BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:The Landau-Alber bifurcation: implications for the analysis of met ocean data\, and open questions in Landau damping - Agis Athanassoulis (Un iversity of Dundee) DTSTART:20220909T093000Z DTEND:20220909T103000Z UID:TALK177866@talks.cam.ac.uk DESCRIPTION:It is famously known that plane wave solutions of the Nonlinea r Schrö\;dinger equation (NLS) are linearly unstable\; this fact is ca lled the modulation instability for the NLS. While recent breakthorughs ha ve provided insights on the qualitative aspects of nonlinear evolution of the modulation instability\, many fundamental questions (such as global ex istence of solutions for general initial perturbations) are still open. Th e NLS is used as an approximate model in ocean waves\, where the modulatio n instability is also known as the Benjamin-Feir instability\, and is link ed with concrete physical phenomena including rogue waves. In this context of ocean engineering\, linear sea states are often used as approximate st ochastic solutions of the NLS\, as only stochastic wavefileds -- and not p lane waves -- are relevant for marine safety applications. \;\nThe mai n points of this talk are the following:\n\nThere exists a 2nd moment theo ry of stochastic wavefileds under the NLS equation that is compatible with the bulk of the metocean data available\, namely the Alber equation. This allows concepts from the NLS to be generalised to a stochastic setting ye t remain tractable. The existing well-posedness theory for the Alber equat ion will be briefly reviewed [1].\nIn the stochastic setting\, the instabi lity of the homogeneous solution (i.e. 'generalised modulation instability ') is not the only possibility. In fact\, many homogeneous sea states will give rise to Landau damping\, unexpectedly stabilising themselves despite the presence of infinite energy and a focusing nonlinearity. Still\, unst able sea states do exist. So a bifurcation between Landau damping and gene ralised modulation instability exists within measured sea states [1-4]. Si nce I. E. Alber was the one who initiated this line of work\, we call this the Landau-Alber bifurcation. The rigorous result on the Landau damping w ill be briefly outlined\, along with its proof which builds crucially on a n esoteric property of the Hilbert transform [1].\nThe linear stability an alysis can surprisingly predict both profiles of rogue waves in the unstab le case\, and the likelihood of extreme events in the stable case. These f indings are in excellent agreement with empirical facts in the ocean engin eering community [2\,5].\n\nA number of open questions will be discussed i n the end.\nIncludes joint work with T. Sapsis (MIT)\, G. Athanassoulis (N TUA)\, M. Ptashnyk (Heriot-Watt) and O. Gramstad (DNV).\nReferences:[1]. A thanassoulis\, Agissilaos G.\, et al. "Strong solutions for the Alber equa tion and stability of unidirectional wave spectra." Kinetic & Related Mode ls \;13.4 (2020).[2]. Athanassoulis\, Agissilaos G.\, and Odin Gramsta d. "Modelling of Ocean Waves with the Alber Equation: Application to Non-P arametric Spectra and Generalisation to Crossing Seas." Fluids \;6.8 ( 2021): 291.[3]. Ribal\, A.\, et al. "Recurrent solutions of the Alber equa tion initialized by Joint North Sea Wave Project spectra." Journal of Flui d Mechanics \;719 (2013): 314-344.[4]. Alber\, I. E. "The effects of r andomness on the stability of two-dimensional surface wavetrains." Proceed ings of the Royal Society of London. A. Mathematical and Physical Sciences  \;363.1715 (1978): 525-546.[5]. Dematteis\, Giovanni\, Tobias Grafke\ , and Eric Vanden-Eijnden. "Rogue waves and large deviations in deep sea."  \;Proceedings of the National Academy of Sciences 115.5 (2018): 855-8 60. LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR