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The aim of this work is multifold. Firstly, it intends to present a complete,quantitative and self-contained description of the periodic traveling wavesolutions and Whitham modulation equations for the Toda lattice, combiningresults from different previous works in the literature. Specifically, weconnect the Whitham modulation equations and a detailed expression for theperiodic traveling wave solutions of the Toda lattice. Along the way, some keydetails are filled in, such as the explicit expression of the characteristicspeeds of the genus-one Toda-Whitham system. Secondly, we use these tools toobtain a detailed quantitative characterization of the dispersive shocks of theToda system. Lastly, we validate the relevant analysis by performing a detailedcomparison with direct numerical simulations.

On the Whitham modulation equations for the Toda lattice and the quantitative characterization of its dispersive shocks