Multi-soliton solutions of KP equation with integrable boundary via $\overline\partial$-dressing method

Abstract: New classes of exact multi-soliton solutions of KP-1 and KP-2 versions of Kadomtsev-Petviashvili equation with integrable boundary condition $u_{y}\big|_{y=0}=0$ by the use of $\overline\partial$-dressing method of Zakharov and Manakov are constructed in the paper. General determinant formula in convenient form for such solutions is derived. It is shown how reality and boundary conditions for the field $u(x,y,t)$ in the framework of $\overline\partial$-dressing method can be satisfied exactly. Explicit examples of two-soliton solutions as nonlinear superpositions of two more simpler \,"deformed"\, one-solitons are presented as illustrations: the fulfillment of boundary condition leads to formation of bound state of two more simpler one-solitons, resonating eigenmodes of $u(x,y,t)$ in semi-plane $y\geq0$ as analogs of standing waves on the string with fixed end points.