Dynamics of KPI lumps

A family of nonsingular rational solutions of the Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions have multiple peaks whose heights are time-dependent and the peak trajectories in the xy-plane are altered after collision. Thus they differ from the standard multi-peaked KPI simple n-lump solutions whose peak heights as well as peak trajectories remain unchanged after interaction. The anomalous scattering occurs due to a non-trivial internal dynamics among the peaks in a slow time scale. This phenomena is explained by relating the peak locations to the roots of complex heat polynomials. It follows from the long time asymptotics of the solutions that the peak trajectories separate as O(root vertical bar t vertical bar) as vertical bar t vertical bar -> infinity, and all the peak heights approach the same constant value corresponding to that of the simple one-lump solution. Consequently, a multi-peaked n-lump solution evolves to a superposition of n one-lump solutions asymptotically as vertical bar t vertical bar -> infinity.

Automated summary

This study investigates a family of nonsingular rational solutions of the KP-I equation that have time-dependent multiple peaks and altered peak trajectories in the xy-plane after collision. The anomalous scattering phenomenon is explained by relating the peak locations to the roots of complex heat polynomials. The long time asymptotics of the solutions show that the peak trajectories separate and all peak heights approach the same constant value corresponding to that of a simple one-lump solution as time tends to infinity.