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

TL;DR

This work analyzes how 3-soliton interactions in the asymmetric ANNV system produce localized stem structures during soliton reconnection under 2-resonance. It develops a two-variable asymptotic framework that treats a spatial variable together with time, yielding accurate arm forms and stem endpoints for both weak ($a_{12}=a_{13}=0$) and strong ($a_{12}=a_{13}=+\infty$) resonance. Key contributions include explicit asymptotic forms of four soliton arms with phase shifts dependent on $a_{23}$, and detailed constructions of variable-length stems with trajectories and length formulas across multiple parameter regimes. The results advance analytic understanding of stem localization and soliton reconnection dynamics in higher-dimensional soliton equations, and point to extensions to higher-order resonances and more complex multi-soliton interactions.

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.

Figures (8)

• Figure 1: The density plots of the weak 2-resonant 3-soliton given by \ref{['uv']} and \ref{['3soliton1']} with $k_1=\frac{1}{2},\,k_2=2,\,k_3=\frac{1}{2},\,p_1=\frac{3}{2},\,p_2=\frac{3}{2},\,p_3=\frac{2}{3},\,\xi_1^0=0,\,\xi_2^0=0,\,\xi_3^0=0$. The lines are the trajectories of the arms and stem structures, and the points are the endpoints of the variable length stem structures.
• Figure 2: (a) (b) The cross-sectional curves \ref{['cross31']}; (c) (d) The cross-sectional curves \ref{['cross32']}. The red points are the endpoints of the variable length stem structures.
• Figure 3: The evolutions of the amputation on the point $R_j$ over time: (a)--(d) correspond to \ref{['amplocaladd1']}\ref{['amplocal1']}\ref{['amplocaladd2']}\ref{['amplocal2']}, respectively.
• Figure 4: The density plots of the weak 2-resonant 3-soliton given by \ref{['uv']} and \ref{['3soliton1']} with $k_1=2,\,k_2=2,\,k_3=1,\,p_1=1,\,p_2=\frac{3}{2},\,p_3=1,\,\xi_1^0=0,\,\xi_2^0=0,\,\xi_3^0=0$. The lines are the trajectories of the arms and stem structures, and the points are the endpoints of the variable length stem structures.
• Figure 5: The density plots of the strong 2-resonant 3-soliton given by \ref{['uv']} and \ref{['3soliton2']} with $k_1=\frac{1}{2},\,k_2=2,\,k_3=-\frac{1}{2},\,p_1=\frac{3}{2},\,p_2=-\frac{3}{2},\,p_3=\frac{2}{3},\,\xi_1^0=0,\,\xi_2^0=0,\,\xi_3^0=0$. The lines are the trajectories of the arms and stem structures, and the points are the endpoints of the variable length stem structures.

Theorems & Definitions (19)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Proposition 3.1
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Proposition 3.2
• Remark 3.6