Based on the Chen--M\"oller--Sauvaget formula, we apply the theory of integrable systems to derive three equations for the generating series of the Masur--Veech volumes ${ \rm Vol} \, \mathcal{ Q}_{ g,n}$ associated with the principal strata of the moduli spaces of quadratic differentials, and propose refinements of the conjectural formulas given in [12,4] for the large genus asymptotics of ${ \rm Vol} \, \mathcal{ Q}_{ g,n}$ and of the associated area Siegel--Veech constants.
Categories: math-ph math.MP nlin.SI