Multi-lump solutions of KP equation with integrable boundary via $\bar{\mathrm{\partial }}$-dressing method

We constructed the new classes of exact multi-lump solutions of KP-1 and KP-2 versions of KP equations with integrable boundary condition $u_y{\mid }_{y=0}=0$ by the use of $\bar{\mathrm{\partial }}$-dressing method of Zakharov and Manakov and derived general determinant formula for such solutions. We demonstrated how reality and boundary conditions for the field $u$ in the framework of $\bar{\mathrm{\partial }}$-dressing method can be exactly satisfied. 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 $y\ge 0$ as analogs of standing waves on string with fixed end points.