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Modulational Instability in Equations of KdV Type

  • Jared C. Bronski
  • Vera Mikyoung Hur
  • Mathew A. JohnsonEmail author
Part of the Lecture Notes in Physics book series (LNP, volume 908)

Abstract

It is a matter of experience that nonlinear waves in a dispersive medium, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics—the amplitude, the phase, the wave number, etc.—slowly vary in large space and time scales. In the 1960s, Whitham developed an asymptotic (WKB) method to study the effects of small “modulations” on nonlinear dispersive waves. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham’s formal theory. We discuss some recent advances in the mathematical understanding of the dynamics, in particular, the instability, of slowly modulated waves for equations of KdV type.

Keywords

Dispersion Relation Solitary Wave Modulational Instability Modulational Stability Cnoidal Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

JCB is supported by the National Science Foundation grant DMS-1211364. VMH is supported by the National Science Foundation grant CAREER DMS-1352597 and an Alfred P. Sloan Foundation Fellowship. MAJ is supported by the National Science Foundation grant DMS-1211183.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Jared C. Bronski
    • 1
  • Vera Mikyoung Hur
    • 1
  • Mathew A. Johnson
    • 2
    Email author
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA

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