Unified Approach to KdV Modulations

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, developed by Deift, Venakides, and Zhou, 1997, and on the method of generating differentials for the KdV-Whitham hierarchy proposed by El, 1996. 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 data.