Waves in periodic structures have attracted a lot of attention in recent years. Such media show interesting macroscopic properties, which may be quite different from those of the individual fluids which constitute the stratified system. On a macroscopic scale, such a system can be considered a sort of fluid metamaterial. Such waves present a peculiar, somehow unexpected, behavior. For example, there is evidence that in spite of the fact that the waves are governed by genuinely quasi-linear hyperbolic system, they do not break and form shocks. As specific case, we study the propagation of small amplitude waves in shallow water over a periodic bathymetry [1]. We assume the wavelength is much larger than the period of the bathymetry. It is shown that an initial pulse of small amplitude will not produce a shock. Indeed, after a long time, an initial Gaussian pulse splits into several waves of various amplitude. The phenomenon is not related to the dispersive waves generated from deep water effect and is explained in terms of dispersive waves satisfying a model system obtained from the original one by suitable asymptotic expansion. The solution to the model system is in good agreement with the detailed numerical solution of the full SW system, agreement that improves with the number of terms in the expansion. Linear stability analysis is adopted to select the more suitable form of the dispersive model. Traveling waves are computed, and compared with the waves that emerge from the detailed dynamics.

Reference
[1] David I. Ketcheson, Lajos L′oczi, Giovanni Russo, An effective medium equation for weakly nonlinear shallow water waves over periodic bathymetry, MMS, submitted.

Professor Giovanni Russo obtained his master's degree in Nuclear Engineering Magna cum Laude at the Polytechnic University of Milano, Italy, in 1982. He obtained a PhD in Physics at the University of Catania in 1987, under the guidance of Prof. Marcello Anile. He spent three years as a Visiting Research Scientist at the Courant Institute in New York. In 1990 he became Assistant Professor in Mathematical Physics at the University of L' Aquila, Italy, where he became associate professor in Numerical Analysis in 1992. Since 2000 he is full professor of Numerical Analysis at the University of Catania, Sicily, Italy. His main research interests cover numerical methods for PDE's, with particular attention to hyperbolic and kinetic problems, multi-scale methods, and various problems related to computational fluid dynamics. His current interest includes asymptotic preserving schemes, ghost fluid methods and multi-scale modelling of sorption kinetics. He has more than one hundred scientific publications in international journals on various topics in applied and computational mathematics, and some books. He has been coordinator of various PhD programs in mathematical subjects for more than 16 years, has been responsible of various national and international research projects, and has been visiting professor of several international institutions. He acts as a member of the Editorial Board of the several journals including: SIAM Journal of Mathematical Analysis, Communication of Mathematical Sciences, Journal of Mathematics in Industry.