The stability of solitary waves,” Proceedings of the Royal Society A, vol. 328, no. 1573, pp. 153–183, 1972.
- R. J. Knops and C. A. Stuart, “Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity,” Archive for Rational Mechanics and Analysis, vol. 86, no. 3, pp. 233–249, 1984.
- R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhuser, Basel, Switzerland, 2nd edition, 1992.
- E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, vol. 118 of Applied Mathematical Sciences, Springer, Berlin, Germany, 1996.
- A. Sjöberg, “Double reduction of PDEs from the association of symmetries with conservation laws with applications,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 608–616, 2007.
- A. H. Bokhari, A. Y. Al-Dweik, A. H. Kara, F. M. Mahomed, and F. D. Zaman, “Double reduction of a nonlinear wave equation via conservation laws,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1244–1253, 2011.
- G. L. Caraffini and M. Galvani, “Symmetries and exact solutions via conservation laws for some partial differential equations of mathematical physics,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1474–1484, 2012.
- E. Noether, “Invariante Variationsprobleme,” Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, vol. 2, pp. 235–257, 1918.
- N. H. Ibragimov, Ed., CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1–3, CRC Press, Boca Raton, Fla, USA, 1994–1996.
- R. Naz, D. P. Mason, and F. M. Mahomed, “Conservation laws and conserved quantities for laminar two-dimensional and radial jets,” Nonlinear Analysis—Real World Applications, vol. 10, no. 5, pp. 2641–2651, 2009.
Copyright © 2014 Ben Muatjetjeja and Chaudry Masood Khalique. 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.