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Wave-packet formation at the zero-dispersion point in the Gardner-Ostrovsky equation

A. J. Whitfield and E. R. Johnson
Phys. Rev. E 91, 051201(R) – Published 18 May 2015

Abstract

The long-time effect of weak rotation on an internal solitary wave is the decay into inertia-gravity waves and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. There is currently no entirely satisfactory explanation as to why these wave packets form. Here the initial value problem is considered within the context of the Gardner-Ostrovsky, or rotation-modified extended Korteweg–de Vries, equation. The linear Gardner-Ostrovsky equation has maximum group velocity at a critical wave number, often called the zero-dispersion point. It is found here that a nonlinear splitting of the wave-number spectrum at the zero-dispersion point, where energy is shifted into the modulationally unstable regime of the Gardner-Ostrovsky equation, is responsible for the wave-packet formation. Numerical comparisons of the decay of a solitary wave in the Gardner-Ostrovsky equation and a derived nonlinear Schrödinger equation at the zero-dispersion point are used to confirm the spectral splitting.

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  • Received 18 March 2015

DOI:https://doi.org/10.1103/PhysRevE.91.051201

©2015 American Physical Society

Authors & Affiliations

A. J. Whitfield* and E. R. Johnson

  • Department of Mathematics, University College London, London WC1E 6BT, United Kingdom

  • *ashley.whitfield.12@ucl.ac.uk
  • e.johnson@ucl.ac.uk

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Issue

Vol. 91, Iss. 5 — May 2015

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