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Regular and Chaotic Dynamics

, Volume 13, Issue 5, pp 424–430 | Cite as

Zero-dispersion limit to the Korteweg-de Vries equation: a dressing chain approach

  • V. Yu. NovokshenovEmail author
Research Articles
  • 33 Downloads

Abstract

An asymptotic solution of the KdV equation with small dispersion is studied for the case of smooth hump-like initial condition with monotonically decreasing slopes. Despite the well-known approaches by Lax-Levermore and Gurevich-Pitaevskii, a new way of constructing the asymptotics is proposed using the inverse scattering transform together with the dressing chain technique developed by A. Shabat [1]. It provides the Whitham-type approximaton of the leading term by solving the dressing chain through a finite-gap asymptotic ansatz. This yields the Whitham equations on the Riemann invariants together with hodograph transform which solves these equations explicitly. Thus we reproduce an uniform in x asymptotics consisting of smooth solution of the Hopf equation outside the oscillating domain and a slowly modulated cnoidal wave within the domain. Finally, the dressing chain technique provides the proof of an asymptotic estimate for the leading term.

Key words

KdV small dispersion limit wave collapse dressing chain 

MSC2000 numbers

34E15 35Q53 35Q51 37K15 

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Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.Institute of MathematicsRussian Academy of SciencesUfaRussia

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