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Whitham modulation theory for the Ostrovsky equation.

A J Whitfield1, E R Johnson1

  • 1Department of Mathematics , University College London , London WC1E 6BT, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|March 8, 2017
PubMed
Summary
This summary is machine-generated.

Researchers analyzed Ostrovsky equation wavepackets using Whitham modulation equations. They found new solutions for anomalous dispersion and described unsteady wavepacket formation for normal dispersion, revealing insights into oceanic and sheared flow dynamics.

Keywords:
Ostrovsky equationmodulation equationswavepackets

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Area of Science:

  • Fluid Dynamics
  • Nonlinear Wave Phenomena
  • Mathematical Physics

Background:

  • The Ostrovsky equation models weakly nonlinear, long waves in rotating fluids.
  • Understanding localized wavepacket solutions is crucial for oceanic and geophysical flows.
  • Previous studies have explored solitary and cnoidal wave solutions, but localized wavepacket dynamics require further investigation.

Purpose of the Study:

  • To derive and apply Whitham modulation equations for the Ostrovsky equation.
  • To analyze localized cnoidal wavepacket solutions in both normal and anomalous dispersion regimes.
  • To investigate the formation and characteristics of these wavepackets from initial solitary wave conditions.

Main Methods:

  • Derivation of Whitham modulation equations for the Ostrovsky equation.
  • Analysis of wavepacket solutions in weak rotation limit for two dispersion regimes.
  • Presentation of new steady, symmetric cnoidal wavepacket solutions for anomalous dispersion.
  • Description of unsteady finite-amplitude wavepacket solutions for normal dispersion.

Main Results:

  • A new steady, symmetric cnoidal wavepacket solution is presented for the Ostrovsky equation with anomalous dispersion.
  • Analytical solutions for the outer regions of these wavepackets are provided for both dispersion regimes.
  • The modulation equations describe the formation of unsteady wavepackets from Korteweg-de Vries solitary waves under normal dispersion.
  • Wavepacket formation is rapid in anomalous dispersion and slower in normal dispersion.

Conclusions:

  • The derived Whitham modulation equations effectively describe localized cnoidal wavepacket solutions of the Ostrovsky equation.
  • The study reveals distinct behaviors and solutions for wavepacket dynamics in normal and anomalous dispersion regimes.
  • Results suggest that unsteady wavepackets are localized, modulated cnoidal wavetrains, offering insights into complex fluid dynamics.