Understanding Nonlinear Surface Water Waves on Deep Water

Oceanographers in the 60s conducted an ambitious experiment in which they tracked waves that were generated by large storms near New Zealand across the Pacific Ocean until they hit the beaches at Alaska. Paradoxically, at about the same time, mathematicians in the Soviet Union, the U.S., and England separately developed mathematical models that predicted such waves to be unstable, meaning that they could not survive to be tracked all the way across the Pacific. In the 70s experimentalists conducted laboratory experiments on these types of waves. They generated waves with a given frequency that propagated down a wavetank, but at the end of the wavetank, the waves had a slightly lower frequency. The mathematical model did not explain this observation. In this talk, we consider these observations and more recent ones within the framework of the mathematical model, the scalar and vector nonlinear Schroedinger equations. These partial differential equations are examples of integrable systems; they also model phenomena observed in optics and in plasmas. They have special mathematical properties, some of which we use to determine when they are adequate models of water waves. We use the field and laboratory experiments to guide variations of the models with the goal of accurately predicting the observations.