The Kadomtsev-Petviashvili Equations In Non-uniform Media

In the dissertation,we first study a noncommutative nonisospectral Kadomtsev-Petviashvili equation. By means of quasideterminant;we derive exact solutions for the noncommutative nonisospectral Kadomtsev-Petviashvili equation through direct verifica-tion. The obtained solutions are expressed in terms of quasidetcrminants. The solutions behavior like line solitons with time-varying amplitudes and velocities, which coincides with nonisospectral properties resulted from time-dependent spectral parameter. Be-sides, we investigate a nonisospectral Kadomtsev-Petviashvili I equation. We derive its N-lump solutions by means of Inverse Scattering Transform procedure, and investigate nonisospectral dynamics of obtained solutions. Since the spectral parameter of the equa-tion is time-dependent, solutions can provide stationary lump waves which arc localized both spatially and temporally. Such solutions behavior like rogue waves. The study of the dissertation is helpful to understanding backgrounds and possible applications of high dimensional integrablc systems.