REMARKS ON DISPERSIONLESS KP, KDV, AND 2D GRAVITY

We use the SDiff(2) framework of Takasaki and Takebe and the (L, M) pr ogram (L is the Lax operator and M psi = psi(lambda) to show that M = semiclassical limit of M is <(xi)over cap> + Sigma(2)(infinity)T-n'lam bda(n-1), where (lambda, -<(xi)over cap>) are action angle variables i n the Gibbons-Kodama theory of Hamilton-Jacobi type for dispersionless KP. We also show <(xi)over cap> is the semiclassical limit of WxW(-1) (W is the gauge operator), where G = WxW(-1) is a quantity studied by the author in an earlier paper in connection with symmetries. We give then a semiclassical version of the Jevicki-Yoneya action principle f or 2D gravity, where again <(xi)over cap> arises in calculations, and this yields directly the Landau-Ginsburg equation that corresponds to the semiclassical limit of an integrated string equation. For KdV we a lso show how inverse scattering data are connected to Hamiltonians for dispersionless KdV. We also discuss Hirota bilinear formulas relative to the dispersionless hierarchies and establish various limiting form ulas.