PDE Seminar (2019/5/31): Mathematical modelling of nonlinear waves

2019-5-31 16:30 - 2019-5-31 17:30
Faculty of Science Building #3, Room 309
Angela Slavova (Bulgarian Academy of Sciences)
The study of water waves involves various disciplines such as mathematics, physics and engineering and within this there are many specific areas of direct or associated interest such as pure mathematics, applied mathematics, modelling, numerical simulation, laboratory experiments, data collection in the field, the design and construction of ships, harbours, the prediction of natural disasters, climate studies and so on.
In this lecture we shall present travelling wave solutions of shallow water waves. Camassa-Holm considered a third order nonlinear PDE of two variables modelling the propagation of unidirectional irrotational shallow water waves over a flat bed, as well as water waves moving over an underlying shear flow. In the special case of the motion of a shallow water over a flat bottom the corresponding system was simplified by Green and Naghdi and related to an appropriate two component first order Camassa-Holm system. Another interesting system of nonlinear PDE is the viscoelastic generalization of Burger's equation. In the above mentioned systems we are looking for travelling wave solutions and we are studying their profiles. To do this we use several results from the classical Analysis of ODE that enable us to give the geometrical picture and in several cases to express the solutions by the inverse of Legendre's elliptic functions.
As an application we shall present propagation of tsunami waves from their small disturbance at the sea level to the size they reach approaching the coast. Even with the aid of the most advanced computers it is not possible to find the exact solutions to the nonlinear governing equations for water waves. For this purpose we introduce Cellular Nonlinear Network (CNN) approach.