Certain (2+1)-dimensional multi-soliton asymptotics in the shallow water

Abstract

This paper investigates the Kadomtsev–Petviashvili I equation, which describes the variation of shallow-water wave amplitude in the transverse direction, making it applicable to the shallow water conditions with strong surface tension., e.g., in the capillary wave problems. With respect to the amplitude of a shallow-water-wave packet, we derive a binary Darboux transformation (DT), and then perform the asymptotic analysis on the $N$-soliton solutions to derive the algebraic expressions of the $N$ soliton components, where $N$ is a positive integer. The asymptotic results indicate the energy stability in the shallow-water soliton interactions under certain conditions. Additionally, each soliton component contributes to the phase shifts of the other soliton components. Taking $N=3$ as an example, we illustrate the 3 interacting solitons via the 3D plots and density plots, which align with our asymptotic-analysis results. Although the asymptotic analysis method used is confined to the binary-DT framework, this paper tries to apply such a method [traditionally used in the $(1+1)$-dimensional systems] to the aforementioned $(2+1)$-dimensional system. Our analysis, which still needs to be confirmed by the relevant numerical simulation and experiments, might offer some explanations for the complex and variable natural mechanisms of the real-world shallow water waves.