Localized stem structures in soliton reconnection of the asymmetric Nizhnik-Novikov-Veselov system

TL;DR

This paper investigates the formation and evolution of localized stem structures during soliton reconnection in the asymmetric Nizhnik-Novikov-Veselov system, revealing phase shifts and detailed asymptotic behaviors of the arms involved.

Contribution

It introduces a two-variable asymptotic analysis method to characterize the asymptotic forms and dynamics of stem structures in soliton reconnection, considering different resonance scenarios.

Findings

Asymptotic forms of soliton arms differ by phase shift as time approaches infinity.

Detailed trajectories, amplitudes, and velocities of soliton arms and stem structures are obtained.

Localization properties of stem structures, including endpoints and lengths, are analyzed in different resonance cases.

Abstract

The reconnection processes of 3-solitons with 2-resonance can produce distinct local structures that initially connect two pairs of V-shaped branches, then disappear, and later re-emerge as new forms. We call such local structures as stem structures. In this paper, we investigate the variable-length stem structures during the soliton reconnection of the asymmetric Nizhnik-Novikov-Veselov system. We consider two scenarios: weak 2-resonances (i.e., $a_{12}=a_{13}=0,\,0<a_{23}<+\infty$) and strong 2-resonances (i.e., $a_{12}=a_{13}=+\infty,\,0<a_{23}<+\infty$). We determine the asymptotic forms of the four arms and their corresponding stem structures using two-variable asymptotic analysis method which is involved simultaneously with one space variable $y$ (or $x$) and one temporal variable $t$. Different from known studies, our findings reveal that the asymptotic forms of the arms $S_2$ and $S_3$ differ by a phase shift as $t\to\pm\infty$. Building on these asymptotic forms, we perform a detailed analysis of the trajectories, amplitudes, and velocities of the soliton arms and stem structures. Subsequently, we discuss the localization of the stem structures, focusing on their endpoints, lengths, and extreme points in both weak and strong 2-resonance scenarios.