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Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg-de Vries equation.

Proceedings. Mathematical, physical, and engineering sciences·2017
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Dispersive dynamics in the characteristic moving frame.

D J Ratliff1

  • 1Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|April 23, 2019
PubMed
Summary
This summary is machine-generated.

Dispersion arises from dispersionless Whitham Modulation equations (WMEs) using a moving frame. This transforms WMEs into the Korteweg-de Vries (KdV) equation with universal coefficients, demonstrated in shallow water and Klein-Gordon systems.

Keywords:
Lagrangian dynamicsWhitham modulationnonlinear waves

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Area of Science:

  • Nonlinear dynamics
  • Mathematical physics

Background:

  • The Whitham Modulation equations (WMEs) traditionally describe wave phenomena without dispersion.
  • Investigating mechanisms for dispersion to emerge from dispersionless systems is crucial for understanding complex wave behaviors.

Purpose of the Study:

  • To present a novel mechanism for generating dispersion from dispersionless Whitham Modulation equations.
  • To demonstrate the transformation of WMEs into the Korteweg-de Vries (KdV) equation.

Main Methods:

  • Employing a moving frame transformation.
  • Linearizing the Whitham system to identify characteristic speeds.
  • Assuming real characteristics to ensure hyperbolicity of WMEs.

Main Results:

  • A method is presented for dispersion to naturally arise from dispersionless WMEs.
  • WMEs are shown to morph into the Korteweg-de Vries (KdV) equation in a boosted coordinate system.
  • The coefficients of the resulting KdV equation are universal, determined by the original Lagrangian density.

Conclusions:

  • The study provides a theoretical framework for deriving the KdV equation from dispersionless systems.
  • The universality of KdV coefficients highlights fundamental properties of the underlying physical system.
  • The method is illustrated with applications to shallow water flows and a complex Klein-Gordon system.