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In this paper we prove a local well-posedness result for a class ofquasi-linear systems of hyperbolic type involving Fourier multipliers. Amongthe physically relevant systems in this class is a family of Whitham-Boussinesqsystems arising in the modeling free-surface water waves. Our result allows toprove the rigorous justification of these systems as approximations to thegeneral water waves system on a relevant time scale, independent of theshallowness parameter.

Local well-posedness result for a class of non-local quasi-linear systems and its application to the justification of Whitham-Boussinesq systems