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. The solution is defined implicitly using solutions to an over-determined system proposed by Tsarev. Moreover the Tsarev solution is shown to correspond to the unique Lax-Levermore zero dispersion weak limit of the Korteweg-de Vries equation and an explicit expression for the Lax-Levermore minimizer is found. This explicit expression for the minimizer can be used to calculate phase information for the actual small dispersion Korteweg-de Vries wave form obtained in Venakides' higher order Lax-Levermore analysis.

Moreover the speeds of the modulation equations of all the commuting flows in the Korteweg-de Vries hierarchy are shown to satisfy the Tsarev over-determined system and in the limit as two Riemann variables come together the one phase speeds approach polynomials in the third Riemann variable. Thus these speeds are building blocks for Tsarev's implicit solution of the modulation equations in the case of analytic initial data. A quick algebro-geometric derivation of the Riemann invariant form of this hierarchy of modulation equations is presented, using polynomial conserved densities arising from a squared eigenfunction. Finally, in the tradition of the one phase numerical results of Gurevich and Pitaevskii, a Fortran code which calculates genuine two phase solutions of the modulation equations is used to examine the interaction of two cubic inflection points.