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The variable-length stem structures in three-soliton resonance of the Kadomtsev-Petviashvili II equation

Feng Yuan, Jingsong He, Yi Cheng

TL;DR

This work analyzes variable-length stem structures that arise during resonant interactions of KPII multi-solitons. Using the KPII equation in its tau-function form and explicit 3-soliton solutions, the authors perform rigorous asymptotic analysis for 2-resonant and 3-resonant cases, deriving trajectories, endpoints, and amplitudes of stem structures and detailing soliton reconnection dynamics. They classify resonances as strong, weak, or mixed according to phase-shift limits $a_{ij}\to0$ or $\infty$, and provide complete asymptotic characterizations, including corrections and extensions to prior results. The findings deepen the understanding of localized structures within KPII resonant networks and offer analytical tools for predicting stem evolution and topological transitions in high-order soliton interactions.

Abstract

The stem structure is a localized feature that arises during high-order soliton interactions, connecting the vertices of two V-shaped waveforms. The interaction of resonant 3-solitons is accompanied by soliton reconnection phenomena, characterized by the disappearance and reconnection of stem structures. This paper investigates variable-length stem structures in resonant 3-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation, focusing on both 2-resonant and 3-resonant 3-soliton cases. Depending on the phase shift tends to plus/minus infinity, different types of resonances are identified, including strong resonance, weak resonance, and mixed (strong-weak) resonance. We derive and analyze the asymptotic forms and explicit expressions for the soliton arm trajectories, velocities, as well as the endpoints, length, and amplitude of the stem structures. A detailed comparison is made between the similarities and differences of the stem structures in the 2-resonant and 3-resonant solitons. In addition, we provide a comprehensive and rigorous analysis of both the asymptotic behavior and the structural properties of the stems.

The variable-length stem structures in three-soliton resonance of the Kadomtsev-Petviashvili II equation

TL;DR

This work analyzes variable-length stem structures that arise during resonant interactions of KPII multi-solitons. Using the KPII equation in its tau-function form and explicit 3-soliton solutions, the authors perform rigorous asymptotic analysis for 2-resonant and 3-resonant cases, deriving trajectories, endpoints, and amplitudes of stem structures and detailing soliton reconnection dynamics. They classify resonances as strong, weak, or mixed according to phase-shift limits or , and provide complete asymptotic characterizations, including corrections and extensions to prior results. The findings deepen the understanding of localized structures within KPII resonant networks and offer analytical tools for predicting stem evolution and topological transitions in high-order soliton interactions.

Abstract

The stem structure is a localized feature that arises during high-order soliton interactions, connecting the vertices of two V-shaped waveforms. The interaction of resonant 3-solitons is accompanied by soliton reconnection phenomena, characterized by the disappearance and reconnection of stem structures. This paper investigates variable-length stem structures in resonant 3-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation, focusing on both 2-resonant and 3-resonant 3-soliton cases. Depending on the phase shift tends to plus/minus infinity, different types of resonances are identified, including strong resonance, weak resonance, and mixed (strong-weak) resonance. We derive and analyze the asymptotic forms and explicit expressions for the soliton arm trajectories, velocities, as well as the endpoints, length, and amplitude of the stem structures. A detailed comparison is made between the similarities and differences of the stem structures in the 2-resonant and 3-resonant solitons. In addition, we provide a comprehensive and rigorous analysis of both the asymptotic behavior and the structural properties of the stems.
Paper Structure (7 sections, 5 theorems, 84 equations, 13 figures, 1 table)

This paper contains 7 sections, 5 theorems, 84 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. The stem structure in 2-resonant 3-soliton
  3. Strong 2-resonant case: $a_{13}$ and $a_{23}\to +\infty$
  4. Weak 2-resonant case: $a_{13}$ and $a_{23} \to 0$
  5. Mixed 2-resonant case: $a_{13} \to +\infty$ and $a_{23} \to 0$
  6. The stem structure in 3-resonance 3-soliton
  7. Conclusions and discussions

Key Result

Proposition 2.1

The asymptotic forms of the strong 2-resonant 3-soliton 3f2r1 with $p_1 = \frac{k_1 \left( k_1 k_3 + k_3^2 + p_3 \right)}{k_3}, \, p_2 = -\frac{k_2 \left( k_2 k_3 + k_3^2 -p_3 \right)}{k_3}$ are as following: Before collision ($t\to-\infty$): After collision ($t\to+\infty$): The stem structures: Here, $S_j,\,S_{i+j},\,S_{i+j+k}$ are the soliton arms and stem structures of the soliton solutions

Figures (13)

  • Figure 1: The density plots of the strong 2-resonant 3-soliton \ref{['3f2r1']} with $k_1=-1,\,k_2=-2,\,k_3=-\frac{4}{3},\,p_1 = \frac{k_1 \left( k_1 k_3 + k_3^2 + p_3 \right)}{k_3}, \, p_2 = -\frac{k_2 \left( k_2 k_3 + k_3^2 -p_3 \right)}{k_3},\,p_3=1$. The lines are the trajectories of the arms and stem structures, and the red points are the endpoints of the variable-length stem structures.
  • Figure 2: Parameters: $k_1=-1,\,k_2=-2,\,k_3=-\frac{4}{3},\,p_1 = \frac{k_1 \left( k_1 k_3 + k_3^2 + p_3 \right)}{k_3}, \, p_2 = -\frac{k_2 \left( k_2 k_3 + k_3^2 -p_3 \right)}{k_3},\,p_3=1$. (a) The cross-sectional curve $u|_{\widehat{l_{1+2+3}}}$ given by \ref{['cross3s01']}; (b) The cross-sectional curve $u|_{l_3}$ given by \ref{['cross3s01']}; (c) The amplitude evolution curves \ref{['up01']}; (d) The cross-sectional curves $u|_{L_{1+2+3}}$ and $\widehat{u}_{1+2+3}|_{L_{1+2+3}}$; (e) The cross-sectional curves $u|_{L_3}$ and $u_3|_{L_3}$.
  • Figure 3: The density plots of the strong 2-resonance 3-soliton \ref{['3f2r1']} with $k_1=-1,\,k_2=\frac{3}{2},\,k_3=2,\,p_1 = \frac{k_1 \left( k_1 k_3 + k_3^2 + p_3 \right)}{k_3}, \, p_2 = -\frac{k_2 \left( k_2 k_3 + k_3^2 -p_3 \right)}{k_3},\,p_3=1$. The lines are the trajectories of the arms and stem structures, and the points are the endpoints of the variable length stem structures.
  • Figure 4: The density plots of the weak 2-resonance 3-soliton with $k_1=1,\,k_2=-1,\,k_3=-2,\,p_1 = -\frac{k_1 \left( k_1 k_3 - k_3^2 - p_3 \right)}{k_3}, \, p_2 = \frac{k_2 \left( k_2 k_3 - k_3^2 +p_3 \right)}{k_3},\,p_3=-\frac{1}{2}$. 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: Parameters: $k_1=1,\,k_2=-1,\,k_3=-2,\,p_1 = -\frac{k_1 \left( k_1 k_3 - k_3^2 - p_3 \right)}{k_3}, \, p_2 = \frac{k_2 \left( k_2 k_3 - k_3^2 +p_3 \right)}{k_3},\,p_3=-\frac{1}{2}$. (a) The cross-sectional curves $u|_{l_{1-3}}$ given by \ref{['cross3s02']}; (b) The cross-sectional curves $u|_{l_{2-3}}$ given by \ref{['cross3s02']}; (c) The amplitude evolution curves \ref{['up02']}; (d) The cross-sectional curves $u|_{L_{1-3}^{(1)}}$ and $u_{1-3}|_{L_{1-3}^{(1)}}$; (e) The cross-sectional curves $u|_{L_{2-3}^{(1)}}$ and $u_{2-3}|_{L_{2-3}^{(1)}}$.
  • ...and 8 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Proposition 3.2