Unified approach to KdV modulations

We develop a unified approach to integrating the Whitham modulation equatio ns. Our approach is based on the formulation of the initial-value problem f or the zero-dispersion KdV as the steepest descent for the scalar Riemann-H ilbert problem [6] and on the method of generating differentials for the Kd V-Whitham. hierarchy [9]. By assuming the hyperbolicity of the zero-dispers ion limit for the KdV with general initial data, we bypass the inverse scat tering transform and produce the symmetric system of algebraic equations de scribing motion of the modulation parameters plus the system of inequalitie s determining the number the oscillating phases at any fixed point on the ( x, t)-plane. The resulting system effectively solves the zero-dispersion Kd V with an arbitrary initial datum. (C) 2001 John Wiley & Sons, Inc.