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Unified approach to KdV modulations
Author(s) -
El Gennady A.,
Krylov Alexander L.,
Venakides Stephanos
Publication year - 2001
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10002
Subject(s) - korteweg–de vries equation , mathematics , scalar (mathematics) , algebraic number , mathematical analysis , zero (linguistics) , initial value problem , inverse scattering transform , method of steepest descent , gradient descent , kdv hierarchy , interior point method , inverse scattering problem , inverse problem , mathematical optimization , geometry , physics , artificial neural network , quantum mechanics , linguistics , philosophy , nonlinear system , machine learning , computer science
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. © 2001 John Wiley & Sons, Inc.