Asymptotic analysis of multi-lumps solutions in the Kadomtsev-Petviashvili-(I) equation

TL;DR

This paper analyzes the long-term behavior of multi-lump solutions in the KP-I equation, revealing how peak locations evolve from vertical to horizontal lines over time, demonstrating a rotation after interaction.

Contribution

It provides an asymptotic analysis of multi-lump solutions in the KP-I equation using Grammian determinants and orthogonal polynomials, highlighting the geometric evolution of peaks.

Findings

Peak locations depend on roots of Wronskian of orthogonal polynomials.

Peaks align vertically as time approaches -∞.

Peaks align horizontally as time approaches ∞, showing a rotation of π/2.

Abstract

Inspired by the works of Y. Ohta and J. Yang, one constructs the lumps solutions in the Kadomtsev-Petviashvili-(I) equation using the Grammian determinants. It is shown that the locations of peaks will depend on the real roots of Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases. Also, one can prove that all the locations of peaks are on a vertical line when time approaches - $\infty$, and then they will be on a horizontal line when time approaches $\infty$, i.e., there is a rotation $\frac{\pi}{2}$ after interaction.