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arXiv · 2003.01716

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

V. G. Dubrovsky·A. V. Topovsky
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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 $u_{y}\big|_{y=0}=0$ 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 $y\geq 0$ as analogs of standing waves on string with fixed end points.

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V. G. Dubrovsky, A. V. Topovsky. 2020-03-03. Multi-lump solutions of KP equation with integrable boundary via $\overline\partial$-dressing method. https://doi.org/10.1016/j.physd.2021.133025

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