Many physical systems admit nonlinear, dispersive waves, for which the speed of propagation depends on both the frequency and amplitude of the wave. These systems are usually dissipative, but the dissipation might be weak. This proposal focusses on nonlinear dispersive waves in the presence of weak dissipation, with special emphasis on surface water waves. Recent work by the PIs of this proposal and others has shown that even weak dissipation can profoundly affect the stability of nonlinear dispersive waves in certain situations. The overall theme of this proposal is to study nonlinear dispersive waves in the presence of weak dissipation. We will use a state-of-the-art laboratory facility to conduct experiments on waves on deep water. These experiments will guide the development of mathematical models and test their validity. Questions we will explore include (a) How does weak dissipation affect the stability and properties of surface wave patterns that are more complicated than ordinary plane waves? (b) What causes the downshifting of the peak of a narrow-banded spectrum of nonlinear, dispersive waves, as observed in experiments on water waves and in optics? (c) Does this theoretical and laboratory work apply to actual ocean waves? (d) What causes the dissipation in surface water waves?
Particular goals of this research include obtaining better predictive models of the dynamics of large-amplitude ocean waves, and of exchange processes that occur at the ocean-atmosphere boundary. Accurate models of these exchange processes will be essential in predicting global warming and related global phenomena. In addition, accurate models of wave dynamics might prove useful in understanding extreme waves such as rogue waves. Much of the work to be conducted is fundamental in nature and the mathematics is shared by many physical systems, so the results obtained for water waves could affect our understanding of many physical systems. These other systems include light waves in an optical fiber, spin waves in a thin magnetic film and others.
