  We study the Cauchy problem for the Korteweg de Vries (KdV) equation with
small dispersion and with monotonically increasing initial data using the
Riemann-Hilbert (RH) approach. The solution of the Cauchy problem, in the zero
dispersion limit, is obtained using the steepest descent method for oscillatory
Riemann-Hilbert problems. The asymptotic solution is completely described by a
scalar function $\g$ that satisfies a scalar RH problem and a set of algebraic
equations constrained by algebraic inequalities. The scalar function $\g$ is
equivalent to the solution of the Lax-Levermore maximization problem. The
solution of the set of algebraic equations satisfies the Whitham equations. We
show that the scalar function $\g$ and the Lax-Levermore maximizer can be
expressed as the solution of a linear overdetermined system of equations of
Euler-Poisson-Darboux type. We also show that the set of algebraic equations
and algebraic inequalities can be expressed in terms of the solution of a
different set of linear overdetermined systems of equations of
Euler-Poisson-Darboux type. Furthermore we show that the set of algebraic
equations is equivalent to the classical solution of the Whitham equations
expressed by the hodograph transformation.
