Multi-lump solutions of KP equation with integrable boundary via [formula omitted]-dressing method

Abstract

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.