Scattering for the KP-II equation

Abstract

We prove scattering in $L^2(\mathbb{R}^2)$ for real-valued solutions of the KP-II equation with localized initial data. The hypotheses cover all Schwartz functions, data with a half spatial moment in $B^2_{1,1}$, and a class of weighted tensor Besov spaces allowing low regularity in either coordinate. The proof uses the scalar $\bar\partial$ equation associated with the heat operator. A uniform inverse estimate and an exact $L^2$ identity give an inverse construction for large amplitudes. We then estimate the nonlinear remainder on the propagation cones of free solutions and identify the limiting spectral mass through the heat equation. No smallness or zero horizontal mean condition is required. To the best of our knowledge, this is the first result establishing scattering for the KP-II equation for every real-valued Schwartz initial datum.