Concentric-ring patterns of higher-order lumps in the Kadomtsev--Petviashvili I equation

TL;DR

This paper analyzes large-time patterns of higher-order lump solutions in the KP-I equation, revealing concentric-ring arrangements of lumps depending on the solution's index vector, with analytical predictions confirmed by numerical comparisons.

Contribution

It provides a detailed characterization of the large-time spatial patterns of higher-order lumps in the KP-I equation, including analytical predictions and numerical validation.

Findings

Lump solutions form concentric rings for specific index vectors.

Patterns depend on the structure of the associated Wronskian-Hermit polynomial.

Predictions match well with actual solutions at large times.

Abstract

Large-time patterns of general higher-order lump solutions in the KP-I equation are investigated. It is shown that when the index vector of the general lump solution is a sequence of consecutive odd integers starting from one, the large-time pattern in the spatial $(x, y)$ plane generically would comprise fundamental lumps uniformly distributed on concentric rings. For other index vectors, the large-time pattern would comprise fundamental lumps in the outer region as described analytically by the nonzero-root structure of the associated Wronskian-Hermit polynomial, together with possible fundamental lumps in the inner region that are uniformly distributed on concentric rings generically. Leading-order predictions of fundamental lumps in these solution patterns are also derived. The predicted patterns at large times are compared to true solutions, and good agreement is observed.