The KPI Equation with Unconstrained Initial Data

Abstract

The solution u(t, x, y) of the Kadomtsev—Petviashvili I (KPI) equation with given initial data u(0,x,y) belonging to the Schwartz space is considered. No additional special constraints, usually considered in the literature as ∫ d xu(0, x, y) = 0 are required to be satisfied by the initial data. The spectral theory associated with KPI is studied in the space of the Fourier transform of the solutions. The variables p = {p1,p2} of the Fourier space are shown to be the most convenient spectral variables to use for spectral data. Spectral data are shown to decay rapidly at large p but to be discontinuous at p = 0. Direct and inverse problems are solved with special attention to the behavior of all the quantities involved in the neighborhood of t = 0 and p = 0. It is shown in particular that the solution u(t,x,y) has a time derivative discontinuous at t = 0 and that at any t ≠ 0 it does not belong to the Schwartz space no matter how small in norm and rapidly decaying at large distances the initial data are chosen.

Mathematics Subject Classifications (1991). 35Q53, 58F07.

Work supported in part by Ministero delle Università e della Ricerca Scientifica e Tecnologica, Italia.