Multi-soliton solutions of KP equation with integrable boundary via ∂¯-dressing method

We construct by the use of ∂¯-dressing method of Zakharov and Manakov new classes of exact multi-soliton solutions of the KP-1 and KP-2 versions of the Kadomtsev–Petviashvili equation with integrable boundary condition uy|y=0=0. We satisfy exactly reality and boundary conditions for the field u(x,y,t) in the framework ∂¯-dressing method and derive for exact solutions a general determinant formula in convenient form. As illustrations we present explicit examples of two-soliton solutions formed by two more simpler deformed one-solitons. The fulfillment of boundary condition in general case leads to the formation of corresponding multi-soliton solutions, i.e. to a certain nonlinear superpositions of an arbitrary number of pairs of bounded with each other one solitons. We interpret constructed multi-soliton solutions of the KP equation with integrable boundary as resonating eigenmodes of the field u(x,y,t) in the half-plane y≥0 — an analogs of standing waves on the string with fixed end points. •Multi-soliton solutions of KP with boundary uy = 0 at y = 0 via ∂¯-dressing.•Illustrative two-solitons as eigen-mode solutions of KP in half-plane.•Applications (possible): rogue waves, nonlinear optics, fluid flows.