On the Inverse Scattering of the Time‐Dependent Schrödinger Equation and the Associated Kadomtsev‐Petviashvili (I) Equation

Abstract

The Kadomtsev‐Petviashvili equation, a two‐spatial‐dimensional analogue of the Korteweg‐deVries equation, arises in physical situations in two different forms depending on a certain sign appearing in the evolution equation. Here we investigate one of the two cases. The initial‐value problem, associated with initial data decaying sufficiently rapidly at infinity, is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation of a nonlocal Riemann‐Hilbert problem in terms of scattering data expressible in closed form in terms of given initial data. The lump solutions, algebraically decaying solitons, are given a definite spectral characterization. Pure lump solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on x, y, t . Many of the above results are also relevant to the problem of inverse scattering for the so‐called time‐dependent Schrödinger equation.