Dynamics of KPI lumps

TL;DR

This paper studies special rational solutions of the KP I equation with multiple peaks that change height and trajectory after collision, revealing anomalous scattering due to internal peak dynamics linked to complex heat polynomial roots.

Contribution

It introduces a new class of nonsingular rational solutions of the KP I equation with time-dependent peak behavior and explains their anomalous scattering phenomena.

Findings

Peak trajectories separate as O(√|t|) at large times.

Peak heights tend to a constant value matching 1-lump solutions.

Multi-peaked solutions asymptotically resemble superpositions of 1-lump solutions.

Abstract

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 $O(\sqrt{|t|})$ as $|t| \to \infty$, and all the peak heights approach the same constant value corresponding to that of the simple 1-lump solution. Consequently, a multi-peaked $n$-lump solution evolves to a superposition of $n$ 1-lump solutions asymptotically as $|t| \to \infty$.