Plenary Lecture

Conservation Laws and Exact Solutions for a Generalized BBM-Like Equation

Professor Maria S. Bruzón
University of Cádiz

Abstract: The Korteweg-de Vries equation arising in the theory of shallow water waves, is one of the most well-known dispersive wave equation in the literature, since the study of its conservation laws has been the starting point of the discovery of a great number of techniques to solve evolutionary equations ut+ux+uux+uxxx=0. Peregrine [1] and subsequently Benjamin, Bona and Mahony [2] proposed the equation ut+ux+uux+utxx=0, as an alternative to the Korteweg-de Vries equation for the description of long nonlinear dispersive waves. In [3] Bruzón and Gandarias determined the subclass of equations which are self-adjoint for a family Benjamin-Bona-Mahony equations. By using a general theorem on conservation laws they derived conservation laws for this equation. First of all, Lie point symmetries have been obtained applying the Lie classical method [4, 5]. Several well-known researchers obtain symmetries because they can be used to obtain systematically exact solutions of the equations. In this work we study a generalized BBM-like equation from the point of view of the theory of symmetry reductions in partial differential equations. The BBM-like equation has been proposed by Saha et all in [6] and it possesses solitons, compactons and undular bores. We obtain the Lie symmetries, then, we use the transformations groups to reduce the equations into ordinary differential equations. Physical interpretation of these reductions and some exact solutions are also provided. Local conservation laws are continuity equations that provide conserved quantities of physical importance for all solutions of a particular equation. In addition, the existence of an infinite hierarchy of local conservation laws of a partial differential equation is a strong indicator of its integrability. For any particular partial differential equation, a complete classification of all local low-order conservation laws can be derived by using the multiplier method [7,8,9]. We also derive all low-order conservation laws for the BBM-like equation by using the multiplier method.
[1] D H Peregrine, J. Fluid Mech. 27, 815 (1967)
[2] T B Benjamin, J L Bona and J J Mahony, Phil. Trans. R. Soc. London Ser. A 272, 47 (1972)
[3] M.S. Bruzón, M.L. Gandarias, WSEAS TRANSACTIONS ON MATHEMATICS, 2012.
[4] Bruzón, M. S., Recio, E., Garrido, T. M. and Márquez, A. P. (2017), Open Physics, 2017.
[5] Garrido, T. M., Kasatkin, A. A., Bruzón, M. S. and Gazizov, R. K. (2017), Journal of Computational and Applied Mathematics 318, (2017) 253-258.
[6] Aparna Saha, B Talukdar, Umapada Das and Supriya Chatterjee, Pramana – J. Phys. (2017) 88: 28
[7] S.C. Anco and G. Bluman, Euro. Jnl of Appl. Math. 13 (2002), 545–566.
[8] S.C. Anco and G. Bluman, Euro. Jnl of Appl. Math. 13 (2002), 567–585.
[9] S.C. Anco, To appear in Fields Institute Communications: Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, arXiv:mathph/1605.08734 (2016).

Brief Biography of the Speaker: Maria Santos Bruzón is Ph.D. in Mathematics. She carried out his studies at the University of Sevilla, Spain. She is currently a Professor of Mathematics with the Department of Mathematics of the University of Cadiz. Hers current research interests include group analysis, methods of group transformations: Lie symmetries, nonclassical methods, direct methods and conservation laws applied to ordinary differential equations and partial differential equations. The research and collaborations have resulted in a large number of publications in leading peer-reviewed journals, over 55, as well as congress proceedings and abstracts. She has also been the author and co-author of some popular science papers. The last publications are: Rosa, María; Camacho Moreno, José Carlos; Bruzón-Gallego, María Santos; Gandarias-Núñez, María Luz. 2017. Classical and potential symmetries for a generalized Fisher equation. Journal of Computational and Applied Mathematics. 318: 181-188. Camacho Moreno, José Carlos; Rosa-Durán, María; Gandarias-Núñez, María Luz; Bruzón-Gallego, María Santos. 2017. Classical symmetries, travelling wave solutions and conservation laws of a Generalized Fornberg-Whitham equation. Journal of Computational and Applied Mathematics. 318: 149-155. De La Rosa -Silva, Rafael, Gandarias-Núñez, María Luz; Bruzón-Gallego, María Santos. 2016. Equivalence transformations and conservation laws for a generalized variable-coefficient Gardner equation. Communications in Nonlinear Science & Numerical Simulation. 40: 71-79. Bruzón-Gallego, María Santos; Garrido-Letran, Tamara Maria; De La Rosa -Silva, Rafael. 2016. Conservation laws and exact solutions of a Generalized Benjamin-Bona-Mahony-Burgers equation. Chaos, Solitons and Fractals. 89: 578-583. Recio-Rodríguez, Elena; Gandarias-Núñez, María Luz; Bruzón-Gallego, María Santos. 2016. Symmetries and conservation laws for a sixth-order Boussinesq equation. Chaos, Solitons and Fractals. 89: 572-577. Garrido-Letran, Tamara Maria; Kasatkin, Alexei; Bruzón-Gallego, María Santos; Gazizov, Rafail K. 2016. Lie symmetries and equivalence transformations for the Barenblatt-Gilman model. Journal of Computational and Applied Mathematics. 318: 253-258. Ibragimov, Nail; Gandarias, Maria Luz; Galiakberova, L.r.; Bruzón-Gallego, María Santos; Avdonina, Elena. 2016. Group classification and conservation laws of anisotropic wave equations with a source. Journal Of Mathematical Physics. 57: 083504-1-083504-9. Tracina-, Rita; Bruzón-Gallego, María Santos; Gandarias-Núñez, María Luz. 2016. On the nonlinear self-adjointness of a class of fourth-order evolution equations. Applied mathematics and computation. 275: 299-304.

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