Rogue waves and large deviations for 2D pure gravity deep water waves
Massimiliano Berti, Ricardo Grande, Alberto Maspero, Gigliola Staffilani
TL;DR
This work rigorously quantifies the tail probability of rogue-wave formation for the 2D pure gravity deep-water equations with random Gaussian initial data. The authors develop a novel large-deviation framework for a quasilinear Hamiltonian PDE by fusing a deterministic paradifferential/Birkhoff normal-form analysis with probabilistic tail methods, and they crucially leverage a random fixed-point argument to achieve phase quasi-synchronization of low Fourier modes. They identify dispersive focusing as the leading formation mechanism up to optimal times, proving both short-time Gaussian-preserving approximations and long-time control via a refined approximate solution η_app2. The results establish sharp exponential decay rates for rogue-wave events and provide a general paradigm for tail-probability analysis of Hamiltonian PDEs without invariant measures, with potential implications for oceanography and nonlinear wave theory. The analysis hinges on rigorous normal-form transformations with Lipschitz properties, a detailed phase-space probabilistic treatment, and a delicate balance between deterministic energy estimates and stochastic dependence across very long times. The findings bridge oceanographic conjectures and mathematical theory by confirming dispersive focusing as the principal rogue-wave mechanism in the weakly nonlinear regime on realistically long timescales.
Abstract
Rogue waves are extreme ocean events characterized by the sudden formation of anomalously large crests, and remain an important subject of investigation in oceanography and mathematics. A central problem is to quantify the probability of their formation under random Gaussian sea initial data. In this work, we rigorously characterize the tail-probability for the formation of rogue waves of the pure gravity water wave equations in deep water, the most accurate quasilinear PDE modeling waves in open ocean. This large deviation result rigorously proves various conjectures from the oceanography literature in the weakly nonlinear regime. Moreover, the result holds up to the optimal timescales allowed by deterministic well-posedness theory. The proof shows that rogue waves most likely arise through "dispersive focusing", where phase quasi-synchronization produces constructive amplification of the water crest. The main difficulty in justifying this mechanism is propagating statistical information over such long timescales, which we overcome by combining normal forms and probabilistic methods. Unlike prior work, this novel approach does not require approximate solutions to be Gaussian. Our general method tracks the tail probability of solutions to Hamiltonian PDEs with an integrable normal form and random Gaussian initial data over very long times, even in the absence of (quasi-)invariant measures.