Strong solutions for the Alber equation and stability of unidirectional wave spectra

Agis Athanassoulis (Lead / Corresponding author), Gerassimos Athanassoulis, Mariya Ptashnyk, Themistoklis Sapsis

Research output: Contribution to journalArticle

Abstract

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 Lspace-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 Op1{1000q; these would be the prime breeding ground for rogue waves.
Original languageEnglish
Pages (from-to)703-737
Number of pages35
JournalKinetic and Related Models
Volume13
Issue number4
DOIs
Publication statusAccepted/In press - 4 Mar 2020

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Keywords

  • Wigner transform
  • Alber equation
  • Landau damping
  • modulation instability
  • Penrose condition
  • ocean wave spectra

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