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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 240797, 5 pages
Conservation Laws for a Generalized Coupled Korteweg-de Vries System
International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa
Received 10 May 2013; Accepted 12 June 2013
Academic Editor: Hossein Jafari
Copyright © 2013 Daniel Mpho Nkwanazana et al. 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.
We construct conservation laws for a generalized coupled KdV system, which is a third-order system of nonlinear partial differential equations. We employ Noether's approach to derive the conservation laws. Since the system does not have a Lagrangian, we make use of the transformation , and convert the system to a fourth-order system in , . This new system has a Lagrangian, and so the Noether approach can now be used to obtain conservation laws. Finally, the conservation laws are expressed in the , variables, and they constitute the conservation laws for the third-order generalized coupled KdV system. Some local and infinitely many nonlocal conserved quantities are found.
The generalized coupled KdV system given by  where , , , , , and are real constants, describes the interaction of two long waves, whose dispersion relations are different. For the case when , soliton solutions have been obtained in [2, 3]. Many other special cases of (1) have been considered in the literature, and various methods have been used to find its exact solutions. See, for example, [4–11].
In this study, we consider a special case of the generalized coupled KdV system given by and construct conservation laws for (2). Recently, the conservation laws of system (2) for special values of the constants and were derived in  using the multiplier approach.
Many nonlinear partial differential equations (PDEs) of mathematical physics and engineering are continuity equations, which express conservation of mass, momentum, energy, or electric charge. It is well known that conservation laws play a crucial role in the solution and reduction of PDEs. For variational problems the conservation laws can be constructed by means of the Noether theorem . The application of the Noether theorem depends upon the existence of a Lagrangian. However, there are nonlinear differential equations that do not have a Lagrangian. In such instances, researchers have developed several methods to derive conserved quantities for such equations. See, for example, [14–20].
The organization of this paper is as follows. In Section 2 we briefly recall some notations and fundamental relations concerning the Noether symmetries approach, which we utilize in the same section to obtain the Noether symmetries and the corresponding conserved vectors. The concluding remarks are summarized in Section 3.
2. Conservation Laws of Coupled KdV Equations
In this section we derive the conservation laws for the generalized coupled KdV system (2). This system does not have a Lagrangian. In order to apply the Noether theorem we transform our system (2) to a fourth-order system, using the transformations and . Then system (2) becomes It can readily be verified that the second-order Lagrangian for system (3) is given by because where and are the standard Euler operators defined by Consider the vector field which has the second-order prolongation defined by Here The Lie-Bäcklund operator defined in (7) is a Noether operator corresponding to the Lagrangian (4) if it satisfies where , are the gauge terms. Expansion of (10) yields The splitting of (11) with respect to different combinations of derivatives of and results in an overdetermined system of PDEs for , and . Solving this system of PDEs we arrive at the following two cases for which Noether symmetries exist.
Case 1. .
In this case we obtain the following Noether symmetries and gauge terms: The above results will now be used to find the components of the conserved vectors for the second-order Lagrangian. Here we can choose , as they contribute to the trivial part of the conserved vector. We recall that the conserved vectors for the second-order Lagrangian are given by [13, 21] Here and are the Lie characteristic functions, given by and . Using (13) together with (12) and , we obtain the following independent conserved vectors for system (2): and for the arbitrary functions and , Conserved vector (14) is a nonlocal conserved vector, and (15) is a local conserved vector for system (2). We now derive two particular cases from conserved vector (16) by letting and , which gives a nonlocal conserved vector and by choosing and , we get the nonlocal conserved vector
Case 2. .
The second case gives the following Noether symmetries and gauge terms: Again we can set and as they contribute to the trivial part of the conserved vector. The independent conserved vectors for system (2), in this case, are and for the arbitrary functions and , we obtain Conserved vectors (20) are nonlocal, whereas (21) is a local conserved vector for system (2). Conserved vector (22) for and gives a nonlocal conserved vector and for and it gives a nonlocal conserved vector We note that for arbitrary values of and infinitely many nonlocal conservation laws exist for system (2).
In this paper we studied the third-order generalized coupled Korteweg-de Vries system (2). This system did not have a Lagrangian. In order to apply Noether theorem the transformations and were utilized, and the system was transformed to fourth-order system (3) in and variables. This system admitted the Lagrangian (4). Noether theorem was then used to derive the conservation laws in and variables. Finally, by reverting back to our original variables and we obtained the conservation laws for the third-order generalized coupled KdV system (2). The conservation laws obtained consisted of some local and infinite number of nonlocal conserved vectors.
Daniel Mpho Nkwanazana would like to thank SANHARP and North-West University, Mafikeng Campus, for financial support. Ben Muatjetjeja would like to thank the Faculty Research Committee of FAST, North-West University, Mafikeng Campus, for their financial support.
- H. Wang, D.-q. Xian, and H.-l. Chen, “Homoclinic breather-wave solutions and doubly periodic wave solutions for coupled KdV equations,” Applied Mathematics and Computation, vol. 218, no. 2, pp. 610–615, 2011.
- R. Hirota and J. Satsuma, “Soliton solutions of a coupled Korteweg-de Vries equation,” Physics Letters A, vol. 85, no. 8-9, pp. 407–408, 1981.
- Q. Chowdhury and R. Mukherjee, “Exact envelope-soliton solution for the KdV equations,” Journal of Physics A, vol. 17, pp. 893–898, 1984.
- H. D. Yu and J. F. Zhang, “Similarity solutions of the super KdV equation,” Applied Mathematics and Mechanics, vol. 16, no. 9, pp. 839–842, 1995.
- X. Q. Liu, “The soliton and elliptical periodic solution of coupled KdV system,” Henan Science, vol. 15, pp. 135–138, 1997.
- X. Q. Liu and Ch. L. Bai, “New soliton solution of coupled KdV equations,” Chinese Journal of Quantum Electronics, vol. 16, pp. 360–364, 1999.
- S.-Y. Lou, X.-Y. Tang, and J. Lin, “Exact solutions of the coupled KdV system via a formally variable separation approach,” Communications in Theoretical Physics, vol. 36, no. 2, pp. 145–148, 2001.
- H. Q. Zhang and Z. Y. Yan, “New explicit exact solutions to nonlinear evolution equations,” Mathematica Applicata, vol. 12, no. 1, pp. 76–79, 1999.
- X. Ch. Yu and Zh. Zh. Dong, “Mutiple traveling wave solutions of the coupled KdV equations,” Journal of Liaocheng University, vol. 18, pp. 31–35, 2005.
- Zh. Zh. Dong, X. Ch. Yu Dong, and W. Ling, “Exact traveling wave solutions of the nonlinear coupled KdV equations,” Chinese Journal of Quantum Electronics, vol. 23, pp. 379–382, 2006.
- B. Xu, X. Q. Liu, and Y. T. Liu, “Symmetry, exact solutions and conservation laws of the coupled KdV equations,” Acta Mathematicae Applicatae Sinica, vol. 33, no. 1, pp. 118–123, 2010.
- R. Naz, “Conservation laws for a complexly coupled KdV system, coupled Burgers' system and Drinfeld-Sokolov-Wilson system via multiplier approach,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1177–1182, 2010.
- E. Noether, “Invariant variation problems,” Transport Theory and Statistical Physics, vol. 1, no. 3, pp. 186–207, 1971, Translated from Nachrichten der Akademie der Wissenschaften in Göttingen II, vol. 2, pp. 235–257, 1918.
- P. S. Laplace, Trait de Mcanique Cleste, vol. 1, Paris, France, 1798, English translation: Celestial Mechanics, New York, NY, USA, 1966.
- A. H. Kara and F. M. Mahomed, “Relationship between symmetries and conservation laws,” International Journal of Theoretical Physics, vol. 39, no. 1, pp. 23–40, 2000.
- A. H. Kara and F. M. Mahomed, “Noether-type symmetries and conservation laws via partial Lagrangians,” Nonlinear Dynamics, vol. 45, no. 3-4, pp. 367–383, 2006.
- H. Steudel, “Über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssätzen,” Zeitschrift für Naturforschung, vol. 17a, pp. 129–132, 1962.
- T. Wolf, A comparison of Lie Groups to Differential Equations, Springer, New York, NY, USA, 1993.
- S. C. Anco and G. Bluman, “Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications,” European Journal of Applied Mathematics, vol. 13, no. 5, pp. 545–566, 2002.
- S. C. Anco and G. Bluman, “Direct construction method for conservation laws of partial differential equations. II. General treatment,” European Journal of Applied Mathematics, vol. 13, no. 5, pp. 567–585, 2002.
- 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.