A study of shallow-water waves

University Of Texas At Arlington, Arlington TX

Investigators

Abstract

Liu DMS-1207840 The investigator studies problems of wave breaking and the stability of peaked solitary waves in mathematical models of nonlinear shallow-water waves, particularly the Camassa-Holm (CH) equation, the Degasperis-Procesi (DP) equation, and the two-component Camassa-Holm (CH2) systems. Shallow-water waves are defined as waves whose wavelengths are far greater than the water depth. Nonlinear wave interactions occur when waves, moving possibly with different speeds and in different directions, intersect. These equations and systems have permanent waves and breaking waves: the wave profile remains bounded, but its slope becomes unbounded in finite time. The CH and DP equations also contain peaked solitary waves, as these wave forms replicate a feature characteristic of the traveling wave solutions to the governing equations for largest possible amplitude. This project provides a greater understanding of wave breaking phenomena and contributes to the extensive modeling of shallow-water waves. Water waves lie at the forefront of modern applied mathematics and theoretical physics. The study of water wave phenomena has been a rich source of mathematical theories for over 200 years and affects a variety of issues pertaining to the ocean and environment. Breaking waves, both whitecaps and surf, are commonly observed in the ocean, but surprisingly, little is known about them. Yet they are important for a variety of reasons. They place large hydrodynamic loads on man-made structures, transfer horizontal momentum to surface currents, provide a source of turbulent energy to mix the upper layers of the ocean, and move sediment in shallow water. The investigator studies mathematical models of nonlinear shallow-water waves, particularly the CH and DP equations as well as the CH2 systems. The CH-type equations might also be relevant to the modeling of an important shallow-water breaking wave phenomenon, the tsunami. A tsunami wave is generated when a large body of water, such as a region in a lake or a sea, becomes rapidly displaced on a massive scale. With typical wavelengths of 200 km, tsunamis are governed by shallow water equations and can be catastrophic when they reach land, as seen in the recent Indonesia and Japan earthquakes. This project seeks further understanding of the dynamics of shallow-water wave breaking, especially before and after breaking has occurred when the flow is steady and more amenable to theoretical and numerical study. In particular, the investigator conducts mathematical analysis pertinent to the modeling of tsunami waves, which in turn helps better predict and understand the waves' characteristics.