The work supported under this grant will address problems in Nonlinear Wave Dynamics, which will be studied using a combination of analytic and numerical methods. For example: (I) Nonlinear interactions of large scale atmospheric waves in the Equator, which play a fundamental role in the global weather. A new mechanism for nonlinear interactions of the waves via topography and convected properties (e.g.: potential vorticity, temperature and humidity) will be investigated. (II) Interaction of nonlinear hyperbolic waves with media nonuniformities. These waves distort as they propagate, leading to the formation of shocks, which are strongly dissipative. Past research shows that nonuniformities or interactions with other waves can stop this process. For example, a large class of new solutions to the Euler equations of Gas Dynamics was found and will be studied. These solutions are attracting for the time evolution and can achieve large pressure variations without breaking and forming shocks. Their existence gives rise to interesting mathematical issues. A better understanding of this phenomenon could provide means to control (or at least diminish) the influence of shocks in some gas flows. (III) Focusing, reflection and refraction of weak shocks. Strong acoustical waves and weak shocks are ubiquitous. A better understanding of their behavior has considerable scientific significance and will shed light into many important practical problems. Many puzzling problems remain in this area. For example, in many situations experiments show three weak shock waves joining at a point in a nearly flat configuration, but theoretical calculations indicate that this is not possible, leading to a contradiction with the experiments. A second related problem occurs with the prediction of infinities in the amplitudes of weak shock waves as they undergo focusing, reflection or refraction under appropriate conditions.
Waves occur in many fields --- such as Acoustics, Atmosphere and Ocean science, Gas Dynamics, Optics, Combustion Theory, etc. --- where they fundamentally affect the dynamics. Many advances have been made in the last few decades, but many open questions remain, both concerning the applications and the mathematics that is used. This award will support research on mathematical models for wave-like phenomena in the global weather, processes in gas dynamics that avoid the formation of shock waves, and the mathematical modeling of complex shock wave interaction.
