Rogue waves and large deviations in nonlinear Schroedinger models

Rogue waves and large deviations in nonlinear Schroedinger models

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Eric Vanden-Eijnden, Courant Institute of Mathematical Sciences, New York University
Fine Hall 322

The appearance of rogue waves in deep sea is investigated using the modified nonlinear Schroedinger (MNLS) equation with random initial conditions that are assumed to be Gaussian distributed, with a spectrum approximating the JONSWAP spectrum obtained from observations of the North Sea. It is shown that by supplementing the incomplete information contained in the JONSWAP spectrum with the MNLS dynamics one can reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. This is achieved by identifying ocean states that are precursors to rogue waves, which also permits their early detection. Our findings indicate that rogue waves in MNLS obey a large deviation principle—i.e., they are dominated by single realizations—which we calculate by solving an optimization problem.  This method generalizes to estimate the probability of extreme events in other deterministic dynamical systems with random initial data and/or parameters, by using prior information about the nature of their statistics.