2018/08/15 by Agissilaos Athanassoulis, G. A. Athanassoulis, Athanassoulis, Agissilaos G. +5 · 1 citation
Earth and Planetary Sciences · #35B35 #Coastal and Marine Dynamics #FOS: Physical sciences #Mathematical Physics (math-ph) #Ocean Waves and Remote Sensing #Oceanographic and Atmospheric Processes #Primary: 35Q35 #Secondary: 81S30
paper · pdf · doi:10.48550/arxiv.1808.05191
openalex publication_date 2018/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Alber equation is a moment equation for the nonlinear Schr "odinger\nequation, formally used in ocean engineering to investigate the stability of\nstationary and homogeneous sea states in terms of their power spectra. In this\nwork we present the first well-posedness theory for the Alber equation with the\nhelp of an appropriate equivalent reformulation. Moreover, we show linear\nLandau damping in the sense that, under a stability condition on the\nhomogeneous background, any inhomogeneities disperse and decay in time. The\nproof exploits novel L2 space-time estimates to control the inhomogeneity\nand our result applies to any regular initial data (without a mean-zero\nrestriction). Finally, the sufficient condition for stability is resolved, and\nthe physical implications for ocean waves are discussed. Using a standard\nreference dataset (the "North Atlantic Scatter Diagram") it is found that the\nvast majority of sea states are stable, but modulationally unstable sea states\ndo appear, with likelihood O(1/1000); these would be the prime breeding\nground for rogue waves.\n