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Dispersive shock wave theory for nonintegrable equations.

A M Kamchatnov1

  • 1Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow 108840, Russia and Moscow Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny, Moscow region 141700, Russia.

Physical Review. E
|February 21, 2019
PubMed
Summary
This summary is machine-generated.

This study presents a new method for calculating dispersive shock wave parameters using Whitham modulation theory for nonintegrable wave equations. The approach offers analytic expressions for wave propagation, confirmed by established integrable models.

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Area of Science:

  • Nonlinear physics
  • Wave phenomena
  • Mathematical modeling

Background:

  • Dispersive shock waves are crucial in nonlinear systems.
  • Whitham modulation theory provides a framework for analyzing wave evolution.
  • Nonintegrable wave equations present challenges for analytical solutions.

Purpose of the Study:

  • To develop a method for calculating dispersive shock wave parameters in nonintegrable wave equations.
  • To apply Whitham modulation theory to a broad class of initial conditions.
  • To derive analytic expressions for wave propagation dynamics.

Main Methods:

  • Utilizing Whitham's "number of waves conservation law".
  • Extending Whitham theory to nonintegrable systems by comparing with integrable counterparts.
  • Calculating limiting characteristic velocities at the pulse boundary.

Main Results:

  • The proposed method allows for the calculation of dispersive shock wave parameters.
  • Explicit analytic expressions for the laws of motion of wave edges were obtained.
  • The method's validity was confirmed through applications to Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations.

Conclusions:

  • The developed method is effective for analyzing dispersive shock waves in nonintegrable systems.
  • The approach provides a powerful tool for theoretical and numerical studies of wave propagation.
  • This work bridges the gap between integrable and nonintegrable wave theories.