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.Comment: 27 pages, Latex, 5 Postscript figures, to be submitted to Comm. Pure.
Appl. Mat
