Stability of periodic traveling flexural‐gravity waves in two dimensions

Olga Trichtchenko (Lead Author), Paul Milewski, Emilian Parau, Jean-Marc Vanden-Broeck

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In this work, we solve the Euler’s equations for periodic waves travelling under a sheet of ice using a reformulation introduced in [1]. These waves are referred to as flexural-gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrödinger equation that describes the modulational instability of periodic travelling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier-Floquet-Hill method to show they agree qualitatively. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyse high frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice.
Original languageEnglish
Pages (from-to)65-90
Number of pages26
JournalStudies in Applied Mathematics
Issue number1
Early online date12 Oct 2018
Publication statusPublished - Jan 2019


  • asymptotic analysis
  • nonlinear waves
  • numerical methods
  • stability of solutions
  • waves under ice

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