Abstract
In this chapter, we consider a generalized coupled Boussinesq system of KdV–KdV type, which belongs to the class of Boussinesq systems modeling two-way propagation of long waves of small amplitude on the surface of an ideal fluid. We obtain conservation laws for this system using Noether theorem. Since this system does not have a Lagrangian, we increase the order of the partial differential equations by using the transformations \(u={U_{x}}\), \(v={V_{x}}\) and convert the Boussinesq system to a fourth-order system in U, V variables, which has a Lagrangian. Consequently, we find infinitely many nonlocal conserved quantities for our original Boussinesq system of KdV–KdV type.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bona, J.L., Chen, M., Saut, J.C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory. J. Nonlinear Sci. 12, 283–318 (2002)
Bona, J.L., Colin, T., Lannes, D.: Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178, 373–410 (2005)
Bona, J.L., Dougalis, V.A., Mitsotakis, D.E.: Numerical solution of Boussinesq systems of KdV–KdV type: II. Evolution of radiating solitary waves. Nonlinearity 21, 1–24 (2008)
Bona, J.L., Dougalis, V.A., Mitsotakis, D.E.: Numerical solution of KdV–KdV systems of Boussinesq equations: I. The numerical scheme and existence of generalized solitary waves. Math. Comput. Simul. 74, 214–228 (2007)
Kara, A.H., Mahomed, F.M.: Relationship between symmetries and conservation laws. Int. J. Theor. Phys. 39, 23–40 (2000)
Kara, A.H., Mahomed, F.M.: Noether-type symmetries and conservation laws via partial Lagragians. Nonlinear Dyn. 45, 367–383 (2006)
Laplace, P.S.: Trait é de M écanique C éleste, vol. 1, Paris (1798). (English transl., Celestial mechanics, New York, (1966))
Naz, R., Mahomed, F.M., Hayat, T.: Conservation laws for third-order variant Boussinesq system. Appl. Math. Lett. 23, 883–886 (2010)
Noether, E.: Invariante variationsprobleme. Nachr. König. Gesell. Wiss. Göttingen Math.-phys. Kl. Heft 2, 235–257 (1918). (The English traslation in transport Theory and Statistical Physics 1, 186–207 (1971))
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)
Pazoto, A.F., Rosier, L.: Stabilization of a Boussinesq system of KdV–KdV type. Syst. Control Lett. 57, 595–601 (2008)
Steudel, H.: Uber die zuordnung zwischen invarianzeigenschaften und erhaltungssatzen. Z. Naturforsch. A 17, 129–132 (1962)
Wolf, T.: A comparison of four approaches to the calculation of conservation laws. Eur. J. Appl. Math. 13, 129–152 (2002)
Acknowledgement
BM and CMK would like to thank the Organizing Committee of “International Conference: AMMCS-2013,” Waterloo, Canada for their kind hospitality during the conference. BM also thanks the Faculty Research Committee of FAST, North-West University, and ETM thanks SANHARP for financial support. We thank the referee for the useful comments, which enhanced the presentation and results of the chapter.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Mogorosi, T., Muatjetjeja, B., Khalique, C. (2015). Conservation Laws for a Generalized Coupled Boussinesq System of KdV–KdV Type. In: Cojocaru, M., Kotsireas, I., Makarov, R., Melnik, R., Shodiev, H. (eds) Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science. Springer Proceedings in Mathematics & Statistics, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-12307-3_45
Download citation
DOI: https://doi.org/10.1007/978-3-319-12307-3_45
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12306-6
Online ISBN: 978-3-319-12307-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)