Nonlinear Wave Problems in Fluid Flows

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0071939MilewskiThe project will consist of the study of three problems in fluid mechanics and nonlinear waves. The first project involves understanding certain aspects of dispersive wave turbulence, that is, the statistical description of a large number of interacting dispersive waves, such as those on the ocean surface. First, a reduced model will be used which contains the fundamental nonlinear processes and can yield the scaling for the energy transfer mechanisms. Second, spectra of two-- and three--dimensional ocean waves with a reduced equation valid for finite depth and deep water will be computed and compared with results from the reduced model. The second project involves the study of three-dimensional solitary waves in regimes where surface tension is an important part of the dynamics. These are waves that can be generated, for example, by flow of a thin fluid layer over a small obstacle. Here, it is proposed to use solutions that have already been computed to find additional solutions in regimes of physical interest, such as increasing depth. The third project is to study the dynamics of reaction-diffusion equations in the presence of spatial inhomogeneities, as for example, in models of certain chemical reactions where the reactant concentration is not uniform in space. In the spatially homogeneous case, one obtains various coherent patterns in the reaction. How these patterns and their boundaries are modified by the inhomogeneities will be studied.The goal of this research is to understand several aspects of wave dynamics in fluids using a combination of theory and advanced computation. There are three distinct phenomena that will be studied. First, the evolution of wave turbulence will be studied: the physical situation in which many waves of different wavelengths and traveling in different directions are superposed. The simplest example is the apparent random mix of waves on the surface of the ocean. The goal is to predict the relative energy in the different waves and the mechanisms by which waves of different sizes exchange energy. These are important predictions whose applications range from understanding satellite remote sensing data to climate dynamics. Second, a class of water waves called lump solitons will be studied: localized coherent waves that travel in a particular direction. The goal is to obtain the range of physical situations in which these waves can exist. This work has implications in a variety of thin film and coating applications. Lastly, the dynamics of the components of biological and chemical reacting systems where the concentration of the reactants vary in space will be studied. The particular case where a catalyst for the reaction is not distributed uniformly and therefore the reaction proceeds differently in different places will be studied. The goal is to understand how the reaction varies from place to place and what happens at the boundaries where the reactions change character.