Near-resonant triad interactions and wave generation theory are investigated for continuously stratified fluids. Interaction equations are derived for spatially-varying wave trains under the inviscid Boussinesq approximation. Rotational effects are included, and properties of the underlying eigenvalue problem are explored. To facilitate a numerical study of the near-resonant interactions, numerical methods are developed and an analysis of wave generation on a periodic domain is performed. Numerical experiments using laboratory and ocean-scale parameters are conducted, and the simulations confirm the validity of the wave forcing theory. Interaction experiments demonstrate a strong tendency for waves to exhibit nonlinear behaviour. While resonant interactions are observed in the laboratory scale simulations, nonlinear steepening effects and the formation of solitary-like waves dominate the ocean-scale experiments. The results suggest that the weakly-nonlinear interaction theory is only appropriate in a limited parameter regime. The problem of analyzing forced wave equations on an infinite domain is also considered. Motivated by the results obtained on a periodic domain, asymptotic analysis is applied to three important wave equations. The method of steepest descents is used to determine the large-time behaviour for the linearized Korteweg-de Vries, Benjamin-Bona-Mahony, and internal gravity wave equations. The asymptotic results are compared with numerical experiments and found to agree to high precision.

Near-resonant triad interactions and wave generation theory are investigated for continuously stratified fluids. Interaction equations are derived for spatially-varying wave trains under the inviscid Boussinesq approximation. Rotational effects are included, and properties of the underlying eigenvalue problem are explored. To facilitate a numerical study of the near-resonant interactions, numerical methods are developed and an analysis of wave generation on a periodic domain is performed. Numerical experiments using laboratory and ocean-scale parameters are conducted, and the simulations confirm the validity of the wave forcing theory. Interaction experiments demonstrate a strong tendency for waves to exhibit nonlinear behaviour. While resonant interactions are observed in the laboratory scale simulations, nonlinear steepening effects and the formation of solitary-like waves dominate the ocean-scale experiments. The results suggest that the weakly-nonlinear interaction theory is only appropriate in a limited parameter regime. The problem of analyzing forced wave equations on an infinite domain is also considered. Motivated by the results obtained on a periodic domain, asymptotic analysis is applied to three important wave equations. The method of steepest descents is used to determine the large-time behaviour for the linearized Korteweg-de Vries, Benjamin-Bona-Mahony, and internal gravity wave equations. The asymptotic results are compared with numerical experiments and found to agree to high precision.

Neste trabalho estudamos o valor mínimo da entropia topológica para uma classe de aplicações isotópicas à identidade em superfícies orientáveis (sem bordo, não necessariamente compactas e possivelmente de tipo finito) sob um ponto de vista estritamente topológico. Este estudo é feito utilizando a nova teoria de forcing para trajetórias transversas de Le Calvez e Tal que se baseia na teoria de Brouwer equivariante, em que é possível folhear superfícies com folhas relacionadas a teoria de Brouwer no plano. O principal resultado deste trabalho é uma melhora na estimativa da entropia topológica obtida por Le Calvez e Tal em um recente trabalho em que os autores buscam ferraduras topológicas em superfícies orientáveis utilizando ferramentas similares apresentadas aqui. Uma aplicação deste resultado acima é feita utilizando aplicações em S^2 que possuam um ponto fixo cuja trajetória pela isotopia deste ponto não seja homotópica a um múltiplo de um loop simples. Com estas hipóteses, melhoramos a estimativa dada por Le Calvez e Tal em que é encontrado um valor mínimo estritamente positivo para a entropia topológica desta aplicação. / In this work we study the minimum topological entropy value for one class of maps isotopics to the identity in oriented surfaces (without border, not necessary compacts and possibly of finite type) under the point of view strictly topological. This study is done using the new forcing theory to transverse trajectories from Le Calvez and Tal which it is based to equivariant Brouwer Theory, on what it is possible to leaf surfaces with leaves related to plane Brouwer theory. The main result in this work is a improvement in the estimates from the topological entropy obtained by Le Calvez and Tal in one recent work where the authors seek topological horseshoes on oriented surfaces using tools very similar to that are shown here. One application of the above result is done using maps on S^2 that have a fixed point whose trajectory by the isotopy of this point do not be homotopic to a multiple of a simple loop. With these hypotheses, we improve the estimates given by Le Calvez and Tal on what is found a strictly positive minimum value to the topological entropy of this map.