双线性方法求n分量非线性薛定谔方程的N孤子解

N-soliton solutions of n-component nonlinear Schrödinger equation via Hirota method

摘要

利用Hirota双线性方法给出了光纤通信中重要模型n分量非线性薛定谔方程（Nonlinear Schrödinger equation, NLSE）的N孤子解一般表达式。作为例子，给出了三分量NLSE和六分量NLSE丰富的解，包括单孤子解、双孤子解和三孤子解，并通过取适当的参数，绘制了孤子解、呼吸解以及相互作用解等的图形，详细讨论了参数对波形的影响。这些结果丰富了n分量NLSE的孤子解的相关研究。

Abstract

The general expression of N-soliton solution of the n-component nonlinear Schrödinger equation (NLSE), an important model in optical fiber communication, was given by using Hirota bilinear method. As an example, a variety of solutions for three-component NLSE and six-component NLSE were given, including single-soliton solution, two-soliton solution and three-soliton solution. The graphs of soliton solutions, breather solutions and interaction solutions were plotted by taking appropriate parameters, and the influence of the parameters on the waveform was discussed in detail. These results could enrich the study of soliton solutions of n-component NLSE.

Graphical abstract

关键词

n分量非线性薛定谔方程 / 双线性形式 / 孤子解 / 呼吸解 / 相互作用解

Key words

n-component nonlinear Schrödinger equation / bilinear form / soliton solution / breather solution / interaction solution

求解非线性发展方程是研究非线性领域的核心内容。由于非线性的复杂性，求精确解还没有统一的方法，常用于求解非线性发展方程的方法有Lie对称法^[1]、反散射法^[2]、Darboux变换法^[3]、Hirota双线性法^[4]、Tanh函数法^[5]、齐次平衡法^[6]等。Hirota双线性方法是一种重要而直接的方法，自1971年Hirota提出双线性方法以来，这一方法一直保持较强的生命力^[7]。基于该方法构造非线性方程的解，对数学、物理、力学等学科研究具有重要的理论和应用价值^[8]。目前随着这些方法的发展，很多非线性发展方程的求解也得到了快速发展，但仍有许多需要深入研究求精确解的工作。

在光纤通信中，NLSE起着至关重要的作用^[9]。光孤子的传播通常可以用NLSE来描述^[10]。NLSE是可积系统里重要的非线性模型。学者们对孤子在单个NLSE和耦合NLSE中的传播问题进行了广泛研究，利用多种方法得到了NLSE的孤子解^[11-12]、呼吸解^[13-14]、怪波解^[15-16]、Lump解^[17-18]、周期波解^[19]和行波解^[20]。但是，目前关于n分量NLSE的N孤子解的研究较少。

综上所述，本文将经典耦合NLSE推广到多个分量相互作用的n分量NLSE。由于孤子的多分量结构，它们的传播和相互作用比单分量或者耦合的孤子更加丰富^[21]，一些物理现象需要三个或者更多分量的NLSE来描述。本文采用Hirota方法给出了n分量NLSE的N孤子解的一般表达式，绘制出了一些重要波解。

1 n 分量NLSE的双线性形式

考虑经典耦合NLSE方程：

\left\{\begin{array}{l}i{q}_{\mathrm{1},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{1},xx}+\sigma \left(|q_{\mathrm{1}}{|}^{\mathrm{2}}+\alpha |q_{\mathrm{2}}{|}^{\mathrm{2}}\right)q_{\mathrm{1}}=\mathrm{0}\\ i{q}_{\mathrm{2},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{2},xx}+\sigma \left(\alpha |q_{\mathrm{1}}{|}^{\mathrm{2}}+|q_{\mathrm{2}}{|}^{\mathrm{2}}\right)q_{\mathrm{2}}=\mathrm{0}\end{array}\right.(1)

其中，q_{\mathrm{1}}和q_{\mathrm{2}}为变量x和t的复函数，\sigma为任意常数，\alpha >\mathrm{0}。上述方程除\alpha =\mathrm{1}外，一般不完全可积，当\alpha =\mathrm{1}时，该方程组为可积Manakov方程^[22]。Manakov首次研究发现耦合NLSE中，当方程耦合仅通过交叉相位调制实现，且自相位调制系数等于交叉相位调制系数时，NLSE是可积的^[23-24]。

\alpha取\mathrm{1}，考虑由式(1)推广的n分量NLSE：

i{q}_{j,t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{j,xx}+\sigma \left({\displaystyle \sum _{l=\mathrm{1}}^n}{q}_l\cdot q_l^{\mathrm{*}}\right)q_j=\mathrm{0},(j=\mathrm{1},\mathrm{2},\dots ,n)(2)

其中，q_j为变量x和t的复函数。符号*表示复共轭。

假设

q_j\left(x,t\right)=\frac{g_j\left(x,t\right)}{f\left(x,t\right)},(j=\mathrm{1},\mathrm{2},\dots ,n)(3)

式(3)代入到方程(2)，得到：

\begin{array}{l}i{\left(\frac{g_j\left(x,t\right)}{f\left(x,t\right)}\right)}_t+\frac{\mathrm{1}}{\mathrm{2}}{\left(\frac{g_j\left(x,t\right)}{f\left(x,t\right)}\right)}_{xx}\\ +\sigma \left({\displaystyle \sum _{r=\mathrm{1}}^n}\frac{g_r\left(x,t\right)}{f\left(x,t\right)}\cdot \frac{g_r^{\mathrm{*}}\left(x,t\right)}{f\left(x,t\right)}\right)\frac{g_j\left(x,t\right)}{f\left(x,t\right)}\\ =\mathrm{0},(j=\mathrm{1},\mathrm{2},\dots ,n)\end{array}(4)

式(4)写成双线性导数形式：

\begin{array}{c}\left(i{D}_t+D_x^{\mathrm{2}}\right)g_j\cdot f+\frac{g_j}{f}\left(-\frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}f\cdot f+\sigma \left({\displaystyle \sum _{r=\mathrm{1}}^n}{g}_r\cdot g_r^{\mathrm{*}}\right)g_j\right)\\ =\mathrm{0},\left(j=\mathrm{1},\mathrm{2},\dots ,n\right)\end{array}(5)

通过分离线性部分和非线性部分并使它们等于零，得到

\left\{\begin{array}{l}\left(i{D}_t+\frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\right)g_j\cdot f=\mathrm{0},\left(j=\mathrm{1},\mathrm{2},\dots ,n\right)\\ \frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}f\cdot f=\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}\left(|g_r{|}^{\mathrm{2}}\right)\end{array}\right.(6)

其中，D_t和D_x^{\mathrm{2}}为双线性导数。双线性导数的定义如下：

\begin{array}{l}{D}_x^{m_{\mathrm{1}}}{D}_t^{m_{\mathrm{2}}}\left(g\cdot f\right)\\ ={\left(\frac{\partial }{\partial x}-\frac{\partial }{\partial x_{\mathrm{1}}}\right)}^{m_{\mathrm{1}}}{\left(\frac{\partial }{\partial t}-\frac{\partial }{\partial t_{\mathrm{1}}}\right)}^{m_{\mathrm{2}}}{\left.g\left(x,t\right)f\left(x_{\mathrm{1}},t_{\mathrm{1}}\right)\right|}_{\begin{array}{l}{x}_{\mathrm{1}}=x,\\ t_{\mathrm{1}}=t\end{array}}\\ ={\displaystyle \sum _{j=\mathrm{1}}^{m_{\mathrm{1}}}}{\displaystyle \sum _{j=\mathrm{1}}^{m_{\mathrm{2}}}}\frac{{\left(-\mathrm{1}\right)}^{m_{\mathrm{1}}+m_{\mathrm{2}}-j-r}{m}_{\mathrm{1}}!}{j!\left(m_{\mathrm{1}}-j\right)!}\frac{m_{\mathrm{2}}!}{r!\left(m_{\mathrm{2}}-r\right)!}\frac{{\partial }^{r+j}g}{{\partial }_{t^r}{\partial }_{x^j}}\frac{{\partial }^{m_{\mathrm{1}}+m_{\mathrm{2}}-r-j}f}{{\partial }_t^{m_{\mathrm{2}}-r}{\partial }_x^{m_{\mathrm{1}}-r}},\end{array}

其中，m和n为整数，g和f分别是关于变量x和t，x_{\mathrm{1}}和t_{\mathrm{1}}的函数。

2 n 分量NLSE的孤子解

为了得到方程(2)的孤子解，还需要函数g_j\left(x,t\right)\left(j=\mathrm{1},\mathrm{2},\dots ,n\right)和f(x,t)的有限摄动展开，其中复函数g_j\left(x,t\right)\left(j=\mathrm{1},\mathrm{2},\dots ,n\right)可由小参数\epsilon的奇次幂展开为级数，实函数f(x,t)由小参数\epsilon的偶次幂展开为级数：

\left\{\begin{array}{l}f\left(x,t\right)=\mathrm{1}+{\epsilon }^{\mathrm{2}}{f}^{(\mathrm{2})}\left(x,t\right)+{\epsilon }^{\mathrm{4}}{f}^{(\mathrm{4})}\left(x,t\right)+{\epsilon }^{\mathrm{6}}{f}^{(\mathrm{6})}\left(x,t\right)+\dots \\ g_j\left(x,t\right)=\epsilon g_j^{(\mathrm{1})}\left(x,t\right)+{\epsilon }^{\mathrm{3}}{g}_j^{(\mathrm{3})}\left(x,t\right)+{\epsilon }^{\mathrm{5}}{g}_j^{(\mathrm{5})}\left(x,t\right)+\dots ,\\ \left(j=\mathrm{1},\mathrm{2},\mathrm{3},\dots ,n\right)\end{array}\right.(7)

将上述展开式代入双线性式(6)，令\epsilon中同次幂系数相等，得到

\begin{array}{l}{\epsilon }^{\mathrm{1}}:\\ \left(i{D}_t+\frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\right)g_j^{(\mathrm{1})}\cdot \mathrm{1}=\mathrm{0}\\ {\epsilon }^{\mathrm{2}}:\\ \frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\left(\mathrm{1}\cdot f^{(\mathrm{2})}+f^{(\mathrm{2})}\cdot \mathrm{1}\right)=\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}\left(g_r^{(\mathrm{1})}\cdot g_r^{{(\mathrm{1})}^{\mathrm{*}}}\right)\\ {\epsilon }^{\mathrm{3}}:\\ \left(i{D}_t+\frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\right)\left(g_j^{\left(\mathrm{1}\right)}\cdot f^{\left(\mathrm{2}\right)}+g_j^{\left(\mathrm{3}\right)}\cdot \mathrm{1}\right)=\mathrm{0}\\ {\epsilon }^{\mathrm{4}}:\\ \frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\left(\mathrm{1}\cdot f^{(\mathrm{4})}+f^{(\mathrm{2})}\cdot f^{(\mathrm{2})}+f^{(\mathrm{4})}\cdot \mathrm{1}\right)=\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}\left(g_r^{(\mathrm{1})}\cdot g_r^{{(\mathrm{3})}^{\mathrm{*}}}+g_r^{(\mathrm{3})}\cdot g_r^{{(\mathrm{1})}^{\mathrm{*}}}\right)\\ {\epsilon }^{\mathrm{5}}:\\ \left(i{D}_t+\frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\right)\left(g_j^{\left(\mathrm{1}\right)}\cdot f^{\left(\mathrm{4}\right)}+g_j^{\left(\mathrm{3}\right)}\cdot f^{\left(\mathrm{2}\right)}+g_j^{\left(\mathrm{5}\right)}\cdot \mathrm{1}\right)=\mathrm{0}\\ {\epsilon }^{\mathrm{6}}:\\ \frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\left(\mathrm{1}\cdot f^{(\mathrm{6})}+f^{(\mathrm{2})}\cdot f^{(\mathrm{4})}+f^{(\mathrm{4})}\cdot f^{(\mathrm{2})}+f^{(\mathrm{6})}\cdot \mathrm{1}\right)\\ =\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}\left(g_r^{\left(\mathrm{1}\right)}\cdot g_r^{{\left(\mathrm{5}\right)}^{*}}+g_r^{\left(\mathrm{3}\right)}\cdot g_r^{{\left(\mathrm{3}\right)}^{*}}+g_r^{\left(\mathrm{5}\right)}\cdot g_r^{{\left(\mathrm{1}\right)}^{*}}\right)\\ \dots \dots \end{array}(8)

2.1 n 分量NLSE的单孤子解

截断由参数\epsilon的摄动展开的级数(6)如下：

\left\{\begin{array}{l}{g}_j=\epsilon g_j^{(\mathrm{1})},\left(j=\mathrm{1},\mathrm{2},\mathrm{3},\dots ,n\right)\\ f=\mathrm{1}+{\epsilon }^{\mathrm{2}}{f}^{(\mathrm{2})}\end{array}\right.(9)

假设

g_j^{(\mathrm{1})}={\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_j},\left(j=\mathrm{1},\mathrm{2},\mathrm{3},\dots ,n\right)(10)

要注意的是{\epsilon }_{\mathrm{1}}=\mathrm{0}，{\epsilon }_{\mathrm{2}},\cdots ,{\epsilon }_n是复数。

{\eta }_{\mathrm{1}}=l_{\mathrm{1}}t+w_{\mathrm{1}}x+{\mu }_{\mathrm{1}},l_{\mathrm{1}}=-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}(11)

其中，l_{\mathrm{1}}、w_{\mathrm{1}}和{\mu }_{\mathrm{1}}均为复数。式(9)~式(11)代入双线性方程(8)，通过计算可以得到

\left\{\begin{array}{l}{f}^{(\mathrm{2})}=c_{\mathrm{11}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}}={\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{13}}+{\tau }^{\mathrm{1}}}\\ c_{\mathrm{11}}={\mathrm{e}}^{{\theta }_{\mathrm{13}}+{\tau }^{\mathrm{1}}}=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}}\end{array}\right.(12)

设\epsilon =\mathrm{1}，n分量NLSE式(2)的单孤子解如下：

\begin{array}{l}{q}_{j,\mathrm{1}}=\frac{g_j^{(\mathrm{1})}}{\mathrm{1}+f^{(\mathrm{2})}}=\frac{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_j}}{\mathrm{1}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{12}}+{\tau }^{\mathrm{1}}}}\\ =\frac{{\mathrm{e}}^{-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}t+w_{\mathrm{1}}x+{\epsilon }_j}}{\mathrm{1}+\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{\mathrm{e}}^{-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}t+w_{\mathrm{1}}x-\frac{i{w}_{\mathrm{1}}^{{\mathrm{*}}^{\mathrm{2}}}}{\mathrm{2}}t+w_{\mathrm{1}}^{\mathrm{*}}x}},\\ \left(j=\mathrm{1},\mathrm{2},\mathrm{3},\dots ,n\right)\end{array}(13)

2.2 n 分量NLSE的双孤子解

截断由参数\epsilon的摄动展开的级数(7)如下：

\left\{\begin{array}{l}{g}_j=\epsilon g_j^{(\mathrm{1})}+{\epsilon }^{\mathrm{3}}{g}_j^{(\mathrm{3})},\left(j=\mathrm{1},\mathrm{2},\mathrm{3},\dots ,n\right)\\ f=\mathrm{1}+{\epsilon }^{\mathrm{2}}{f}^{(\mathrm{2})}+{\epsilon }^{\mathrm{4}}{f}^{(\mathrm{4})}\end{array}\right.(14)

假设

g_j^{(\mathrm{1})}={\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\epsilon }_j},\left(j=\mathrm{1},\mathrm{2},\mathrm{3},\dots ,n\right)(15)

要注意的是{\epsilon }_{\mathrm{1}}=\mathrm{0},{\epsilon }_{\mathrm{2}},\cdots ,{\epsilon }_n是复数。

\left\{\begin{array}{l}{\eta }_{\mathrm{1}}=l_{\mathrm{1}}t+w_{\mathrm{1}}x+{\mu }_{\mathrm{1}},l_{\mathrm{1}}=-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}\\ {\eta }_{\mathrm{2}}=l_{\mathrm{2}}t+w_{\mathrm{2}}x+{\mu }_{\mathrm{2}},l_{\mathrm{2}}=-\frac{i{w}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}}\end{array}\right.(16)

其中，l_{\mathrm{1}}、l_{\mathrm{2}}、w_{\mathrm{1}}、w_{\mathrm{2}}、{\mu }_{\mathrm{1}}和{\mu }_{\mathrm{2}}均为复数。有

\left\{\begin{array}{l}{f}^{(\mathrm{2})}=c_{\mathrm{11}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}}+c_{\mathrm{21}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}}+c_{\mathrm{12}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}}+c_{\mathrm{22}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}}\\ g_j^{(\mathrm{3})}=a_{\mathrm{121}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j}+a_{\mathrm{122}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j},\left(j=\mathrm{1},\mathrm{2},\dots ,n\right)\\ f^{(\mathrm{4})}=c_{\mathrm{1212}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}}\end{array}\right.(17)

其中，c_{\mathrm{11}},c_{\mathrm{12}},c_{\mathrm{21}},c_{\mathrm{22}},c_{\mathrm{1212}}是待确定系数。式(14)~式(17)代入双线性方程(8)可以得到

\left\{\begin{array}{l}{c}_{\mathrm{11}}={\mathrm{e}}^{{\theta }_{\mathrm{13}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}}\\ c_{\mathrm{12}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}}\\ c_{\mathrm{21}}={\mathrm{e}}^{{\theta }_{\mathrm{23}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}}\\ c_{\mathrm{22}}={\mathrm{e}}^{{\theta }_{\mathrm{24}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}}\\ a_{\mathrm{121}}={\mathrm{e}}^{{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{12}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}}\\ a_{\mathrm{122}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}}\\ c_{\mathrm{1212}}={\mathrm{e}}^{{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{34}}+{\tau }^{\mathrm{2}}}\\ =\frac{{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}}\end{array}\right.(18)

同样地，取\epsilon =\mathrm{1}，n分量NLSE的双孤子解如下：

\begin{array}{l}{q}_{j,\mathrm{2}}=\frac{\begin{array}{l}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\tau }^{\mathrm{1}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\tau }^{\mathrm{1}}}\end{array}}{\begin{array}{l}\mathrm{1}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{13}}+{\tau }^{\mathrm{1}}}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{23}}+{\tau }^{\mathrm{1}}}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\tau }^{\mathrm{1}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{24}}+{\tau }^{\mathrm{1}}}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\tau }^{\mathrm{2}}}\end{array}},\\ \left(j=\mathrm{1},\mathrm{2},\mathrm{3},\dots ,n\right)\end{array}(19)

2.3 n 分量NLSE的三孤子解

截断由参数\epsilon的摄动展开的级数(7)如下：

\left\{\begin{array}{l}{g}_j=\epsilon g_j^{(\mathrm{1})}+{\epsilon }^{\mathrm{3}}{g}_j^{(\mathrm{3})}+{\epsilon }^{\mathrm{5}}{g}_j^{(\mathrm{5})},\left(j=\mathrm{1},\mathrm{2},\dots ,n\right)\\ f_{\mathrm{3}}=\mathrm{1}+{\epsilon }^{\mathrm{2}}{f}^{(\mathrm{2})}+{\epsilon }^{\mathrm{4}}{f}^{(\mathrm{3})}+{\epsilon }^{\mathrm{6}}{f}^{(\mathrm{6})}\end{array}\right.(20)

假设

g_j^{(\mathrm{1})}={\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\epsilon }_j},\left(j=\mathrm{1},\mathrm{2},\dots ,n\right)(21)

要注意的是{\epsilon }_{\mathrm{1}}=\mathrm{0}，{\epsilon }_{\mathrm{2}},{\epsilon }_{\mathrm{3}}是复数。

\left\{\begin{array}{l}{\eta }_{\mathrm{1}}=l_{\mathrm{1}}t+w_{\mathrm{1}}x+{\mu }_{\mathrm{1}},l_{\mathrm{1}}=-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}\\ {\eta }_{\mathrm{2}}=l_{\mathrm{2}}t+w_{\mathrm{2}}x+{\mu }_{\mathrm{2}},l_{\mathrm{2}}=-\frac{i{w}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}}\\ {\eta }_{\mathrm{3}}=l_{\mathrm{3}}t+w_{\mathrm{3}}x+{\mu }_{\mathrm{3}},l_{\mathrm{3}}=-\frac{i{w}_{\mathrm{3}}^{\mathrm{2}}}{\mathrm{2}}\end{array}\right.(22)

其中，l_{\mathrm{1}}、l_{\mathrm{2}}、l_{\mathrm{3}}、w_{\mathrm{1}}、w_{\mathrm{2}}、w_{\mathrm{3}}、{\mu }_{\mathrm{1}}、{\mu }_{\mathrm{2}}和{\mu }_{\mathrm{3}}均为复数。

\left\{\begin{array}{l}{f}^{(\mathrm{2})}=c_{\mathrm{11}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}}+c_{\mathrm{12}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}}+c_{\mathrm{13}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}}\\ +c_{\mathrm{21}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}}+c_{\mathrm{22}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}}+c_{\mathrm{23}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}}\\ +c_{\mathrm{31}}{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}}+c_{\mathrm{32}}{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}}+c_{\mathrm{33}}{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}}\\ g_j^{(\mathrm{3})}=a_{\mathrm{121}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j}+a_{\mathrm{122}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j}+a_{\mathrm{123}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j}\\ +a_{\mathrm{131}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j}+a_{\mathrm{132}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j}+a_{\mathrm{133}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j}\\ +a_{\mathrm{231}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j}+a_{\mathrm{232}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j}+a_{\mathrm{233}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j}\\ f^{(\mathrm{4})}=c_{\mathrm{1212}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}}+c_{\mathrm{1213}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}}+c_{\mathrm{1223}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}}\\ +c_{\mathrm{1312}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}}+c_{\mathrm{1313}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}}+c_{\mathrm{1323}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}}\\ +c_{\mathrm{2312}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}}+c_{\mathrm{2313}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}}+c_{\mathrm{2323}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}}\\ g_j^{(\mathrm{5})}=a_{\mathrm{12312}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j}+a_{\mathrm{12313}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j}\\ +a_{\mathrm{12323}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j}\\ f^{(\mathrm{6})}=c_{\mathrm{123123}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}}\end{array}\right.(23)

经过计算确定未确定系数：

\begin{array}{c}{c}_{\mathrm{11}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}},\\ c_{\mathrm{12}}={\mathrm{e}}^{{\theta }_{\mathrm{15}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}},\\ c_{\mathrm{13}}={\mathrm{e}}^{{\theta }_{\mathrm{16}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}},\\ c_{\mathrm{21}}={\mathrm{e}}^{{\theta }_{\mathrm{24}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}},\\ c_{\mathrm{22}}={\mathrm{e}}^{{\theta }_{\mathrm{25}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}},\\ c_{\mathrm{23}}={\mathrm{e}}^{{\theta }_{\mathrm{26}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{2}}+w_{\mathrm{3}}^{\mathrm{*}}},\\ c_{\mathrm{31}}={\mathrm{e}}^{{\theta }_{\mathrm{34}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{3}}+w_{\mathrm{1}}^{\mathrm{*}}},\\ c_{\mathrm{32}}={\mathrm{e}}^{{\theta }_{\mathrm{35}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{3}}+w_{\mathrm{2}}^{\mathrm{*}}},\\ c_{\mathrm{33}}={\mathrm{e}}^{{\theta }_{\mathrm{36}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}}{w_{\mathrm{3}}+w_{\mathrm{3}}^{\mathrm{*}}},\end{array}

\begin{array}{l}{a}_{\mathrm{121}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ a_{\mathrm{122}}={\mathrm{e}}^{{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{12}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ a_{\mathrm{123}}={\mathrm{e}}^{{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{12}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ a_{\mathrm{131}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{13}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ a_{\mathrm{132}}={\mathrm{e}}^{{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{13}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ a_{\mathrm{133}}={\mathrm{e}}^{{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{13}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ a_{\mathrm{231}}={\mathrm{e}}^{{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{23}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{2}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ a_{\mathrm{232}}={\mathrm{e}}^{{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{23}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{2}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ a_{\mathrm{233}}={\mathrm{e}}^{{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{23}}+\tau }=\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}{\left(w_{\mathrm{2}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{2}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ c_{\mathrm{1212}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ =\frac{{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ c_{\mathrm{1213}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ =\frac{{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\end{array}

\begin{array}{l}{c}_{\mathrm{1223}}={\mathrm{e}}^{{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\\ =\frac{{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}\left(w_{\mathrm{2}}^{\mathrm{*}}-w_{\mathrm{3}}^{\mathrm{*}}\right)}{{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\end{array}

\begin{array}{l}{c}_{\mathrm{1312}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ =\frac{{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ c_{\mathrm{1313}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ =\frac{{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\end{array}

\begin{array}{l}{c}_{\mathrm{1323}}={\mathrm{e}}^{{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\\ =\frac{{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}^{\mathrm{*}}-w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ c_{\mathrm{2312}}={\mathrm{e}}^{{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ =\frac{{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ c_{\mathrm{2313}}={\mathrm{e}}^{{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ =\frac{{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ c_{\mathrm{2323}}={\mathrm{e}}^{{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\\ =\frac{{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}^{\mathrm{*}}-w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}},\\ a_{\mathrm{12312}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ =\frac{\begin{array}{l}{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}\\ {\left(w_{\mathrm{2}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}\end{array}}{\begin{array}{l}{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}\\ {\left(w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}\end{array}},\end{array}

\begin{array}{l}{a}_{\mathrm{12313}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ =\frac{\begin{array}{l}{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}\\ {\left(w_{\mathrm{2}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}\end{array}}{\begin{array}{l}{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}\\ {\left(w_{\mathrm{2}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}\end{array}},\end{array}

\begin{array}{l}{a}_{\mathrm{12323}}={\mathrm{e}}^{{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\\ =\frac{\begin{array}{l}{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}\\ {\left(w_{\mathrm{2}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}^{\mathrm{*}}-w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}\end{array}}{\begin{array}{l}{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}\\ {\left(w_{\mathrm{2}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}\end{array}},\\ c_{\mathrm{123123}}={\mathrm{e}}^{{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{45}}+{\theta }_{\mathrm{56}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{3}}}\\ =\frac{\begin{array}{l}{\left(\sigma {\displaystyle \sum _{r=\mathrm{1}}^n}{\mathrm{e}}^{{\epsilon }_r+{\epsilon }_r^{\mathrm{*}}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}-w_{\mathrm{3}}\right)}^{\mathrm{2}}\\ {\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}^{\mathrm{*}}-w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}^{\mathrm{*}}-w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}\end{array}}{\begin{array}{l}{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}\\ {\left(w_{\mathrm{2}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{2}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{1}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{2}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}{\left(w_{\mathrm{3}}+w_{\mathrm{3}}^{\mathrm{*}}\right)}^{\mathrm{2}}\end{array}}\end{array}(24)

取\epsilon =\mathrm{1}，n分量NLSE的三孤子解为

q_j=\frac{g_j^{(\mathrm{1})}+g_j^{(\mathrm{3})}+g_j^{(\mathrm{5})}}{\mathrm{1}+f^{(\mathrm{2})}+f^{(\mathrm{4})}+f^{(\mathrm{6})}},\left(j=\mathrm{1},\mathrm{2},\dots ,n\right)(25)

2.4 n 分量NLSE的 N 孤子解

如果取幂级数展开式(7)中：

\left\{\begin{array}{l}{g}_j^{\left(\mathrm{1}\right)}={\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\epsilon }_j}+\dots +{\mathrm{e}}^{{\eta }_N+{\epsilon }_j},\left(j=\mathrm{1},\mathrm{2},\dots ,n\right)\\ {\eta }_N=l_{N}t+w_{N}x+{\mu }_N,l_N=-\frac{i{w}_N^{\mathrm{2}}}{\mathrm{2}},\left(N=\mathrm{1},\mathrm{2},\dots ,N\right)\end{array}\right.(26)

n分量NLSE式(2)的N孤子解q_{j,N}\left(j=\mathrm{1},\mathrm{2},\dots ,n\right)，所对应的分母与分子可以表示为

(27)

其中

(28)

A_{\mathrm{1}}\left(\mu \right)与A_{\mathrm{2}}\left(\mu \right)表示当{\mu }_j\left(j=\mathrm{1},\mathrm{2},\dots ,N\right)取所有可能的\mathrm{0}和\mathrm{1}时，还需要分别满足条件

{\displaystyle \sum _{k=\mathrm{1}}^N}{\mu }_k={\displaystyle \sum _{k=\mathrm{1}}^N}{\mu }_{{}_{N+k}},{\displaystyle \sum _{k=\mathrm{1}}^N}{\mu }_k={\displaystyle \sum _{k=\mathrm{1}}^N}{\mu }_{N+k}+\mathrm{1}(29)

3 举例

3.1 三分量NLSE的孤子解

式(2)中，当n=\mathrm{3}时，有以下三分量NLSE:

\left\{\begin{array}{l}i{q}_{\mathrm{1},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{1},xx}+\sigma \left(|q_{\mathrm{1}}{|}^{\mathrm{2}}+|q_{\mathrm{2}}{|}^{\mathrm{2}}+|q_{\mathrm{3}}{|}^{\mathrm{2}}\right)q_{\mathrm{1}}=\mathrm{0}\\ i{q}_{\mathrm{2},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{2},xx}+\sigma \left(|q_{\mathrm{1}}{|}^{\mathrm{2}}+|q_{\mathrm{2}}{|}^{\mathrm{2}}+|q_{\mathrm{3}}{|}^{\mathrm{2}}\right)q_{\mathrm{2}}=\mathrm{0}\\ i{q}_{\mathrm{3},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{2},xx}+\sigma \left(|q_{\mathrm{1}}{|}^{\mathrm{2}}+|q_{\mathrm{2}}{|}^{\mathrm{2}}+|q_{\mathrm{3}}{|}^{\mathrm{2}}\right)q_{\mathrm{3}}=\mathrm{0}\end{array}\right.(30)

三分量NLSE的双线性形式为

\left\{\begin{array}{l}\left(i{D}_t+\frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\right)g_{\mathrm{1}}\cdot f=\mathrm{0}\\ \left(i{D}_t+\frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\right)g_{\mathrm{2}}\cdot f=\mathrm{0}\\ \left(i{D}_t+\frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\right)g_{\mathrm{3}}\cdot f=\mathrm{0}\\ \frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}f\cdot f=\sigma \left(|g_{\mathrm{1}}{|}^{\mathrm{2}}+|g_{\mathrm{2}}{|}^{\mathrm{2}}+|g_{\mathrm{3}}{|}^{\mathrm{2}}\right)\end{array}\right.(31)

3.1.1 三分量NLSE的单孤子解

根据式(26)，可以得到式(30)的单孤子解如下：

\left\{\begin{array}{l}{q}_{\mathrm{1},\mathrm{1}}=\frac{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_{\mathrm{1}}}}{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{12}}+\tau }}\\ =\frac{{\mathrm{e}}^{-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}t+w_{\mathrm{1}}x+{\epsilon }_{\mathrm{1}}}}{\mathrm{1}+\frac{\left(\mathrm{1}+{\mathrm{e}}^{{\epsilon }_{\mathrm{2}}+{\epsilon }_{\mathrm{2}}^{\mathrm{*}}}+{\mathrm{e}}^{{\epsilon }_{\mathrm{3}}+{\epsilon }_{\mathrm{3}}^{\mathrm{*}}}\right)\sigma }{w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}}{\mathrm{e}}^{-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}t+w_{\mathrm{1}}x-\frac{i{w}_{\mathrm{1}}^{{\mathrm{*}}^{\mathrm{2}}}}{\mathrm{2}}t+w_{\mathrm{1}}^{\mathrm{*}}x}}\\ q_{\mathrm{2},\mathrm{1}}=\frac{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_{\mathrm{2}}}}{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{12}}+\tau }}\\ =\frac{{\mathrm{e}}^{-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}t+w_{\mathrm{1}}x+{\epsilon }_{\mathrm{2}}}}{\mathrm{1}+\frac{\left(\mathrm{1}+{\mathrm{e}}^{{\epsilon }_{\mathrm{2}}+{\epsilon }_{\mathrm{2}}^{\mathrm{*}}}+{\mathrm{e}}^{{\epsilon }_{\mathrm{3}}+{\epsilon }_{\mathrm{3}}^{\mathrm{*}}}\right)\sigma }{w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}}{\mathrm{e}}^{-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}t+w_{\mathrm{1}}x-\frac{i{w}_{\mathrm{1}}^{{\mathrm{*}}^{\mathrm{2}}}}{\mathrm{2}}t+w_{\mathrm{1}}^{\mathrm{*}}x}}\\ q_{\mathrm{3},\mathrm{1}}=\frac{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_{\mathrm{3}}}}{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{12}}+\tau }}\\ =\frac{{\mathrm{e}}^{-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}t+w_{\mathrm{1}}x+{\epsilon }_{\mathrm{3}}}}{\mathrm{1}+\frac{\left(\mathrm{1}+{\mathrm{e}}^{{\epsilon }_{\mathrm{2}}+{\epsilon }_{\mathrm{2}}^{\mathrm{*}}}+{\mathrm{e}}^{{\epsilon }_{\mathrm{3}}+{\epsilon }_{\mathrm{3}}^{\mathrm{*}}}\right)\sigma }{w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}}{\mathrm{e}}^{-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}t+w_{\mathrm{1}}x-\frac{i{w}_{\mathrm{1}}^{{\mathrm{*}}^{\mathrm{2}}}}{\mathrm{2}}t+w_{\mathrm{1}}^{\mathrm{*}}x}}\end{array}\right.(32)

q_{\mathrm{2},\mathrm{1}}通过取适当的参数w_{\mathrm{1}}=-\mathrm{1}+\mathrm{0.5}i，{\mu }_{\mathrm{1}}=\mathrm{0.3}-i，{\epsilon }_{\mathrm{2}}=\mathrm{0.5}+\mathrm{0.25}i，{\epsilon }_{\mathrm{3}}=\mathrm{0.5}+\mathrm{0.3}i，\sigma =\mathrm{0.5}，利用mathematica可以画出单亮孤子解，如图1所示。

3.1.2 三分量NLSE的双孤子解

根据式(26)，可以得到式(30)的双孤子解如下：

\left\{\begin{array}{l}{q}_{\mathrm{1},\mathrm{2}}=\frac{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_{\mathrm{1}}}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\epsilon }_{\mathrm{1}}}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{*}+{\epsilon }_{\mathrm{1}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{12}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{*}+{\epsilon }_{\mathrm{1}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+\tau }}{\mathrm{1}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{*}+{\theta }_{\mathrm{13}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}^{*}+{\theta }_{\mathrm{14}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{*}+{\theta }_{\mathrm{23}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{*}+{\theta }_{\mathrm{24}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{*}+{\eta }_{\mathrm{2}}^{*}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{34}}+{\tau }^{\mathrm{2}}}}\\ q_{\mathrm{2},\mathrm{2}}=\frac{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_{\mathrm{2}}}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\epsilon }_{\mathrm{2}}}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{*}+{\epsilon }_{\mathrm{2}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{12}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{*}+{\epsilon }_{\mathrm{2}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+\tau }}{\mathrm{1}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{*}+{\theta }_{\mathrm{13}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}^{*}+{\theta }_{\mathrm{14}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{*}+{\theta }_{\mathrm{23}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{*}+{\theta }_{\mathrm{24}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{*}+{\eta }_{\mathrm{2}}^{*}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{34}}+{\tau }^{\mathrm{2}}}}\\ q_{\mathrm{3},\mathrm{2}}=\frac{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_{\mathrm{3}}}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\epsilon }_{\mathrm{3}}}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{*}+{\epsilon }_{\mathrm{3}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{12}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{*}+{\epsilon }_{\mathrm{3}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+\tau }}{\mathrm{1}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{*}+{\theta }_{\mathrm{13}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}^{*}+{\theta }_{\mathrm{14}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{*}+{\theta }_{\mathrm{23}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{*}+{\theta }_{\mathrm{24}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{*}+{\eta }_{\mathrm{2}}^{*}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{34}}+{\tau }^{\mathrm{2}}}}\end{array}\right.(33)

其中

(34)

接下来通过取不同参数，利用mathematica可以绘制出一些不同的解。第一种情形：q_{\mathrm{2},\mathrm{2}}取w_{\mathrm{1}}=\mathrm{1.01}，w_{\mathrm{2}}=\mathrm{1}，{\mu }_{\mathrm{1}}=\mathrm{0.3}-i，{\mu }_{\mathrm{2}}=\mathrm{0.25}-i，{\epsilon }_{\mathrm{2}}=\mathrm{0.5}+\mathrm{0.2}i，{\epsilon }_{\mathrm{3}}=\mathrm{0.5}+\mathrm{0.3}i，\sigma =\mathrm{0.5}，图2(a)中显示了平行双亮孤子解。第二种情形：q_{\mathrm{3},\mathrm{2}}取w_{\mathrm{1}}=\mathrm{1.8}，w_{\mathrm{2}}=\mathrm{0.8}，{\mu }_{\mathrm{1}}=\mathrm{0.3}-i，{\mu }_{\mathrm{2}}=\mathrm{0.25}-i，{\epsilon }_{\mathrm{2}}=\mathrm{0.5}+\mathrm{0.2}i，{\epsilon }_{\mathrm{3}}=\mathrm{0.5}+\mathrm{0.3}i，\sigma =\mathrm{0.5}，图2(b)中显示了呼吸解。第三种情形：q_{\mathrm{3},\mathrm{2}}取w_{\mathrm{1}}=\mathrm{0.5}+i，w_{\mathrm{2}}=\mathrm{0.5}-\mathrm{1.25}i，{\mu }_{\mathrm{1}}=\mathrm{0.3}-i，{\mu }_{\mathrm{2}}=\mathrm{0.25}-i，{\epsilon }_{\mathrm{2}}=\mathrm{0.5}-i，{\epsilon }_{\mathrm{3}}=\mathrm{0.5}-\mathrm{0.3}i，\sigma =\mathrm{0.1}，得到双亮孤子解转向呼吸解传播过程，如图3所示。

3.1.3 三分量NLSE的三孤子解

根据式(26)，可以得到式(30)的三孤子解如下：

\begin{array}{l}{q}_{j,\mathrm{3}}=\frac{g_{j,\mathrm{3}}}{f}=\\ \frac{\begin{array}{l}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{12}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{12}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{13}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{13}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{13}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{23}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{23}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{23}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\end{array}}{\begin{array}{l}\mathrm{1}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{15}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{16}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{24}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{25}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{26}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{34}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{35}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{36}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{\begin{array}{l}{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}\\ +{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{45}}+{\theta }_{\mathrm{56}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{3}}\end{array}}\end{array}},\\ \left(j=\mathrm{1},\mathrm{2},\mathrm{3}\right)\end{array}(35)

其中

(36)

q_{\mathrm{1},\mathrm{3}}通过取适当的参数w_{\mathrm{1}}=\mathrm{1}-i，w_{\mathrm{2}}=\mathrm{1}+i，w_{\mathrm{3}}=\mathrm{1}，{\mu }_{\mathrm{1}}=\mathrm{0.3}-i，{\mu }_{\mathrm{2}}=\mathrm{0.25}-i，{\mu }_{\mathrm{3}}=\mathrm{0.1}+i，{\epsilon }_{\mathrm{2}}=\mathrm{0.5}-\mathrm{0.2}i，{\epsilon }_{\mathrm{3}}=\mathrm{0.6}-\mathrm{0.3}i，\sigma =\mathrm{0.5}，可以得到三亮孤子解，如图4(a)所示。q_{\mathrm{3},\mathrm{3}}取参数w_{\mathrm{1}}=\mathrm{1}-i，w_{\mathrm{2}}=\mathrm{1.25}-i，w_{\mathrm{3}}=\mathrm{0.75}+i，{\mu }_{\mathrm{1}}=\mathrm{0.3}-i，{\mu }_{\mathrm{2}}=\mathrm{0.25}-i，{\mu }_{\mathrm{3}}=\mathrm{0.1}+i，{\epsilon }_{\mathrm{2}}=\mathrm{0.5}-\mathrm{0.2}i，{\epsilon }_{\mathrm{3}}=\mathrm{0.6}-\mathrm{0.3}i，\sigma =\mathrm{0.75}，可以得到三亮孤子解，如图4(b)所示。

3.2 六分量NLSE的孤子解

式(2)中，当n=\mathrm{6}时，有以下六分量NLSE：

\left\{\begin{array}{l}i{q}_{\mathrm{1},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{1},xx}+\sigma \left({\displaystyle \sum _{r=\mathrm{1}}^{\mathrm{6}}}{q}_l\cdot q_l^{\mathrm{*}}\right)q_{\mathrm{1}}=\mathrm{0}\\ i{q}_{\mathrm{2},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{2},xx}+\sigma \left({\displaystyle \sum _{r=\mathrm{1}}^{\mathrm{6}}}{q}_l\cdot q_l^{\mathrm{*}}\right)q_{\mathrm{2}}=\mathrm{0}\\ i{q}_{\mathrm{3},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{3},xx}+\sigma \left({\displaystyle \sum _{r=\mathrm{1}}^{\mathrm{6}}}{q}_l\cdot q_l^{\mathrm{*}}\right)q_{\mathrm{3}}=\mathrm{0}\\ i{q}_{\mathrm{4},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{4},xx}+\sigma \left({\displaystyle \sum _{r=\mathrm{1}}^{\mathrm{6}}}{q}_l\cdot q_l^{\mathrm{*}}\right)q_{\mathrm{4}}=\mathrm{0}\\ i{q}_{\mathrm{5},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{5},xx}+\sigma \left({\displaystyle \sum _{r=\mathrm{1}}^{\mathrm{6}}}{q}_l\cdot q_l^{\mathrm{*}}\right)q_{\mathrm{5}}=\mathrm{0}\\ i{q}_{\mathrm{6},t}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{\mathrm{6},xx}+\sigma \left({\displaystyle \sum _{r=\mathrm{1}}^{\mathrm{6}}}{q}_l\cdot q_l^{\mathrm{*}}\right)q_{\mathrm{6}}=\mathrm{0}\end{array}\right.(37)

上式的双线性形式为

\left\{\begin{array}{l}\left(i{D}_t+\frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}\right)g_j\cdot f=\mathrm{0},\left(j=\mathrm{1},\mathrm{2},\dots ,\mathrm{6}\right)\\ \frac{\mathrm{1}}{\mathrm{2}}{D}_x^{\mathrm{2}}f\cdot f=\sigma \left({\displaystyle \sum _{r=\mathrm{1}}^{\mathrm{6}}}{g}_l\cdot g_l^{\mathrm{*}}\right)\end{array}\right.(38)

3.2.1 六分量NLSE的单孤子解

根据式(26)，可以得到式(37)的单孤子解：

\begin{array}{l}{q}_{j,\mathrm{1}}=\frac{g_{j,\mathrm{1}}}{f_{\mathrm{1}}}=\frac{{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_j}}{\mathrm{1}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{12}}+{\tau }^{\mathrm{1}}}}\\ =\frac{{\mathrm{e}}^{-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}t+w_{\mathrm{1}}x+{\epsilon }_j}}{\mathrm{1}+\frac{\sigma {\displaystyle \sum _{r=\mathrm{1}}^{\mathrm{6}}}{\mathrm{e}}^{{\epsilon }_j+{\epsilon }_j^{\mathrm{*}}}}{{\left(w_{\mathrm{1}}+w_{\mathrm{1}}^{\mathrm{*}}\right)}^{\mathrm{2}}}{\mathrm{e}}^{-\frac{i{w}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{2}}t+w_{\mathrm{1}}x-\frac{i{w}_{\mathrm{1}}^{{\mathrm{*}}^{\mathrm{2}}}}{\mathrm{2}}t+w_{\mathrm{1}}^{\mathrm{*}}x}},\\ \left(j=\mathrm{1},\mathrm{2},\dots ,\mathrm{6}\right)\end{array}(39)

3.2.2 六分量NLSE的双孤子解

根据式(26)，可以得到式(37)的双孤子解：

\begin{array}{l}{q}_{j,\mathrm{2}}=\frac{g_{j,\mathrm{2}}}{f}\\ =\frac{\begin{array}{l}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{12}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+\tau }\end{array}}{\begin{array}{l}\mathrm{1}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{13}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{23}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{24}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{34}}+{\tau }^{\mathrm{2}}}\end{array}},\\ \left(j=\mathrm{1},\mathrm{2},\dots ,\mathrm{6}\right)\end{array}(40)

其中

(41)

q_{\mathrm{5},\mathrm{2}}通过取适当的参数w_{\mathrm{1}}=\mathrm{1}-i，w_{\mathrm{2}}=-\mathrm{1}，{\mu }_{\mathrm{1}}=\mathrm{0.3}-i，{\mu }_{\mathrm{2}}=\mathrm{0.25}-i，{\epsilon }_{\mathrm{2}}=\mathrm{0.5}-\mathrm{0.2}i，{\epsilon }_{\mathrm{3}}=\mathrm{0.3}-\mathrm{0.3}i，{\epsilon }_{\mathrm{4}}=\mathrm{0.7}-\mathrm{0.4}i，{\epsilon }_{\mathrm{5}}=\mathrm{0.7}-\mathrm{0.5}i，{\epsilon }_{\mathrm{6}}=\mathrm{0.7}-\mathrm{0.6}i，\sigma =\mathrm{0.5}，得到如图5(a)所示的双孤子解。通过取适当的参数w_{\mathrm{1}}=\mathrm{1}-i，w_{\mathrm{2}}=-\mathrm{1}，{\mu }_{\mathrm{1}}=\mathrm{0.3}-i，{\mu }_{\mathrm{2}}=\mathrm{0.25}-i，{\epsilon }_{\mathrm{2}}=\mathrm{0.25}+i，{\epsilon }_{\mathrm{3}}=\mathrm{1.5}-\mathrm{0.3}i，{\epsilon }_{\mathrm{4}}=\mathrm{0.5}-\mathrm{0.4}i，{\epsilon }_{\mathrm{5}}=\mathrm{1.25}-\mathrm{0.5}i，{\epsilon }_{\mathrm{6}}=\mathrm{1.75}-\mathrm{0.6}i，\sigma =\mathrm{0.5}，得到图5(b)所示的孤子和呼吸子相互作用解。

3.2.3 六分量NLSE的三孤子解

根据式(26)，可以得到式(37)的三孤子解：

\begin{array}{l}{q}_{j,\mathrm{3}}=\frac{g_{j,\mathrm{3}}}{f}=\\ \frac{\begin{array}{l}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\epsilon }_j}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{12}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{12}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{12}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{13}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{13}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{13}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{23}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{23}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{23}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\epsilon }_j+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\end{array}}{\begin{array}{l}\mathrm{1}+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{15}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{16}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{24}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{25}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{26}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\theta }_{\mathrm{34}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{35}}+\tau }+{\mathrm{e}}^{{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{36}}+\tau }\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{45}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{56}}+{\tau }^{\mathrm{2}}}\\ +{\mathrm{e}}^{\begin{array}{l}{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}+{\eta }_{\mathrm{3}}+{\eta }_{\mathrm{1}}^{\mathrm{*}}+{\eta }_{\mathrm{2}}^{\mathrm{*}}+{\eta }_{\mathrm{3}}^{\mathrm{*}}+{\theta }_{\mathrm{14}}+{\theta }_{\mathrm{24}}+{\theta }_{\mathrm{34}}+{\theta }_{\mathrm{15}}+{\theta }_{\mathrm{25}}+{\theta }_{\mathrm{35}}\\ +{\theta }_{\mathrm{16}}+{\theta }_{\mathrm{26}}+{\theta }_{\mathrm{36}}+{\theta }_{\mathrm{12}}+{\theta }_{\mathrm{13}}+{\theta }_{\mathrm{23}}+{\theta }_{\mathrm{45}}+{\theta }_{\mathrm{56}}+{\theta }_{\mathrm{46}}+{\tau }^{\mathrm{3}}\end{array}}\end{array}},\\ \left(j=\mathrm{1},\mathrm{2},\dots ,\mathrm{6}\right)\end{array}(42)

其中

(43)

q_{\mathrm{6},\mathrm{2}}通过取适当的参数w_{\mathrm{1}}=-\mathrm{1}+\mathrm{0.5}i，w_{\mathrm{2}}=-\mathrm{1.25}-i，w_{\mathrm{3}}=-\mathrm{0.75}+i，{\mu }_{\mathrm{1}}=-\mathrm{0.3}-i，{\mu }_{\mathrm{2}}=\mathrm{0.25}-i，{\mu }_{\mathrm{3}}=\mathrm{0.1}+i，{\epsilon }_{\mathrm{2}}=\mathrm{0.5}-\mathrm{0.2}i，{\epsilon }_{\mathrm{3}}=\mathrm{0.3}-\mathrm{0.3}i，{\epsilon }_{\mathrm{4}}=\mathrm{0.7}-\mathrm{0.4}i，{\epsilon }_{\mathrm{5}}=\mathrm{0.7}-\mathrm{0.5}i，{\epsilon }_{\mathrm{6}}=\mathrm{0.7}-\mathrm{0.6}i，\sigma =\mathrm{0.3}，得到如图6所示的三孤子解。

4 总结

本文基于n分量NLSE的Hirota双线性形式，得到了n分量NLSE的N孤子解的一般表达式。并由三维图像展现了孤子和呼吸子的传播过程以及孤子和呼吸子的交互行为。发现在孤子解中当参数w_i为实数时，可以导出特殊孤子，如图2平行双亮孤子解和呼吸解。当所有参数取相同的数值，各分量间的波形相差不大。图5描述了不同{\epsilon }_i值时，孤子和呼吸子之间的相互作用。因此，分量越多，相互作用会更加丰富，波形结构主要受参数w_i和{\epsilon }_i的影响，参数\sigma的微小差异对波形结构的影响较小。本文所获得的结果对物理学或非线性光学等相关领域的研究具有一定的启示意义。

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基金资助

国家自然科学基金项目(12262027)

内蒙古自治区自然科学基金项目(2023LHMS01006)