Unified approach to KdV modulations

Gennady A. El*, Alexander L. Krylov, Stephanos Venakides

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Citations (Scopus)
18 Downloads (Pure)


We develop a unified approach to integrating the Whitham modulation equations. Our approach is based on the formulation of the initial-value problem for the zero-dispersion KdV as the steepest descent for the scalar Riemann-Hilbert problem [6] and on the method of generating differentials for the KdV-Whitham hierarchy [9]. By assuming the hyperbolicity of the zero-dispersion limit for the KdV with general initial data, we bypass the inverse scattering transform and produce the symmetric system of algebraic equations describing motion of the modulation parameters plus the system of inequalities determining the number the oscillating phases at any fixed point on the (x, t)-plane. The resulting system effectively solves the zero-dispersion KdV with an arbitrary initial datum.

Original languageEnglish
Pages (from-to)1243-1270
Number of pages28
JournalCommunications on Pure and Applied Mathematics
Issue number10
Early online date31 Jul 2001
Publication statusPublished - 1 Oct 2001
Externally publishedYes


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