Construction of periodic solutions of the modified Kadomtsev-Petviashvili equation via the dibar-dressing method

New periodical solutions ( , , ) u x y t of modified Kadomtsev-Petviashvili-1 (mKP-1) equation with the integrable boundary condition = 0 y u by the use of the Zakharov-Manakov  -dressing(dibar-dressing) method are constructed. A general determinant formula for these solutions is derived. The restrictions from reality and boundary conditions for solutions via a general determinant formula are exactly satisfied by an appropriate choice of corresponding parameters. It is shown how the imposition of a boundary condition leads to the formation of some kind of eigen modes of normal oscillations of the field ( , , ) u x y t in the semi-plane 0 y  . An explicit example of a two-periodic solution with an integrable boundary condition as some nonlinear superposition of two simple one-periodical (linear-periodic) solutions and point singularities