Asymptotic dynamics of higher-order lumps in the Davey-Stewartson II equation

TL;DR

This paper analyzes the asymptotic behavior of higher-order rational lumps in the Davey-Stewartson II equation, revealing their time-dependent heights, arbitrary scattering angles, and their interpretation as superpositions of fundamental lumps at large times.

Contribution

It introduces a detailed asymptotic analysis of higher-order lumps on non-zero background in the DS II equation, highlighting their scattering properties and superposition structure.

Findings

Heights of lumps approach a constant value over time.

Scattering angles can vary within (rac{ o rac{ ext{pi}}{2}}, ext{pi}) interval.

Higher-order lumps can be viewed as superpositions of first-order lumps asymptotically.

Abstract

A family of higher-order rational lumps on non-zero constant background of Davey-Stewartson (DS) II equation are investigated. These solutions have multiple peaks whose heights and trajectories are approximately given by asymptotical analysis. It is found that the heights are time-dependent and for large time they approach the same constant height value of the first-order fundamental lump. The resulting trajectories are considered and it is found that the scattering angle can assume arbitrary values in the interval of $(\frac{\pi}{2}, \pi)$ which is markedly distinct from the necessary orthogonal scattering for the higher-order lumps on zero background. Additionally, it is illustrated that the higher-order lumps containing multi-peaked $n$-lumps can be regarded as a nonlinear superposition of $n$ first-order ones as $|t|\rightarrow\infty$.