The characteristic solution of the initial value problem for the Riemann invariant form of the Korteweg-de Vries modulation equations is obtained locally through first breaking in the case of initial data which is analytic near a cubic inflection point. Tsarev's (see [11]) system is used to define the solution implicitly in terms of the speeds of the generalized KdV modulation equations. These “higher flow” speeds satisfy the same derivative identities obtained by Levermore for the KdV flow (see [6]).

Moreover, by explicitly constructing the Lax-Levermore (see [5]) minimizer the Tsarev solution is shown to be the unique solution of Whitham's equations corresponding to the KdV zero dispersion limit. Also the Tsarev system possesses at most one three-sheeted solution in a uniform neighborhood of the inflection point. Finally the explicit minimizer provides phase information for the Venakides' small dispersion KdV waveform through first breaking; see [12].