Strong solutions for the Alber equation and stability of unidirectional wave spectra
The Alber equation is a moment equation for the nonlinear Schrödinger 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 wit...
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ftdatacite:10.48550/arxiv.1808.05191 2023-05-15T17:33:10+02:00 Strong solutions for the Alber equation and stability of unidirectional wave spectra Athanassoulis, Agissilaos G. Athanassoulis, Gerassimos A. Ptashnyk, Mariya Sapsis, Themistoklis 2018 https://dx.doi.org/10.48550/arxiv.1808.05191 https://arxiv.org/abs/1808.05191 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Mathematical Physics math-ph FOS Physical sciences Primary 35Q35, 35B35, Secondary 81S30 Preprint Article article CreativeWork 2018 ftdatacite https://doi.org/10.48550/arxiv.1808.05191 2022-04-01T09:30:31Z The Alber equation is a moment equation for the nonlinear Schrödinger 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 $L^2$ 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. Report North Atlantic DataCite Metadata Store (German National Library of Science and Technology) |
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Mathematical Physics math-ph FOS Physical sciences Primary 35Q35, 35B35, Secondary 81S30 |
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Mathematical Physics math-ph FOS Physical sciences Primary 35Q35, 35B35, Secondary 81S30 Athanassoulis, Agissilaos G. Athanassoulis, Gerassimos A. Ptashnyk, Mariya Sapsis, Themistoklis Strong solutions for the Alber equation and stability of unidirectional wave spectra |
topic_facet |
Mathematical Physics math-ph FOS Physical sciences Primary 35Q35, 35B35, Secondary 81S30 |
description |
The Alber equation is a moment equation for the nonlinear Schrödinger 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 $L^2$ 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. |
format |
Report |
author |
Athanassoulis, Agissilaos G. Athanassoulis, Gerassimos A. Ptashnyk, Mariya Sapsis, Themistoklis |
author_facet |
Athanassoulis, Agissilaos G. Athanassoulis, Gerassimos A. Ptashnyk, Mariya Sapsis, Themistoklis |
author_sort |
Athanassoulis, Agissilaos G. |
title |
Strong solutions for the Alber equation and stability of unidirectional wave spectra |
title_short |
Strong solutions for the Alber equation and stability of unidirectional wave spectra |
title_full |
Strong solutions for the Alber equation and stability of unidirectional wave spectra |
title_fullStr |
Strong solutions for the Alber equation and stability of unidirectional wave spectra |
title_full_unstemmed |
Strong solutions for the Alber equation and stability of unidirectional wave spectra |
title_sort |
strong solutions for the alber equation and stability of unidirectional wave spectra |
publisher |
arXiv |
publishDate |
2018 |
url |
https://dx.doi.org/10.48550/arxiv.1808.05191 https://arxiv.org/abs/1808.05191 |
genre |
North Atlantic |
genre_facet |
North Atlantic |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.1808.05191 |
_version_ |
1766131582017470464 |