Stability of Kadomtsev-Petviashvili multi-line solitons

We prove the long-standing inverse scattering theory (IST) of perturbed Kadomtsev Petviashvili multi-line solitons. Our work is the first rigorous IST of a multi-dimensional integrable system when both continuous and discrete scattering data are present, and the support of continuous scattering data does not degenerate into contours in the complex plane. As an application, an $L^\infty$-stability theorem of the Kadomtsev Petviashvili multi-line solitons is justified.