Multi-lump formations from lump chains and plane solitons in the KP1 equation

Multi-lump formations from lump chains and plane solitons in the KP1 equation

We show that complex higher-order lump patterns can be constructed in two different ways within the Kadomtsev–Petviashvili equation which describes nonlinear wave processes in media with positive dispersion. In the first approach, we start with solutions describing stationary moving higher-order lump chains. By degenerating these solutions, we obtain first coupled lump chains which reduce then to multi-lump bound states in the limit when the period of lump chains goes to infinity. In another approach, a skillful technique is exploited to derive multi-lump bound states directly from the N-soliton solution of the KP1 equation presented in the Hirota form. It is shown then that through the proper selection of soliton parameters, these higher-order solutions can be reduced to the various lump patterns, such as triangular, polygonal patterns, and so on. The suggested approaches can be extended to other one- and multi-dimensional integrable systems to derive complex bound states and rogue waves.

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