On Dispersionless Hirota Type Equations

Various connections between 2-D gravity and KdV, dKdV, inverse scattering, etc. are established. For KP we show how to extract from the dispersionless limit of the Fay differential identity of Takasaki-Takebe the collection of differential equations for F = log(τ ) which play the role of Hirota type equations in the dispersionless theory. 1. HIROTA EQUATIONS In [7] we showed how second derivatives of F = log(τ ) survive in the dispersionless limit of the KP Hirota equations and in [31] it was shown how such second derivatives are contained in the dispersionless limit of the Fay differential identity (cf. [1]). We show here how to extract all such dispersionless Hirota type equations from the limiting Fay identity and indicate the pattern. Thus, referring to [1,6-9,31,32] for background in KP theory, we recall first some basic relations involving the Lax and gauge operator. Thus