Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Localized wave solutions of three-component defocusing Kundu-Eckhaus equation with 4x4 matrix spectral problem

Yanan Wang, Min Xue

TL;DR

This work addresses the defocusing three-component Kundu-Eckhaus equation by developing a binary Darboux transformation based on a $4\times4$ Lax pair and a gauge-reduced AKNS form to generate $N$-fold solutions. It yields explicit vector dark solitons, including second-order cases, and performs an asymptotic analysis to obtain far-field soliton components, confirming elastic inter-component interactions. Additionally, the method produces breather and Y-shaped breather solutions that arise from inter-component coupling, a feature absent in single-component systems. The results deepen the understanding of coupling-driven nonlinear wave phenomena in defocusing multi-component systems and provide a robust theoretical framework for future studies.

Abstract

This work focuses on three-component defocusing Kundu-Eckhaus equation, which serves as a significant coupled model for describing complex wave propagation in nonlinear optical fibers. By employing binary Darboux transformation based on 4x4 matrix spectral problem, we derive vector dark soliton solutions, and meanwhile, the exact expressions of asymptotic dark soliton components are obtained through an asymptotic analysis method. Furthermore, breather and Y-shaped breather solutions, absent from single-component defocusing kundu-Eckhaus systems, are obtained due to the mutual coupling effects between different components. The results significantly advance our understanding nonlinear wave phenomenon induced by coupling effects and provide a theoretical reference for subsequent studies on defocusing multi-component systems.

Localized wave solutions of three-component defocusing Kundu-Eckhaus equation with 4x4 matrix spectral problem

TL;DR

This work addresses the defocusing three-component Kundu-Eckhaus equation by developing a binary Darboux transformation based on a Lax pair and a gauge-reduced AKNS form to generate -fold solutions. It yields explicit vector dark solitons, including second-order cases, and performs an asymptotic analysis to obtain far-field soliton components, confirming elastic inter-component interactions. Additionally, the method produces breather and Y-shaped breather solutions that arise from inter-component coupling, a feature absent in single-component systems. The results deepen the understanding of coupling-driven nonlinear wave phenomena in defocusing multi-component systems and provide a robust theoretical framework for future studies.

Abstract

This work focuses on three-component defocusing Kundu-Eckhaus equation, which serves as a significant coupled model for describing complex wave propagation in nonlinear optical fibers. By employing binary Darboux transformation based on 4x4 matrix spectral problem, we derive vector dark soliton solutions, and meanwhile, the exact expressions of asymptotic dark soliton components are obtained through an asymptotic analysis method. Furthermore, breather and Y-shaped breather solutions, absent from single-component defocusing kundu-Eckhaus systems, are obtained due to the mutual coupling effects between different components. The results significantly advance our understanding nonlinear wave phenomenon induced by coupling effects and provide a theoretical reference for subsequent studies on defocusing multi-component systems.

Paper Structure

This paper contains 5 sections, 1 theorem, 31 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Binary Darboux transformation for the system \ref{['eq']} with $\sigma=-1$
  3. Dark solitons and asymptotic analysis
  4. Breather solutions
  5. Conclusions and discussions

Key Result

Proposition 1

Let $\Phi_{j}=(\phi_{j1},\phi_{j2},\phi_{j3},\phi_{j4})^T,(j=1,2,\cdots,N)$ be $N$ linearly independent solutions of the spectral problem lax1 under the spectral parameters $\lambda_j,(j=1,2,\cdots,N)$, respectively. The $N$-fold binary DT for system eq2 is given as follows. with $H=(\Phi_1,\Phi_2,\cdots,\Phi_N)$ and where $\dagger$ denotes the Hermitian conjugate, $\Omega(\Phi_j,\Phi_k)=\frac{\

Figures (5)

  • Figure 1: The single dark soliton solution of the system \ref{['eq']} when $\sigma=-1$ with $a_1=-1,a_2=1,a_3=2,\rho=1,c_1=c_2=c_3=1,\gamma=0,\lambda_1=-\frac{1}{2}$.
  • Figure 2: The second-order dark soliton solution of the system \ref{['eq']} when $\sigma=-1$ with $a_1=-1,a_2=1,a_3=2,\rho=1,c_1=c_2=c_3=1,\gamma=0,\lambda_1=-\frac{1}{2},\lambda_2=-\frac{1}{8}$.
  • Figure 3: (a)(b) The comparison of the asymptotic solitons and the exact solution for the dark soliton component $q_1$. Blue: asymptotic soliton $[q_1]_{\mathrm{I}}$; Red: asymptotic soliton $[q_1]_{\mathrm{II}}$; Green: the exact solution \ref{['exact']}; (c) The wave crest line graph of the asymptotic solitons for the dark soliton component $q_1$.
  • Figure 4: The breather solution for the system \ref{['eq']} when $\sigma=-1$ with $a_1=a_2=a_3=1,\rho=1,c_1=c_2=c_3=1,\lambda_1=\mathrm{i}$.
  • Figure 5: The Y-shaped breather solution for the system \ref{['eq']} when $\sigma=-1$ with $a_1=-1,a_2=1,a_3=0,\rho=1,c_1=c_2=c_3=1,\lambda_1=\frac{\mathrm{i}}{2}$.

Theorems & Definitions (1)

  • Proposition 1