Water Waves with Weak Dispersion and Nonlinearity

This research proposal is directed at the study of surface waves on bodies of water, such as oceans rivers and lakes. In order to understand the motion of surface waves on water, in principle the motion of the whole body of fluid has to be computed. Howev er, there are simplifying assumptions, such as assuming that the governing differential equations are linear which yield simplified model equations which can be solved numerically in a reasonable amount of time. One particular set of simplifying assumptio ns is the so-called Boussinesq approximation, where the waves are supposed to have long wavelength, and small amplitude. The resulting set of equations is not linear, but can still be handled numerically without resorting to supercomputers. The particula r aim of this study is to understand the influence of weak nonlinearity and weak dispersion on the properties of model equations for surface water waves, when the assumption of long wavelength is relaxed. It will be a further objective of this study to in vestigate the influence of transverse effects, leading to short-crested waves, and long-wave / short-wave interactions. We hope to be able to combine the expertise of the applicant and the experience of the fluid mechanics group at the mathematics depart ment at the University of Bergen, which has a strong record of producing Ph.D. students and is internationally reknowned, to obtain new insights into models and model properties for weakly transverse long-wave-short-wave interactions as well as weakly no nlinear models.