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

Abstract

We constructed the new classes of exact multi-lump solutions of KP-1 and KP-2 versions of KP equations with integrable boundary condition $u_y{|}_{y=0}=0$ by the use of $\overline{\partial }$-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 $\overline{\partial }$-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.

arXiv admin note: text overlap with arXiv:2003.01715

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