Modeling rogue waves in deep water.

Abstract: Rogue waves are isolated large surface ocean waves that can have destructive effects. In this talk I will discuss rogue waves in the framework of the nonlinear Schrodinger (NLS) equation and correlate their development in oceanic sea states characterized by the Joint North Sea Wave Project spectrum (JONSWAP) with the proximity to homoclinic solutions of the NLS equation. (These are "large" amplitude solutions which approach a lower amplitude background wave in the limit of large time, positive and negative.)

The machinery of inverse spectral theory for the NLS equation associates to each solution the spectrum of a certain linear operator (known as the AKNS system). I will introduce the "splitting distance" between simple points in the discrete spectrum of the AKNS system, and use it to determine the proximity in spectral space to instabilities of solutions of the NLS equation and their homoclinic orbits.

The results of several hundred simulations, where the parameters and phases in the JONSWAP initial data are varied, indicate that rogue waves develop whenever the splitting distance is small, and do not when the splitting distance is large.