Mathematical Sciences: Analysis on Waves in Stratified Fluids of Infinite Depth

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9623060 Sun The purpose of this project is to present a mathematical and numerical study of solitary and periodic waves in stratified fluids of infinite depth subject to the gravitational force. The proposal consists of two parts. The first part intends to give a rigorous answer to an open question whether there exist solitary internal waves with algebraic decay at infinity in a continuously stratified fluid that is bounded only by a rigid bottom and has a constant density except for a layer with continuous density stratification. The proposed research will give a rigorous justification of the existence of solitary waves derived from a formal model equation, called the Benjamin-Ono equation. The second part deals with the existence of periodic waves of large amplitude in a two-layer fluid without boundaries. A new formulation of the problem will be introduced, which transforms the governing equations into a single integral equation. Then the solutions of the integral equation will be studied numerically and theoretically for any density ratio without restrictions on the amplitude of solutions. In this project, the main thrust is to show that the fully nonlinear governing equations for stratified fluids have solutions of finite and large amplitude and give the various properties of the solutions using numerical and theoretical approaches. An interplay of theories in differential equations and functional analysis will be essential to obtaining the rigorous justifications, while numerical computation gives some crucial information on the solutions. %%% The gravity waves of large amplitude in a fluid of density variation, also called a stratified fluid, with great depth are of considerable geophysical interest. Large amplitude internal wave disturbances are common features in the oceans as well as in the lower atmosphere. In particular, a solitary wave, whose form is a localized single hump, and a periodic wave, which always has a same form after a certain distance, are relevant to various oceanic and atmospherical phenomena. The solitary waves of large amplitude have been associated with the formation of tornados in the atmosphere and the enormous transport of momentum and energy within the oceans. This work focuses on obtaining the conditions under which the solitary or periodic waves can exist and predicting how large the amplitude of the waves can be if they exist. It will also try to capture the qualitative features of these waves using numerical computations. The results obtained from this research may provide some theoretical explanation of the formation of these waves and give more understanding of certain wave motions in the oceans and atmosphere so that these waves may be either utilized or avoided in different physical applications. ***