Some new methods and results in the theory of ($2+1$)-dimensional integrable equations

The general resolvent scheme for solving nonlinear integrable evolution equations is formulated. Special attention is paid for the problem of nontrivial dressing and corresponding transformation of spectral data. Kadomtsev–Petviashvili equation is considered as the standard example of integrable models in $2+1$ dimensions. Properties of the solution $u(t,x,y)$ of the Kadomtsev–Petviashvili I equation as well as corresponding Jost solutions and spectral data with given initial data $u(0,x,y)$ belonging to the Schwartz space are presented.