Pattern transformation in higher-order lumps of the Kadomtsev-Petviashvili I equation

TL;DR

This paper analytically studies pattern formation in higher-order lumps of the KP-I equation at large times, revealing two main pattern types and their evolution, with predictions validated against true solutions.

Contribution

It introduces a detailed analytical description of pattern transformations in higher-order lumps of the KP-I equation using special polynomials, expanding understanding of their large-time behavior.

Findings

Triangular patterns described by Yablonskii-Vorob'ev polynomials.

Non-triangular outer patterns described by Wronskian-Hermit polynomials.

Excellent agreement between predicted and true solutions.

Abstract

Pattern formation in higher-order lumps of the Kadomtsev-Petviashvili I equation at large time is analytically studied. For a broad class of these higher-order lumps, we show that two types of solution patterns appear at large time. The first type of patterns comprise fundamental lumps arranged in triangular shapes, which are described analytically by root structures of the Yablonskii-Vorob'ev polynomials. As time evolves from large negative to large positive, this triangular pattern reverses itself along the x-direction. The second type of patterns comprise fundamental lumps arranged in non-triangular shapes in the outer region, which are described analytically by nonzero-root structures of the Wronskian-Hermit polynomials, together with possible fundamental lumps arranged in triangular shapes in the inner region, which are described analytically by root structures of the Yablonskii-Vorob'ev polynomials. When time evolves from large negative to large positive, the non-triangular pattern in the outer region switches its x and y directions, while the triangular pattern in the inner region, if it arises, reverses its direction along the x-axis. Our predicted patterns at large time are compared to true solutions, and excellent agreement is observed.