6 The Cauchy Problem on the Plane for the Dispersionless Kadomtsev - Petviashvili Equation

We construct the formal solution of the Cauchy problem for the dispersionless Kadomtsev Petviashvili equation as application of the Inverse Scattering Transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy problem for the Kadomtsev Petviashvili equation, associated with the Inverse Scattering Transform of the time dependent Schrödinger operator for a quantum particle in a timedependent potential. 1. Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDEs) arise in various problems of Mathematical Physics and are intensively studied in the recent literature (see, f.i., [1, 2, 3, 4, 5]). In particular, a quasi-classical dressing has been developed [4] for the prototypical example of the dispersionless Kadomtsev Petviashvili (dKP) (or KhokhlovZabolotskaya) equation: utx + uyy + (uux)x = 0, u = u(x, y, t) ∈ R, x, y, t ∈ R. (1) In this paper we construct the formal solution of the Cauchy problem on the plane for the following system of PDEs in 2+1 dimensions: uxt + uyy = −(uux)x − vxuxy + vyuxx, u, v ∈ R, x, y, t ∈ R, vxt + vyy = −uvxx − vxvxy + vyvxx (2) and for its v = 0 reduction, the dKP equation (1), as application of the recently developed Inverse Scattering Transform (IST) for vector fields [6].