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

We constructed new classes of exact multi-lump solutions of KP-1 and KP-2 versions of KP equations with integrable boundary condition uy|y=0=0 by the use of ∂¯-dressing method of Zakharov and Manakov and derived general determinant formula for such solutions. We demonstrated how reality and boundary conditions for the field u can be exactly satisfied in the framework of ∂¯-dressing method. Here we present explicit examples of two-lump solutions with integrable boundary as nonlinear superpositions of two more simpler deformed one-lump solutions: the fulfillment of boundary condition leads to formation of certain eigenmodes of the field u(x,y,t) in semiplane y≥0 as analogs of standing waves on a string with fixed end points. •Multi-lump solutions of KP with boundary uy=0 at y=0 via ∂¯ -dressing.•Illustrative two- and four-lumps as eigen-mode solutions of KP in semiplane.•Applications (possible): Rogue waves, nonlinear optics, fluid flows.