IST of KPII equation for perturbed multisoliton solutions

The Direct and the Inverse Scattering Problems for the heat operator with a potential being a perturbation of an arbitrary $N$ soliton potential are formulated. We introduce Jost solutions and spectral data and present their properties. Then, giving the time evolution of the spectral data, the initial value problem of the Kadomtsev–Petviashvili II equation for a solution describing $N$ solitons perturbed by a generic smooth fast decaying potential is linearized.