Bilinear Differential-Difference Equations and One-Point Distributions of Some KPZ-Class Models

Abstract: We introduce a collection of nonlinear integrable partial differential-difference equations that are satisfied by the one-point distribution functions of some classical integrable KPZ models. Moreover, these equations can be regarded as reparametrizations or as scaling limits of the Hirota bilinear difference equation (HBDE), a canonical discretization for many important integrable systems such as the Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviashvili (KP) equation, and the two-dimensional Toda lattice (2DTL). Our contributions are threefold: (i) general Fredholm determinant solutions; (ii) verification that known formulas for classical integrable KPZ models fit within our framework; and (iii) zero-curvature/Lax pair formulations. As an application, we derive formal scaling limits of the equations, including the KP limit under 1:2:3 KPZ scaling.