Wave breaking is the main mechanism that dissipates energy input into ocean waves by wind and transferred across the spectrum by nonlinearity. It determines the properties of a sea state and plays a crucial role in ocean-atmosphere interaction, ocean pollution, and rogue waves. Owing to its turbulent nature, wave breaking remains too computationally demanding to solve using direct numerical simulations except in simple, short-duration circumstances. To overcome this challenge, we present a blended machine learning framework in which a physics-based nonlinear evolution model for deep-water, non-breaking waves and a recurrent neural network are combined to predict the evolution of breaking waves. We use wave tank measurements rather than simulations to provide training data and use a long short-term memory neural network to apply a finite-domain correction to the evolution model. Our blended machine learning framework gives excellent predictions of breaking and its effects on wave evolution, including for external data.

Stationary wave groups exist in a range of nonlinear dispersive media, including optics, Bose-Einstein condensates, plasma, and hydrodynamics. We report experimental observations of nonlinear surface gravity X waves, i.e., X-shaped wave envelopes that propagate over long distances with constant form. These can be described by the 2D+1 nonlinear Schrödinger equation, which predicts a balance between dispersion and diffraction when the envelope (the arms of the X) travel at ±arctan(1/2)≈±35.26° to the carrier wave. Our findings may help improve understanding the lifetime of extremes in directional seas and motivate further studies in other nonlinear dispersive media.

Solitons and breathers are nonlinear modes that exist in a wide range of physical systems. They are fundamental solutions of a number of nonlinear wave evolution equations, including the unidirectional nonlinear Schrödinger equation (NLSE). We report the observation of slanted solitons and breathers propagating at an angle with respect to the direction of propagation of the wave field. As the coherence is diagonal, the scale in the crest direction becomes finite; consequently, beam dynamics form. Spatiotemporal measurements of the water surface elevation are obtained by stereo-reconstructing the positions of the floating markers placed on a regular lattice and recorded with two synchronized high-speed cameras. Experimental results, based on the predictions obtained from the (2D + 1) hyperbolic NLSE equation, are in excellent agreement with the theory. Our study proves the existence of such unique and coherent wave packets and has serious implications for practical applications in optical sciences and physical oceanography. Moreover, unstable wave fields in this geometry may explain the formation of directional large-amplitude rogue waves with a finite crest length within a wide range of nonlinear dispersive media, such as Bose–Einstein condensates, solids, plasma, hydrodynamics, and optics.