Strong solutions for the Alber equation and stability of unidirectional wave spectra
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document.pdf
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Published version
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2.32 MB
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Author(s)
Athanassoulis, Agissilaos G.
Athanassoulis, Gerassimos A.
Ptashnyk, Mariya
Sapsis, Themistoklis
Date Issued
2020
Journal
Kinetic & Related Models
Publisher
American Institute of Mathematical Sciences (AIMS)
Version
Final published version
Abstract
© American Institute of Mathematical Sciences. The Alber equation is a moment equation for the nonlinear Schrodinger equation, formally used in ocean engineering to investigate the stability of stationary and homogeneous sea states in terms of their power spectra. In this work we present the first well-posedness theory for the Alber equation with the help of an appropriate equivalent reformulation. Moreover, we show linear Landau damping in the sense that, under a stability condition on the homogeneous background, any inhomogeneities disperse and decay in time. The proof exploits novel L2 space-time estimates to control the inhomogeneity and our result applies to any regular initial data (without a mean-zero restriction). Finally, the sufficient condition for stability is resolved, and the physical implications for ocean waves are discussed. Using a standard reference dataset (the \North Atlantic Scatter Diagram") it is found that the vast majority of sea states are stable, but modulationally unstable sea states do appear, with likelihood O(1/1000); these would be the prime breeding ground for rogue waves.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
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DOI of Published Version
https://doi.org/10.3934/KRM.2020024
