数学物理学报, 2026, 46(6): 2183-2195

广义耦合导数非线性薛定谔方程的孤子分子、孤子共振以及 breather-positon 解

曾平平,1, 曾平安,2, 许国安,3,*

1 浙江金融职业学院信息技术学院 杭州 310000

2 浙江工业大学之江学院理学院 浙江绍兴 312000

3 华侨大学数学科学学院 福建泉州 362021

Soliton Molecules, Soliton Resonances and Breather Positon Solutions for the General Coupled Derivative Nonlinear Schrödinger Equation

Zeng Pingping,1, Zeng Pingan,2, Xu Guoan,3,*

1 Institute of Information Technology, Zhejiang Financial College, Hangzhou 310000

2 School of Science, Zhijiang College of Zhejiang University of Technology, Zhejiang Shaoxing 312000

3 School of Mathematical Sciences, Huaqiao University, Fujian Quanzhou 362021

通讯作者: 许国安, E-mail: xga99163@163.com

收稿日期: 2025-07-27   修回日期: 2026-03-26  

基金资助: 国家自然科学基金(12271096)
福建省自然科学基金(2025J01165)

Received: 2025-07-27   Revised: 2026-03-26  

Fund supported: NSFC(12271096)
NSFFP(2025J01165)

作者简介 About authors

曾平平,E-mail:1360544995@qq.com;

曾平安,E-mail:460877822@qq.com

摘要

该文聚焦于广义耦合导数非线性薛定谔方程, 该方程以窄脉冲的非线性自变陡特性为显著特征. 基于 Lax 对, 构建行列式形式的 $N$ 重 Darboux 变换. 随后, 借助 Darboux 变换与极限技巧, 系统研究了广义耦合导数非线性薛定谔方程的孤子分子、孤子共振及 breather-positon 解. 当选取零种子解时, 可得到不同类型的 $N$ 孤子分子以及孤子分子与暗孤子/类呼吸子孤子的孤子共振. 当选取非零种子解时, 能够推导出高阶 breather-positon 解. 值得注意的是, breather-positon 解的中心区域呈现出怪波, 这促使作者提出一种猜想, 即其或许能够阐明怪波生成的潜在机制. 该文获得孤子相互作用解和 breather-positon 解的方法, 可为物理现象研究领域提供助力.

关键词： 广义耦合导数非线性薛定谔方程; Darboux 变换; 孤子分子; 孤子共振; breather-positon 解

Abstract

In this paper, we focus on the general coupled derivative nonlinear Schrödinger equation which is distinguished by the nonlinear self-steepening of narrow pulses. Based on the Lax pair, the $N$-fold Darboux transformation in the form of determinants. Subsequently, using the Darboux transformation and limiting techniques, we can systematically study the soliton molecules, solitons resonances and breather positons for the general coupled derivative nonlinear Schrödinger equation. By choosing the zero seed solutions, different types of $N$-soliton molecules and the soliton resonances with soliton molecules and the dark soliton/breather-like soliton are obtained. Then by choosing the nonzero seed solutions, the higher-order breather positons can be derived. It is noteworthy that the central region of breather positons manifests rogue waves, thus prompting the proposal that they may elucidate the underlying mechanism of rogue wave generation. The approach shown in this paper to gain the soliton interaction solutions and breather positons can contribute to the research field of physical phenomena.

Keywords： the general coupled derivative nonlinear Schrödinger equation; Darboux transformation; soliton molecules; soliton resonances; breather positon

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本文引用格式

曾平平, 曾平安, 许国安. 广义耦合导数非线性薛定谔方程的孤子分子、孤子共振以及 breather-positon 解[J]. 数学物理学报, 2026, 46(6): 2183-2195

Zeng Pingping, Zeng Pingan, Xu Guoan. Soliton Molecules, Soliton Resonances and Breather Positon Solutions for the General Coupled Derivative Nonlinear Schrödinger Equation[J]. Acta Mathematica Scientia, 2026, 46(6): 2183-2195

1 引言

众所周知,偏微分方程 (PDEs) 在物理和工程领域占据关键地位, 众多学术著作中对此有广泛记载[1,2,3,4]. 鉴于非线性的普遍性, 含空间和时间自变量的非线性偏微分方程, 有时也被称作非线性演化方程. 例如, 在流体研究中, 非线性偏微分方程是描述非线性流体动力学的基础, 涵盖湍流、冲击波、流体不稳定性等现象[5,6,7]. 在光学领域, 它们对于建模自聚焦、光孤子、非线性波传播等非线性光学现象至关重要[8,9,10]. 此外, 在电磁学背景下, 非线性偏微分方程对理解和预测非线性电磁行为及相互作用意义重大[11,12]. 探寻偏微分方程的非线性局域解备受关注, 可采用多种方法求解,如 Hirota 双线性方法[13,14,15], Riemann-Hilbert 方法[16,17], Darboux 变换[18,19,20,21,22].

"孤子分子" 的概念最初是为了阐释凝聚态物质中一类特殊的平行孤子而提出的, 这类孤子以其具有弹性、 自持和束缚的状态为特征[23,24]. 孤子分子的一个显著特征是, 基础孤子应具备相等的群速度. 孤子分子的一个显著特点是, 其基本构成单元 (基态孤子) 必须具有相同的传播速度. 显然, 孤子分子在实验和理论研究领域都已成为一个备受关注的课题[25,26,27,28,29,30]. 在物理现象的语境中, 共振被定义为这样一种现象: 当某些物理量 (例如波频率) 被赋予特定值时, 一个物理系统能够产生幅度更大的振动[31]. 当孤子——那些极其稳定的波包——处于一种高度特定的相互作用状态时, 共振便会出现. 这种相互作用会导致它们集体行为的显著放大, 常常引发新颖且复杂的波图样的出现[32,33]. 通常而言, positon 可视为孤子的长程类比物, 其特点是具有振荡行为和强度 (或振幅) 的逐渐衰减[34]. 在大多数情况下, 这些存在于非零背景上的非奇异 positons 被称为简并孤子或平滑波势. 值得注意的是, 在非零背景上激发、并在各种非线性薛定谔方程中构造的 breather-positon, 其本质与 positon 不同[35,36]. 更重要的是, 研究表明, breather-positon 的中心区域会呈现出怪波, 即可以通过特定的极限过程将其转化为怪波[37]. 在非线性可积方程中关于平滑 positon 和 breather-positon 的研究活动数量已显著增加[38,39,40,41,42,43,44,45].

本文主要研究广义耦合导数非线性薛定谔方程

$\begin{equation} \begin{split} &{\rm i}q_{1t}+q_{1xxx}+\frac{2}{3}{\rm i}[(a|q_1|^2+c|q_2|^2+bq_1q_2^*+b^*q_1^*q_2)q_{1}]_x=0,\\ &{\rm i}q_{2t}+q_{2xxx}+\frac{2}{3}{\rm i}[(a|q_1|^2+c|q_2|^2+bq_1q_2^*+b^*q_1^*q_2)q_{2}]_x=0, \end{split} \end{equation}$

其中 $q_1=q_1(x,t)$, $q_2=q_2(x,t)$ 是复函数, 符号 $*$ 代表了复共轭, $a$ 和 $c$ 是实数, $b$ 是任意复数. 从物理视角分析, $a$ 和 $c$ 描述了自相位调制和交叉相位调制, $b$ 和 $b^*$ 则对应四波混频效应. 方程 (1.1) 源于对耦合导数非线性薛定谔方程的物理深化与数学推广. 当 $a=c$ 以及 $b=0$ 时, 方程 (1.1) 可退化为耦合导数非线性薛定谔方程. 受到耦合导数非线性薛定谔方程的启发可知, 方程 (1.1) 在多个物理领域扮演着关键的理论模型角色, 其核心价值在于精准描述了多种波场在非线性、色散及耦合效应共同作用下的演化动力学. 在非线性光学中, 该方程可用于研究双折射光纤或双核波导中超短脉冲传输, 其中不同偏振态或空间模式的光场通过交叉相位调制相互耦合. 在等离子体领域, 该方程可被用于刻画多组分磁流体中具有不同圆偏振态的阿尔文波的非线性相互作用, 其耦合项反映了波与波之间的能量交换, 对于理解空间等离子体环境中相干结构的形成与稳定性至关重要. 此外, 该模型在描述具有自旋-轨道耦合的玻色-爱因斯坦凝聚体中不同超精细态原子的动力学方面也展现出应用潜力, 为研究多组分量子气体中的非线性现象提供了理论框架. 目前, 针对该方程的研究仅局限于孤子解[46], 其在孤子分子、孤子共振及 breather-positon 等高阶非线性结构方面的系统性研究尚属空白. 本文基于该方程的 Lax 对, 构建了行列式形式的 $N$ 重 Darboux 变换, 并进一步结合极限技巧, 系统推导出了三种类型的解: i) $N$-孤子分析, 揭示了多孤子在速度共振条件下的束缚机制; ii) 孤子共振态, 展示了孤子分子与暗孤子/类呼吸子之间的非弹性相互作用; iii) 高阶 breather-positon 解, 其中心区域在特定极限下可演化为怪波结构.

文章结构如下. 第 2 节基于 Lax 对, 以行列式形式构建了广义耦合导数非线性薛定谔方程 (1.1) 的 $N$ 重 Darboux 变换. 第 3 节通过选取零种子解, 推导出 $N$-孤子分子及孤子共振解, 同时通过复杂的数学推导给出孤子分子与孤子共振的产生条件. 第 4 节采用非零种子解, 结合达布变换与极限技术, 得到高阶 breather-positon 解. 最后一节总结全文核心研究成果.

2 $N$ 重 Darboux 变换

方程 (1.1) 的 Lax 对可写为

$\begin{equation} \begin{split} &\Psi_x=({\rm i}\lambda^2U_0+\lambda U_1)\Psi,\\ &\Psi_t=[{\rm i}\lambda^4V_0+3\lambda^3 U_1-{\rm i}\lambda^2V_1U_1^2+{\rm i}\lambda(V_1U_{1x}-\frac{2}{3}{\rm i}Q^3)]\Psi, \end{split} \end{equation}$

其中

$\begin{equation}\nonumber \begin{split} &U_0= \begin{pmatrix} -2 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix},\quad U_1= \begin{pmatrix} 0 & q_1 & q_2\\ -aq_1^*-bq_2^* & 0 & 0\\ -b^*q_1^*-cq_2^* & 0 & 0 \end{pmatrix},\\ &V_0= \begin{pmatrix} -9 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix},\quad V_1= \begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}, \end{split} \end{equation}$

$\Psi$ 是位势函数, $\lambda$ 是一个谱参数.

接下来, 我们构造方程 (1.1) 的 $N$ 重 Darboux 变换.

令 $\Psi_1=(\psi_1^{(1)},\psi_2^{(1)},\psi_3^{(1)})^T$ 是 Lax 对 (2.1) 关于 $\lambda=\lambda_1$ 的一个基解, 那么可以引入以下 1 重 Darboux 变换

$\begin{equation} T[1]=\frac{\lambda^2-\lambda_1^{*2}}{\lambda_1^{*2}}+\frac{\lambda_1^{*2}-\lambda_1^2}{\lambda_1^{*2}}M(\lambda) \begin{pmatrix} \lambda\varphi_1^{(1)} & \lambda_1^*\varphi_2^{(1)} & \lambda_1^*\varphi_3^{(1)}\\ \lambda_1^*\varphi_1^{(1)} & \lambda\varphi_2^{(1)} & \lambda\varphi_3^{(1)}\\ \lambda_1^*\varphi_1^{(1)} & \lambda\varphi_2^{(1)} & \lambda\varphi_3^{(1)}\\ \end{pmatrix}, \end{equation}$

其中

$\begin{equation} \varphi_1^{(1)}=\psi_1^{(1)*},\;\varphi_2^{(1)}=\frac{-c\psi_2^{(1)*}+b^*\psi_3^{(1)*}}{|b|^2-ac},\;\varphi_3^{(1)}=\frac{b\psi_2^{(1)*}-a\psi_3^{(1)*}}{|b|^2-ac}, \end{equation}$
$\begin{equation} M(\lambda)={\rm diag}\left(\frac{\lambda\psi_1^{(1)}}{\lambda_1D},\frac{\lambda\psi_2^{(1)}}{\lambda_1D^*},\frac{\lambda\psi_3^{(1)}}{\lambda_1D^*}\right), \;D=\lambda_1|\psi_1^{(1)}|^2+\lambda_1^*(|\psi_2^{(1)}|^2+|\psi_3^{(1)}|^2). \end{equation}$

由此, 新旧函数之间的关系可表示为

$\begin{equation} \begin{split} &q_1[ 1]=q_1-\frac{\lambda_1^{*2}-\lambda_1^2}{|\lambda_1|^2}\left(\frac{\varphi_2^{(1)}\psi_1^{(1)}}{D}\right)_x,\\ &q_2[1]=q_2-\frac{\lambda_1^{*2}-\lambda_1^2}{|\lambda_1|^2}\left(\frac{\varphi_3^{(1)}\psi_1^{(1)}}{D}\right)_x. \end{split} \end{equation}$

通过迭代上述 Darboux 变换 $N$ 次, 就能获得 $N$ 重 Darboux 变换. 设 $\Psi_k=(\psi_1^{(k)},\psi_2^{(k)},\psi_3^{(k)})^T(1\leq k \leq N)$ 是 Lax 对 (2.1) 关于 $\lambda=\lambda_k$ 的一个基解. 定义

$\begin{equation} T[k]=\frac{\lambda^2-\lambda_k^{*2}}{\lambda_k^{*2}}+\frac{\lambda_k^{*2}-\lambda_k^2}{\lambda_k^{*2}}M_k(\lambda) \begin{pmatrix} \lambda\varphi_1^{(k)}[k-1] & \lambda_k^*\varphi_2^{(k)}[k-1] & \lambda_k^*\varphi_3^{(k)}[k-1]\\ \lambda_k^*\varphi_1^{(k)}[k-1] & \lambda\varphi_2^{(k)}[k-1] & \lambda\varphi_3^{(k)}[k-1]\\ \lambda_k^*\varphi_1^{(k)}[k-1] & \lambda\varphi_2^{(k)}[k-1] & \lambda\varphi_3^{(k)}[k-1]\\ \end{pmatrix}, \end{equation}$

其中

$ \begin{align*} &M_{k}(\lambda) = \text{diag}\left(\frac{\lambda\psi_{1}^{(k)}[k - 1]}{\lambda_{k}D[k]}, \frac{\lambda\psi_{2}^{(k)}[k - 1]}{\lambda_{k}D^*[k]}, \frac{\lambda\psi_{3}^{(k)}[k - 1]}{\lambda_{k}D^*[k]}\right),\\ &\Psi_{k}[k - 1] = (\psi_{1}^{(k)}[k - 1],\psi_{2}^{(k)}[k - 1],\psi_{3}^{(k)}[k - 1])^{T}\\ &\qquad\qquad= T[k - 1]T[k - 2]\cdots T[1]|_{\lambda_{i}=\lambda_{k}}\Psi_{k},\\ & \Phi_k[k-1]=(\varphi_{1}^{(k)}[k - 1],\varphi_{2}^{(k)}[k - 1],\varphi_{3}^{(k)}[k - 1])^T \\ &\qquad\qquad= B^{-1}\Psi^*_{k}[k - 1],\\ &B = \begin{pmatrix} 1 & 0 & 0\\ 0 & a & b\\ 0 & b^* & c \end{pmatrix}. \end{align*} $

迭代 $N$ 次, 可得

$\begin{equation} T_N=T[N]T[N-1]\cdots T[2]T[1]. \end{equation}$

令

$\begin{equation} \widetilde{\Psi}_{k}^{1}=(-\varphi_{2}^{(k)},\varphi_{1}^{(k)},0)^{T}, \quad \widetilde{\Psi}_{k}^{2}=(-\varphi_{3}^{(k)},0,\varphi_{1}^{(k)})^{T}. \end{equation}$
$\begin{equation} \widetilde{\Psi}_{k}^{l}[k - 1]=T_{k - 1}|_{\lambda=\lambda_{k}}\widetilde{\Psi}_{k}^{l}. \end{equation}$

验证

$\begin{equation} T_{k}|_{\lambda=\lambda_{k}}\widetilde{\Psi}_{k}^{l}=0, \quad l = 1,2. \end{equation}$

因此, 我们得到

$\begin{equation} T_{k}|_{\lambda=\lambda_{k}}\widetilde{\Psi}_{k}^{l}=T[k]|_{\lambda=\lambda_{k}}\widetilde{\Psi}_{k}^{l}[k - 1]. \end{equation}$

根据 $T[k]$ 的表达式, 可知

$\begin{equation} T_{N}|_{\lambda=\lambda_{k}}\Psi_{k}=0,\quad T_{N}|_{\lambda=\lambda_{k}}\widetilde{\Psi}_{k}^{1}=0,\quad T_{N}|_{\lambda=\lambda_{k}}\widetilde{\Psi}_{k}^{2}=0,\quad (k=1,2,...,N). \end{equation}$

基于上述讨论, $q_1[N]$ 和 $q_2[N]$ 可以用行列式的形式表示为

$\begin{equation} q_{1}[N]=q_{1}-\left(\frac{\Omega_{1}}{\Omega}\right)_{x},\quad q_{2}[N]=q_{2}-\left(\frac{\Omega_{2}}{\Omega}\right)_{x}, \end{equation}$

其中

$\begin{equation} \Omega=\begin{vmatrix} \lambda_{1}^{2N}\psi_{1}^{(1)} & \lambda_{1}^{2N - 1}\psi_{2}^{(1)} & \lambda_{1}^{2N - 1}\psi_{3}^{(1)} & \cdots & \lambda_{1}^{2}\psi_{1}^{(1)} & \lambda_{1}\psi_{2}^{(1)} & \lambda_{1}\psi_{3}^{(1)} \\ -\lambda_{1}^{*2N}\varphi_{2}^{(1)} & \lambda_{1}^{*2N - 1}\varphi_{1}^{(1)} & 0 & \cdots & -\lambda_{1}^{*2}\varphi_{2}^{(1)} & \lambda_{1}^*\varphi_{1}^{(1)} & 0 \\ -\lambda_{1}^{*2N}\varphi_{3}^{(1)} & 0 & \lambda_{1}^{*2N - 1}\varphi_{1}^{(1)} & \cdots & -\lambda_{1}^{*2}\varphi_{3}^{(1)} & 0 & \lambda_{1}^*\varphi_{1}^{(1)} \\ \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \\ \lambda_{N}^{2N}\varphi_{1}^{(N)} & \lambda_{N}^{2N - 1}\psi_{2}^{(N)} & \lambda_{N}^{2N - 1}\psi_{3}^{(N)} & \cdots & \lambda_{N}^{2}\psi_{1}^{(N)} & \lambda_{N}\psi_{2}^{(N)} & \lambda_{N}\psi_{3}^{(N)} \\ -\lambda_{N}^{*2N}\varphi_{2}^{(N)} & \lambda_{N}^{*2N-1}\varphi_{1}^{(N)} & 0 & \cdots & -\lambda_{N}^{*2}\varphi_{2}^{(N)} & \lambda_{N}^*\varphi_{1}^{(N)} & 0 \\ -\lambda_{N}^{*2N}\varphi_{3}^{(N)} & 0 & \lambda_{N}^{*2N - 1}\varphi_{1}^{(N)} & \cdots & -\lambda_{N}^{*2}\varphi_{3}^{(N)} & 0 & \lambda_{N}^*\varphi_{1}^{(N)} \end{vmatrix}, \end{equation}$

$\Omega_1$ 和 $\Omega_2$ 分别通过用列向量

$\begin{equation} (\psi_{1}^{(1)},-\varphi_{2}^{(1)},-\varphi_{3}^{(1)},\cdots,\psi_{1}^{(N-1)},-\varphi_{2}^{(N-1)},-\varphi_{3}^{(N-1)},\psi_{1}^{(N)},-\varphi_{2}^{(N)},-\varphi_{3}^{(N)})^{T} \end{equation}$

替换 $\Omega$ 的第 $(3N-1)$ 和 第 $3N$ 列给出.

3 $N$-孤子分子和孤子共振

在此基础上, $N$-孤子分子和孤子共振可以被导出. 选择零种子解 $q_1=q_2=0$ 和 $\lambda_k(k=1,2,...,N)=m_k+{\rm i}n_k$, Lax 对的一个解可表示为

$\begin{equation} \begin{split} &\Psi_k= \left( \begin{array}{c} \psi_1^{(k)}\\ \psi_2^{(k)}\\ \psi_3^{(k)} \end{array} \right) = \left( \begin{array}{c} c_1\mathrm{e}^{-2{\rm i}\lambda_k^2x-9{\rm i}\lambda_k^4t}\\ c_2\mathrm{e}^{{\rm i}\lambda_k^2x}\\ c_3\mathrm{e}^{{\rm i}\lambda_k^2x} \end{array} \right),\\ &\Phi_k=\left( \begin{array}{c} \varphi_1^{(k)}\\ \varphi_2^{(k)}\\ \varphi_3^{(k)} \end{array} \right) =B^{-1}\Psi_k^*, \end{split} \end{equation}$

这里的 $c_j(j=1,2,3)$ 是复数, $m_k$, $n_k$ 是实数. 将方程 (3.1) 代入上述 $N$ 重 Darboux 变换中, 可得 $N$-孤子解.

当 $N=1$ 时, 1-孤子的精确表达式可写为

$\begin{equation} \begin{split} &q_1[ 1]=B_1\frac{(ac-|b|^2)|c_1|^2\lambda_1^*{\rm e}^{8m_1n_1x+72m_1n_1(m_1^2-n_1^2)t}+A\lambda_1{\rm e}^{-4m_1n_1x}}{(ac-|b|^2)|c_1|^2\lambda_1{\rm e}^{8m_1n_1x+72m_1n_1(m_1^2-n_1^2)t}+A\lambda_1^*{\rm e}^{-4m_1n_1x})^2},\\ &q_2[1]=B_2\frac{(ac-|b|^2)|c_1|^2\lambda_1^*{\rm e}^{8m_1n_1x+72m_1n_1(m_1^2-n_1^2)t}+A\lambda_1{\rm e}^{-4m_1n_1x}}{(ac-|b|^2)|c_1|^2\lambda_1{\rm e}^{8m_1n_1x+72m_1n_1(m_1^2-n_1^2)t}+A\lambda_1^*{\rm e}^{-4m_1n_1x})^2}, \end{split} \end{equation}$

其中

$\begin{equation} \begin{split} &A=a|c_3|^2+c|c_2|^2-b^*c_2c_3^*-bc_2^*c_3,\\ &B_1=12c_1(cc_2^*-b^*c_3^*)m_1n_1{\rm e}^{2m_1n_1x+36m_1n_1(m_1^2-n_1^2)t-3i((m_1^2-n_1^2)x+3(m_1^4-6m_1^2n_1^2+n_1^4)t)},\\ &B_2=-6c_1(ac_3^*-bc_2^*)m_1n_1{\rm e}^{2m_1n_1x+36m_1n_1(m_1^2-n_1^2)t-3i((m_1^2-n_1^2)x+3(m_1^4-6m_1^2n_1^2+n_1^4)t)}.\\ \end{split} \end{equation}$

显然, 1-孤子的速度为

$\begin{equation} v_1=6(n_1^2-m_1^2). \end{equation}$

接下来, 我们讨论 $N\geqslant2$ 的情况. 当两个孤子以相同的速度运动, 就会产生 1-孤子分子. 换言之, 当满足条件 $n_k^2-m_k^2=n_l^2-m_l^2(k\neq l,k,l=1,2,...,N)$ 时, $N$-孤子分子就会出现. 但是, 当满足条件 $n_k^2-m_k^2\neq n_l^2-m_l^2(k\neq l,k,l=1,2,...,N)$ 时, 那么呈现的将会是孤子共振.

当 $N=2$ 时, 我们得到了 2-孤子分子和孤子共振的表达式, 如下:

$\begin{equation} q_{1}[2]=-\left(\frac{\Omega_{1}}{\Omega}\right)_{x},\quad q_{2}[2]=-\left(\frac{\Omega_{2}}{\Omega}\right)_{x}, \end{equation}$

其中

$\begin{equation} \Omega=\begin{vmatrix} \lambda_{1}^{4}\psi_{1}^{(1)} & \lambda_{1}^{3}\psi_{2}^{(1)} & \lambda_{1}^{3}\psi_{3}^{(1)} & \lambda_{1}^{2}\psi_{1}^{(1)} & \lambda_{1}\psi_{2}^{(1)} & \lambda_{1}\psi_{3}^{(1)} \\ -\lambda_{1}^{*4}\varphi_{2}^{(1)} & \lambda_{1}^{*3}\varphi_{1}^{(1)} & 0 & -\lambda_{1}^{*2}\varphi_{2}^{(1)} & \lambda_{1}^*\varphi_{1}^{(1)} & 0 \\ -\lambda_{1}^{*4}\varphi_{3}^{(1)} & 0 & \lambda_{1}^{*3}\varphi_{1}^{(1)} & -\lambda_{1}^{*2}\varphi_{3}^{(1)} & 0 & \lambda_{1}^*\varphi_{1}^{(1)} \\ \lambda_{2}^{4}\varphi_{1}^{(2)} & \lambda_{2}^{3}\psi_{2}^{(2)} & \lambda_{2}^{3}\psi_{3}^{(2)} & \lambda_{2}^{2}\psi_{1}^{(2)} & \lambda_{2}\psi_{2}^{(2)} & \lambda_{2}\psi_{3}^{(2)} \\ -\lambda_{2}^{*4}\varphi_{2}^{(2)} & \lambda_{2}^{*3}\varphi_{1}^{(2)} & 0 & -\lambda_{2}^{*2}\varphi_{2}^{(2)} & \lambda_{2}^*\varphi_{1}^{(2)} & 0 \\ -\lambda_{2}^{*4}\varphi_{3}^{(2)} & 0 & \lambda_{2}^{*3}\varphi_{1}^{(2)} & -\lambda_{2}^{*2}\varphi_{3}^{(2)} & 0 & \lambda_{2}^*\varphi_{1}^{(2)} \end{vmatrix}, \end{equation}$

$\Omega_1$ 和 $\Omega_2$ 分别通过用列向量

$\begin{equation} \widetilde{\Omega}=(\psi_{1}^{(1)},-\varphi_{2}^{(1)},-\varphi_{3}^{(1)},\psi_{1}^{(2)},-\varphi_{2}^{(2)},-\varphi_{3}^{(2)})^{T} \end{equation}$

替换 $\Omega$ 的第 5 和 第 6 列给出.

与 1-孤子情形类似, 2-孤子中的两个孤子的速度为

$\begin{equation} v_k=6(n_k^2-m_k^2)(k=1,2). \end{equation}$

因此, 当 $n_2^2-m_2^2=n_1^2-m_1^2$, 我们就可以导出两种类型的 2-孤子分子, 见图1-2. 图1 展示了类呼吸型孤子. 图2呈现了带有束缚态结构的孤子分子.

图1

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图1   2-孤子分子, 参数为 $a=\frac{1}{2}$, $b=1+{\rm i}$, $c=\frac{1}{2}$, $c_1=1+{\rm i}$, $c_2=1$, $c_3=1$, $m_1=\frac{1}{2}$, $n_1=1$, $m_2=\frac{3}{2}$, $n_2=\sqrt{3}$.


图2

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图2   2-孤子分子, 参数为 $a=\frac{1}{2}$, $b=1+{\rm i}$, $c=\frac{1}{2}$, $c_1=1+\frac{{\rm i}}{2}$, $c_2=1$, $c_3=1$, $m_1=\frac{\sqrt{2}}{2}$, $n_1=\frac{\sqrt{3}}{2}$, $m_2=1$, $n_2=\frac{\sqrt{5}}{2}$.


同理, 在 Darboux 变换 (2.13)-(2.14) 中取 $N=3$ 并令 $n_3^2-m_3^2=n_2^2-m_2^2=n_1^2-m_1^2$, 两种类型的 3-孤子分子呈现于图2-4. 从图3 中可以发现平行的类呼吸子型孤子分子. 在图4 中, 观察到 3-孤子分子由一个带有束缚态结构的 2-孤子分子和类呼吸子型的孤子分子组成. 若令 $n_3^2-m_3^2\neq n_2^2-m_2^2\neq n_1^2-m_1^2$, 孤子共振就产生了, 见图5-6. 图5 所示为由 2-孤子分子和1-暗孤子所组成的孤子共振, 而图6展示的是由 2-孤子分子和 1-类呼吸子型孤子组成的孤子共振.

图3

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图3   3-孤子分子, 参数为 $a=\frac{1}{2}$, $b=1+{\rm i}$, $c=\frac{1}{2}$, $c_1=1+{\rm i}$, $c_2=1$, $c_3=1$, $m_1=1$, $n_1=-1$, $m_2=\frac{1}{2}$, $n_2=\frac{1}{2}$, $m_3=2$, $n_3=-2$.


图4

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图4   3-孤子分子, 参数为 $a=\frac{1}{2}$, $b=1+{\rm i}$, $c=\frac{1}{2}$, $c_1=1+{\rm i}$, $c_2=1$, $c_3=1$, $m_1=1$, $n_1=\frac{1}{2}$, $m_2=\frac{\sqrt{6}}{2}$, $n_2=\frac{3}{2}$, $m_3=\frac{\sqrt{2}}{2}$, $n_3=\frac{\sqrt{5}}{2}$.


图5

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图5   2-孤子分子与 1-暗孤子的孤子共振, 参数为 $a=\frac{1}{2}$, $b=1+{\rm i}$, $c=\frac{1}{2}$, $c_1=1+{\rm i}$, $c_2=1$, $c_3=1$, $m_1=1$, $n_1=-1$, $m_2=\frac{2}{3}$, $n_2=-1$, $m_3=-\frac{1}{4}$, $n_3=\frac{1}{2}$.


图6

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图6   2-孤子分子与 1-类呼吸子型孤子的孤子共振, 参数为 $a=\frac{1}{2}$, $b=1+{\rm i}$, $c=\frac{1}{2}$, $c_1=1+{\rm i}$, $c_2=1$, $c_3=1$, $m_1=\frac{1}{10}$, $n_1=\frac{3}{10}$, $m_2=\frac{4}{5}$, $n_2=-1$, $m_3=-\frac{1}{5}$, $n_3=1$.


4 高阶 breather-positon 解

这一部分, 借助上述 Darboux 变换和极限技巧, 构造了高阶 breather-positon 解. 选取非零种子解

$\begin{equation} q_1=a_1\mathrm{e}^{-\frac{2}{3}{\rm i}\theta x},\quad q_2=a_2\mathrm{e}^{-\frac{2}{3}{\rm i}\theta x},\quad \theta=aa_1^2+a_1a_2{\rm Re}(b)+ca_2^2, \end{equation}$

这里的 $a_1$ 和 $a_2$ 是实数.

通过符号计算, Lax 对 (2.1) 的解可表示为

$\begin{equation} \Psi_{1}(x,t,\lambda)=\begin{pmatrix} (l_{1}\mathrm{e}^{\Xi_{1}+\Xi_{2}} - l_{2}\mathrm{e}^{\Xi_{1}-\Xi_{2}})\mathrm{e}^{-\frac{{\rm i}}{3}\theta x} \\ \rho_{1}(l_{1}\mathrm{e}^{\Xi_{1}-\Xi_{2}}+l_{2}\mathrm{e}^{\Xi_{1}+\Xi_{2}})\mathrm{e}^{\frac{{\rm i}}{3}\theta x} \\ \rho_{2}(l_{1}\mathrm{e}^{\Xi_{1}-\Xi_{2}} - l_{2}\mathrm{e}^{\Xi_{1}+\Xi_{2}})\mathrm{e}^{\frac{{\rm i}}{3}\theta x} \end{pmatrix}, \end{equation}$

其中

$ \begin{align*} &\Xi_{1} = -\frac{{\rm i}}{2}\lambda^{2}(x + 9\lambda^{2}t), \\ &\Xi_{2} = \frac{{\rm i}}{6}\sqrt{81\lambda^{4}+4\theta^{2}}\left(3\lambda^{2}t + x + \sum_{k = 1}^{N} (p_{k}+{\rm i}q_{k})\delta^{k}\right), \\ &l_{1} = \frac{{\rm i}(9\lambda^{2}-2\theta-\sqrt{81\lambda^{4}+4\theta^{2}})^{\frac{1}{2}}}{\sqrt{81\lambda^{4}+4\theta^{2}}}, \\ &l_{2} = -\frac{{\rm i}(9\lambda^{2}-2\theta+\sqrt{81\lambda^{4}+4\theta^{2}})^{\frac{1}{2}}}{\sqrt{81\lambda^{4}+4\theta^{2}}}, \\ &\tau = a_{1}^{2}+a_{2}^{2} \quad \rho_{1} = \frac{a_{1}}{\sqrt{\tau}}, \rho_{2} = \frac{a_{2}}{\sqrt{\tau}}, \end{align*} $

$p_k, q_k$$(k=1,2,...,N)$ 均为实数.

令 $\lambda=\lambda_1+\epsilon$, $\lambda_1=m_1+{\rm i}n_1$, $\Psi_1$ 在 $\epsilon=0$ 处泰勒展开

$ \begin{align*} (\lambda_{1}+\epsilon)^{j}\psi_{1}&=\psi_1[j,0]+\psi_1[j,1]\epsilon+\cdots+\psi_1[j,k]\epsilon^{k}+\cdots, \\ (\lambda_{1}+\epsilon)^{j}\psi_{2}&=\psi_2[j,0]+\psi_2[j,1]\epsilon+\cdots+\psi_2[j,k]\epsilon^{k}+\cdots, \\ (\lambda_{1}+\epsilon)^{j}\psi_{3}&=\psi_3[j,0]+\psi_3[j,1]\epsilon+\cdots+\psi_3[j,k]\epsilon^{k}+\cdots, \end{align*} $

其中

$ \begin{align*} \psi_1[j,k]&=\frac{1}{k!}\frac{\partial^{k}}{\partial\epsilon^{k}}\left[(\lambda_{1}+\epsilon)^{j}\psi_1(\lambda_{1}+\epsilon)\right],\\ \psi_2[j,k]&=\frac{1}{k!}\frac{\partial^{k}}{\partial\epsilon^{k}}\left[(\lambda_{1}+\epsilon)^{j}\psi_2(\lambda_{1}+\epsilon)\right],\\ \psi_3[j,k]&=\frac{1}{k!}\frac{\partial^{k}}{\partial\epsilon^{k}}\left[(\lambda_{1}+\epsilon)^{j}\psi_3(\lambda_{1}+\epsilon)\right]. \end{align*} $

基于上述讨论, 我们呈现了高阶 breather-positon 解

$\begin{equation} q_{1}[N]=q_{1}-\left(\frac{\Omega_{1}}{\Omega}\right)_{x},\quad q_{2}[N]=q_{2}-\left(\frac{\Omega_{2}}{\Omega}\right)_{x}, \end{equation}$

其中

$\begin{equation} \Omega=\begin{vmatrix} \lambda_{1}^{2N}\psi_{1}[0,0] & \lambda_{1}^{2N - 1}\psi_{2}[0,0] & \lambda_{1}^{2N - 1}\psi_{3}[0,0] & \cdots & \lambda_{1}^{2}\psi_{1}[0,0] & \lambda_{1}\psi_{2}[0,0] & \lambda_{1}\psi_{3}[0,0] \\ -\lambda_{1}^{*2N}\varphi_{2}[0,0] & \lambda_{1}^{*2N - 1}\varphi_{1}[0,0] & 0 & \cdots & -\lambda_{1}^{*2}\varphi_{2}[0,0] & \lambda_{1}^{*}\varphi_{1}[0,0] & 0 \\ -\lambda_{1}^{*2N}\varphi_{3}[0,0] & 0 & \lambda_{1}^{*2N - 1}\varphi_{1}[0,0] & \cdots & -\lambda_{1}^{*2}\varphi_{3}[0,0] & 0 & \lambda_{1}^{*}\varphi_{1}[0,0] \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ \psi_{1}[N - 1] & \psi_{2}[N - 1] & \psi_{3}[N - 1] & \cdots & \psi_{1}[N - 1] & \psi_{2}[N - 1] & \psi_{3}[N - 1] \\ -\varphi_{2}[N - 1] & \varphi_{1}[N - 1] & 0 & \cdots & -\varphi_{2}[N - 1] & \varphi_{1}[N - 1] & 0 \\ -\varphi_{3}[N - 1] & 0 & \varphi_{1}[N - 1] & \cdots & -\varphi_{3}[N - 1] & 0 & \varphi_{1}[N - 1] \end{vmatrix}, \end{equation}$

$\Omega_1$ 和 $\Omega_2$ 分别通过用列向量

$ \begin{align*} \widetilde{\Omega}=&(\psi_{1}[0,0],-\varphi_{2}[0,0],-\varphi_{3}[0,0],...,\psi_{1}[N-2],\\ &-\varphi_{2}[N-2],-\varphi_{3}[N-2],\psi_{1}[N-1],-\varphi_{2}[N-1],-\varphi_{3}[N-1])^{T} \end{align*} $

替换 $\Omega$ 的第 $(3N-1)$ 和 第 $3N$ 列给出.

当 $N=1$ 时, 从图7-8中可以看到两种不同类型的一阶 breather-positon 解. 值得注意的是, 不同于呼吸子, 一阶 breather-positon 解的表达式可表示为有理函数与指数函数的组合. 一方面, 能观察到图7中的一阶 breather-positon 解关于 $t$ 轴呈周期性, 而图8 中的一阶 breather-positon 解既关于 $x$ 轴, 也关于 $t$ 轴呈周期性. 另一方面, 当参数发生变化时, 一阶 breather-positon 解的振幅会发生变化.

图7

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图7   一阶 breather-positon 解, 参数为 $a_1=1$, $a_2=1$, $a=\frac{1}{2}$, $b=1+{\rm i}$, $c=\frac{1}{2}$, $m_1=1$, $n_1=-1$.


图8

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图8   一阶 breather-positon 解, 参数为 $a_1=1$, $a_2=1$, $a=\frac{1}{2}$, $b=1+{\rm i}$, $c=\frac{1}{2}$, $m_1=1$, $n_1=-\frac{1}{2}$.


接下来, 我们以图8 中的一阶 breather-positon 解 (分量 $q_1$) 为例, 进行数值模拟探究其在极限行为下的演化过程. 图9 展示了 breather-positon 解在时空上的完整演化结构. 可以观察到, 通过参数极限, breather-positon 解中心区域的振幅增强, 同时振荡结构逐渐平滑, 最终演化为类怪波结构, 其呈现出典型的时空局域化特征, 最大振幅达到了 2.39, 且位于 (-0.03, -0.05). 这种强烈的振幅放大效应证实了解在特定参数条件下向怪波模式演化的潜在能力.

图9

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图9   一阶 breather-positon 演化为类怪波过程,参数与图8相同.


当 $N=2$ 时, 可推导出二阶 breather-positon 解, 见图10-图11. 有趣的是, 随着 $p_k$, $q_k$ 的值变化, 两个 breather-positon 解之间的距离会变大. 由此可推断, $p_k,q_k(k=1,2,...,N)$ 的值均会对高阶 breather-positon 的结构产生显著影响.

图10

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图10   二阶 breather-positon 解, 参数为 $a_1=1$, $a_2=1$, $a=\frac{1}{2}$, $b=1+{\rm i}$, $c=\frac{1}{2}$, $m_1=1$, $n_1=-1$, $p_k=0$, $q_k=0$.


图11

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图11   二阶 breather-positon 解, 参数为 $a_1=1$, $a_2=1$, $a=\frac{1}{2}$, $b=1+{\rm i}$, $c=\frac{1}{2}$, $m_1=1$, $n_1=-1$, $p_k=200$, $q_k=200$.


5 结论

本文系统研究了广义耦合导数非线性薛定谔方程, 该方程在描述窄脉冲非线性自陡峭效应方面具有重要作用. 基于 Lax 对, 成功构建了行列式形式的 $N$ 重 Darboux 变换, 不仅为该系统的可积性提供了结构支撑, 也为系统性地构造高阶精确解奠定了坚实的理论基础. 结合极限技巧, 推导出多种类型的孤子分子、孤子共振及高阶 breather-positon 解. 具体而言, 在零种子解背景下, 通过精确控制速度共振条件, 实现了 $N$-孤子分子的稳定束缚, 并进一步观察到其与暗孤子或类呼吸子孤子之间的非弹性共振现象, 展现了多孤子系统中丰富的相互作用机制. 在非零种子解背景下, 所构造的高阶 breather-positon 解兼具时空局域振荡与代数衰减特性, 且其中心区域可通过参数极限演化为类怪波结构, 为探索怪波的激发机制提供了理论视角与潜在路径. 本工作为非线性光学、流体动力学等物理领域中的复杂波结构的调控与观测提供了理论依据. 未来研究将进一步考察该类解的动力学稳定性, 探索其在更多变系数或非均匀可积模型中的推广, 并推动在超短脉冲激光及深水波等物理场景中的实验验证与应用.

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Solitons, particle-like excitations ubiquitous in many fields of physics, have been shown to exhibit bound states akin to molecules. The formation of such temporal soliton bound states and their internal dynamics have escaped direct experimental observation. By means of an emerging time-stretch technique, we resolve the evolution of femtosecond soliton molecules in the cavity of a few-cycle mode-locked laser. We track two- and three-soliton bound states over hundreds of thousands of consecutive cavity roundtrips, identifying fixed points and periodic and aperiodic molecular orbits. A class of trajectories acquires a path-dependent geometrical phase, implying that its dynamics may be topologically protected. These findings highlight the importance of real-time detection in resolving interactions in complex nonlinear systems, including the dynamics of soliton bound states, breathers, and rogue waves.Copyright © 2017, American Association for the Advancement of Science.

Liu X M, Yao X K, Cui Y D.

Real-time observation of the buildup of soliton molecules

Phys Rev Lett, 2018, 121: 23905

DOI:10.1103/PhysRevLett.121.023905      PMID:30085749      [本文引用: 1]

Real-time spectroscopy access to ultrafast fiber lasers opens new opportunities for exploring complex soliton interaction dynamics. Here, we have reported the first observation, to the best of our knowledge, of the entire buildup process of soliton molecules (SMs) in a mode-locked laser. We have observed that the birth dynamics of a stable SM experiences five different stages, i.e., the raised relaxation oscillation (RO) stage, beating dynamics stage, transient single pulse stage, transient bound state, and finally the stable bound state. We have discovered that the evolution of pulses in the raised RO stage follows a law that only the strongest one can ultimately survive and, meanwhile, the pulses periodically appear at the same temporal positions for all lasing spikes during the same RO stage (named as memory ability) but they lose such ability between different RO stages. Moreover, we have found that the buildup dynamics of SMs is quite sensitive to both the polarization state of intracavity light and the fluctuation of pump power. These results provide new perspectives into the ultrafast transient process in mode-locked lasers and the dynamics of complex nonlinear systems.

Lou S Y.

Soliton molecules and asymmetric solitons in three fifth order systems via velocity resonance

J Phys Commun, 2020, 4: 041002

DOI:10.1088/2399-6528/ab833e      [本文引用: 1]

Soliton molecules can be formed in some possible mechanisms both theoretically and experimentally. In this paper, we introduce a new mechanism, namely the velocity resonant, to find soliton molecules. Under the velocity resonance mechanism, two solitons can form a kink-antikink molecule, an asymmetric soliton, a two-peak soliton and/or a far away bounded molecule depended on the selections of the wave numbers and the distance between two solitons of a molecule. The results are exhibited via three well known fifth order integrable systems which serve as a general fluid model, as well as models many other physical fields.

Yan Z W, Lou S Y.

Soliton molecules in Sharma-Tasso-Olver-Burgers equation

Appl Math Lett, 2020, 104: 106271

DOI:10.1016/j.aml.2020.106271      URL     [本文引用: 1]

Kopidakis G, Aubry S, Tsironis G P.

Targeted energy transfer through discrete breathers in nonlinear systems

Phys Rev Lett, 2001, 87: 165501

DOI:10.1103/PhysRevLett.87.165501      URL     [本文引用: 1]

Li B Q, Ma Y L.

Soliton resonances and soliton molecules of pump wave and Stokes wave for a transient stimulated Raman scattering system in optics

Eur Phys J Plus, 2022, 137: 1227

DOI:10.1140/epjp/s13360-022-03455-3      [本文引用: 1]

Li B Q, Ma Y L.

Optical soliton resonances and soliton molecules for the Lakshmanan-Porsezian-Daniel system in nonlinear optics

Nonlinear Dyn, 2023, 111: 6689-6699

DOI:10.1007/s11071-022-08195-8      [本文引用: 1]

Matveev V B.

Positons: Slowly decreasing analogues of solitons

Theoret Math Phys, 2022, 131: 483-497

DOI:10.1023/A:1015149618529      [本文引用: 1]

Kedziora D J, Ankiewicz A, Akhmediev N.

Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits

Phys Rev E, 2012, 85: 066601

DOI:10.1103/PhysRevE.85.066601      URL     [本文引用: 1]

Chowdury A, Krolikowski W, Akhmediev A.

Breather solutions of a fourth-order nonlinear Schrödinger equation in the degenerate, soliton, and rogue wave limits

Phys Rev E, 2017, 96: 042209

DOI:10.1103/PhysRevE.96.042209      URL     [本文引用: 1]

Wang L H, He J S, Xu H, et al.

Generation of higher-order rogue waves from multibreathers by double degeneracy in an optical fiber

Phys Rev E, 2017, 95: 042217

DOI:10.1103/PhysRevE.95.042217      URL     [本文引用: 1]

Stahlhofen A A.

Positons of the modified Korteweg-de Vries equation

Ann Phys Berlin, 1992, 504: 554-569

DOI:10.1002/andp.v504:7      URL     [本文引用: 1]

Song W J, Xu S W, Li M H, He J S.

Generating mechanism and dynamic of the smooth positons for the derivative nonlinear Schrödinger equation

Nonlinear Dyn, 2019, 97: 2135-2145

DOI:10.1007/s11071-019-05111-5      [本文引用: 1]

Li P, He J S, Li M H.

The higher-order positon and breather-positon solutions for the complex short pulse equation

Nonlinear Dyn, 2024, 112: 10239-10258

DOI:10.1007/s11071-024-09503-0      [本文引用: 1]

Liu W, Zhang Y S, He J S.

Dynamics of the smooth positons of the complex modified KdV equation

Wave Random Complex, 2018, 28: 203-214

DOI:10.1080/17455030.2017.1335916      URL     [本文引用: 1]

Zhang Z, Li B, Chen J C, Guo Q.

Construction of higher-order smooth positons and breather positons via Hirota's bilinear method

Nonlinear Dyn, 2021, 105: 2611-2618

DOI:10.1007/s11071-021-06751-2      [本文引用: 1]

Lv N N, Huang L.

Breather-soliton molecules and breather-positons for the extended complex modified KdV equation

Commun Nonlinear Sci Numer Simul, 2022, 107: 106148

DOI:10.1016/j.cnsns.2021.106148      URL     [本文引用: 1]

Yang J, Tian H J.

Nth-order smooth positon and breather-positon solutions for the generalized integrable discrete nonlinear Schrödinger equation

Nonliear Dyn, 2023, 111: 5629-5639

[本文引用: 1]

Xu T, Zhu J Y.

Soliton molecules and breather positon solutions for the coupled modified nonlinear Schrödinger equation

Wave Motion, 2024, 129: 103347

DOI:10.1016/j.wavemoti.2024.103347      URL     [本文引用: 1]

Kuang Y H.

A general coupled derivative nonlinear Schrödinger system: Darboux transformation and soliton solutions

J Nonlinear Math Phys, 2024, 31: 46

DOI:10.1007/s44198-024-00212-1      [本文引用: 1]

In this work we present a general coupled derivative nonlinear Schrödinger system. We construct the corresponding N-fold Darboux transform and generalized Darboux transform. Under this construction, we give different soliton solutions and plot their figures describing the soliton characteristics and dynamical behaviors, including higher-order soliton and rouge wave solution etc.

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