Nonlinear Wave Interaction in Coastal and Open Seas: Deterministic and Stochastic Theory

Raphael Stuhlmeier*, Teodor Vrecica, Yaron Toledo

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


We review the theory of wave interaction in finite and infinite depth. Both of these strands of water-wave research begin with the deterministic governing equations for water waves, from which simplified equations can be derived to model situations of interest, such as the mild slope and modified mild slope equations, the Zakharov equation, or the nonlinear Schrödinger equation. These deterministic equations yield accompanying stochastic equations for averaged quantities of the sea-state, like the spectrum or bispectrum. We discuss several of these in depth, touching on recent results about the stability of open ocean spectra to inhomogeneous disturbances, as well as new stochastic equations for the nearshore.

Original languageEnglish
Title of host publicationNonlinear Water Waves
Subtitle of host publicationAn Interdisciplinary Interface
EditorsDavid Henry, Konstantinos Kalimeris, Emilian I. Părău, Jean-Marc Vanden-Broeck, Erik Wahlén
Place of PublicationCham
Number of pages31
ISBN (Electronic)978-3-030-33536-6
ISBN (Print)978-3-030-33535-9
StatePublished - 2019

Publication series

NameTutorials, Schools, and Workshops in the Mathematical Sciences
ISSN (Print)2522-0969
ISSN (Electronic)2522-0977


  • Deep water
  • Kinetic equations
  • Mild-slope equation
  • Nearshore
  • Nonlinear interaction
  • Nonlinear Schrödinger equation
  • Resonant interaction
  • Shoaling
  • Water waves
  • Wave forecasting
  • Zakharov equation


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