  We study the Cauchy problem for the Whitham modulation equations for monotone
increasing smooth initial data. The Whitham equations are a collection of
one-dimensional quasi-linear hyperbolic systems. This collection of systems is
enumerated by the genus g=0,1,2,... of the corresponding hyperelliptic Riemann
surface. Each of these systems can be integrated by the so called hodograph
transform introduced by Tsarev. A key step in the integration process is the
solution of the Tsarev linear overdetermined system. For each $g>0$, we
construct the unique solution of the Tsarev system, which matches the genus
$g+1$ and $g-1$ solutions on the transition boundaries. Next we characterize
initial data such that the solution of the Whitham equations has genus $g\leq
N$, $N>0$, for all real $t\geq 0$ and $x$.
