Plenary Lecture

Exact Solutions for a Model Equations for Long Waves in Nonlinear Dispersive Systems

Professor Maria S. Bruzón
University of Cádiz

Abstract: Benjamin, Bona, Mahony studied several topics concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular kind. It is a well-known equation of Korteweg-de Vries which arises in the theory of shallow water waves and it is an example of the propagation of weakly dispersive and weakly nonlinear waves in many physical systems. It has been used to account adequately for observable phenomena such as the interaction of solitary waves and dissipationless, undular shocks. 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. 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. 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 and, in addition, the existence of an infinite hierarchy of local conservation laws of a partial differential equation is a strong indicator of its integrability, a complete classification of all local low-order conservation laws can be derived by using the multiplier method. We also derive all low-order conservation laws for the BBM-like equation by using the multiplier method.

Brief Biography of the Speaker: Maria Santos Bruzón carried out her studies at the University of Seville, Spain. She is a Full Professor of the Department of Mathematics of the University of Cadiz. Her current research interests include group analysis, methods of group transformations: classical 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 80) as well as congress proceedings and abstracts.

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