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The zero dispersion limit for the Korteweg-deVries KdV equation.

P D Lax1, C D Levermore

  • 1Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012.

Proceedings of the National Academy of Sciences of the United States of America
|August 1, 1979
PubMed
Summary

This study uses the inverse scattering method to find the weak limit of Korteweg-deVries equation solutions as dispersion approaches zero. The results are valid for all time and connect to quadratic programming and averaged equations.

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Area of Science:

  • Nonlinear partial differential equations
  • Mathematical physics
  • Fluid dynamics

Background:

  • The Korteweg-deVries (KdV) equation models phenomena like shallow water waves.
  • Understanding the behavior of KdV solutions as dispersion becomes negligible is crucial.
  • Previous studies have explored approximations for KdV dynamics.

Purpose of the Study:

  • To determine the weak limit of Korteweg-deVries equation solutions as dispersion tends to zero.
  • To characterize this limit using a solvable mathematical problem.
  • To analyze the long-time behavior of these solutions.

Main Methods:

  • Application of the inverse scattering method.
  • Characterization of the weak limit via a quadratic programming problem.

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  • Utilizing function theoretic methods for problem-solving.

    • Main Results:

    • The weak limit of Korteweg-deVries solutions is determined for all time.
    • The limit is expressible as a quadratic programming problem.
    • For large times, solutions exhibit behavior described by Whitham's averaged equations and Flaschka et al.'s equations.

    Conclusions:

    • The inverse scattering method provides a framework for analyzing KdV equation limits.
    • The derived limit offers insights into the transition from dispersive to non-dispersive regimes.
    • The long-time dynamics reveal a complex interplay between different averaging approaches.