1970年以来，孤立子现象的研究受到物理学家和数学家的高度重视。孤立子可以保持其形状和速度不变，从而实现远距离保真通信^[1-3]，因此在光纤通信领域中有着非常重要的研究意义。纤维介质中多个脉冲的共同传播现象可以用聚焦Manakov方程组描述^[4]。本文研究下述更一般的变系数聚焦Manakov方程组：

$ \begin{array}{l} i\frac{{\partial {u_k}}}{{\partial t}} + r{\nabla ^2}{u_k} + 2nr\left( {\sum\limits_{j = 1}^m {{{\left| {{u_j}} \right|}^2}} } \right){u_k} + \\ \sum\limits_{i = 1}^n {\left( {{F_k} + \psi {x_i}} \right){u_k} = 0} \\ k = 1,2, \cdots ,m \end{array} $

(1)

方程中i为虚数单位，函数u_k=u_k(x₁, x₂, …, x_n, t)是n+1维复函数，系数r=r(t), F_k=F_k(t), ψ=ψ(t)是t的任意实函数，拉普拉斯算子${\nabla ^2}{u_k} = \sum\limits_{i = 1}^n {\frac{{{\partial ^2}{u_k}}}{{\partial x_i^2}}} $。当n=1, m=2时，变系数聚焦Manakov方程组(1)又称为可积耦合非线性Schrödinger方程^[4]。

1 双线性形式

设g(x, t), f(x, t)为变量t与x的可微函数，定义函数对(g, f)上的双线性微分算子^[5]为：

$ \begin{array}{l} D_t^mD_x^ng \cdot f = \frac{{{\partial ^m}}}{{\partial {s^m}}}\frac{{{\partial ^n}}}{{\partial {y^n}}}g(x + y,t + s)f(x + y,\\ t + s)\left| {_{y = 0,s = 0}} \right. \end{array} $

(2)

由定义可得双线性微分算子具有如下性质：

$ \begin{array}{*{20}{l}} {{D_t}g \cdot f = gf - {f_t}g}\\ {{D_x}g \cdot f = {g_x}f - {f_x}g}\\ {D_x^2g \cdot f = {g_{xx}}f - 2{g_x}{f_x} + {f_{xx}}g}\\ {D_x^2f \cdot f = {f_{xy}}f - 2{f_x}{f_x} + {f_{xy}}f} \end{array} $

(3)

这里的下标表示偏导数。特别的，双线性微分算子还具有下列重要性质：

$ D_t^mD_x^n\exp \delta \cdot \exp \delta = 0 $

(4)

为了求解方程组(1)，我们先作如下有理变换：

$ {u_k}\left( {{x_1},{x_2}, \cdots ,{x_n},t} \right) = \frac{{{g_k}\left( {{x_1},{x_2}, \cdots ,{x_n},t} \right)}}{{f\left( {{x_1},{x_2}, \cdots ,{x_n},t} \right)}} $

(5)

其中f(x₁, x₂, …, x_n, t)是实函数，g_k(x₁, x₂, …, x_n, t)是复函数。

将有理变换(5)代入方程组(1)中，整理可得：

$ \begin{array}{l} i\left( {\frac{{\partial {g_k}}}{{\partial t}}f - \frac{{\partial f}}{{\partial t}}{g_k}} \right) + \sum\limits_{i = 1}^n r \left( {\frac{{{\partial ^2}{g_k}}}{{\partial x_i^2}}f - 2\frac{{\partial {g_k}}}{{\partial {x_i}}}\frac{{\partial f}}{{\partial {x_i}}} + } \right.\\ \left. {\frac{{{\partial ^2}f}}{{\partial x_i^2}}{g_k}} \right) + \frac{1}{{{f^2}}}\left[ {\sum\limits_{i = 1}^n r \left( {2{g_k}f{{\left( {\frac{{{\partial ^2}f}}{{\partial {x_i}}}} \right)}^2} - 2{g_k}{f^2}\frac{{{\partial ^2}f}}{{\partial x_i^2}}} \right) + } \right.\\ \left. {2nr\left( {\sum\limits_{j = 1}^m {{g_j}} {g_j}} \right){g_k}f} \right] + \sum\limits_{i = 1}^n {\left( {{F_k} + \psi {x_i}} \right)} {g_k}f = 0\\ \;\;\;\;\;\;k = 1,2, \cdots ,m \end{array} $

(6)

借助双线性微分算子定义及其性质，可将方程组(6)写成如下两个双线性形式的方程：

$ \begin{array}{l} \sum\limits_{i = 1}^n {D_{{x_i}}^2} f \cdot f - 2n\sum\limits_{j = 1}^m {\left( {{g_j}g_j^*} \right)} = 0\\ i{D_t}{g_k} \cdot f + r\left( {\sum\limits_{i = 1}^n {D_{{x_i}}^2{g_k}} \cdot f} \right) + \sum\limits_{i = 1}^n {\left( {{F_k} + } \right.} \\ \left. {\psi {x_i}} \right){g_k}f = 0\\ k = 1,2, \cdots ,m \end{array} $

(7)

2 摄动方法

通过Pade近似^[6]的方法，将g_k和f展成小参数ε的幂级数：

$ \begin{array}{*{20}{l}} {f = 1 + {f_2}{\varepsilon ^2} + {f_4}{\varepsilon ^4} + \cdots }\\ {{g_k} = {g_{k1}}\varepsilon + {g_{k3}}{\varepsilon ^3} + \cdots } \end{array} $

(8)

将摄动展开式(8)代入到双线性形式(7)中，并对比ε的同次幂系数，可得如下系数方程组：

$ \begin{array}{l} \begin{array}{*{20}{l}} {i\frac{{\partial {g_{k1}}}}{{\partial t}} + r\sum\limits_{i = 1}^n {\frac{{\partial {g_{k1}}}}{{\partial {x_i}}}} + \sum\limits_{i = 1}^n {\left( {{F_k}(t) + \psi {x_i}} \right)} {g_{k1}} = 0}\\ {\sum\limits_{i = 1}^n {\frac{{{\partial ^2}{f_2}}}{{\partial x_i^2}}} - n\sum\limits_{j = 1}^m {\left( {{g_{j1}}g_{j1}^*} \right)} = 0} \end{array}\\ i\left( {\frac{{\partial {g_{k1}}}}{{\partial t}}{f_2} - {g_{k1}}\frac{{\partial {f_2}}}{{\partial t}} + \frac{{\partial {g_{k3}}}}{{\partial t}}} \right) + \sum\limits_{i = 1}^n {\left( {{F_k} + \psi {x_i}} \right)} \\ \begin{array}{*{20}{l}} {\left( {{g_{k1}}{f_2} + {g_{k3}}} \right) + r\sum\limits_{i = 1}^n {\left( {\frac{{{\partial ^2}{g_{k1}}}}{{\partial x_i^2}}{f_2} - 2\frac{{\partial {g_{k1}}}}{{\partial {x_i}}} + {g_{k1}}\frac{{{\partial ^2}{f_2}}}{{\partial x_i^2}} + } \right.} }\\ {\left. {\frac{{{\partial ^2}{g_{k3}}}}{{\partial x_i^2}}} \right) = 0} \end{array}\\ \;\;\;\;\;\;\;\sum\limits_{i = 1}^n {\left( {\frac{{{\partial ^2}{f_4}}}{{\partial x_i^2}} + {f_2}\frac{{{\partial ^2}{f_2}}}{{\partial x_i^2}} - {{\left( {\frac{{\partial {f_2}}}{{\partial {x_i}}}} \right)}^2}} \right)} - n\sum\limits_{j = 1}^m {\left( {{g_{j1}}g_{j3}^* + } \right.} \\ \left. {{g_{j3}}g_{j1}^*} \right) = 0\\ \;\;\;\;\;\;\;\; \cdots \cdots \\ \;\;\;\;\;\;\;\;k = 1,2, \cdots ,m \end{array} $

(9)

对双线性形式(7)作摄动方法时，根据双线性微分算子的性质(4)，选择指数函数解^[5]作为解的N阶近似时，摄动展开(8)在有限项处截断，使得εⁱ, i=N+1, N+2, …项的系数全为零，从而得到方程组(1)的精确解。这里得到的精确解在传播时不改变形状，即是一种孤立波解，称这样的解为方程组的N-孤子解。

3 精确解

将系数方程组(9)中的函数展成指数函数的幂级数：

$ {g_{k1}} = \sum\limits_{j = 1}^N {{a_{kj}}} \mathit{exp}\left( {{b_{kj}}\left( {{x_1},{x_2}, \ldots ,{x_n},t} \right)} \right) $

(10)

其中a_kj是任意非零常数，b_kj(x₁, x₂, ..., x_n, t)是待定的复函数，j=1, 2, …, N。将幂级数(10)代入系数方程组(9)，求解得到方程组(1)的精确解：

$ \begin{array}{l} {u_k} = \frac{{\sum\limits_{j = 1}^N {{a_{kj}}} \exp \left( {{b_{kj}}} \right) + \sum\limits_{j \in \Lambda } {\exp } \left( {\sum\limits_{l \in \Lambda } {{b_{kjl}}} + \sum\limits_{l \in \Lambda } {b_{kjl}^*} + \sum\limits_{l \in \Lambda } {{b_l}} } \right)}}{{1 + \sum\limits_{j = 1}^N {\exp } \left( {{b_{kj}} + b_{kj}^* + {b_j}} \right) + \sum\limits_{l \in \Lambda } {\exp } \left( {\sum\limits_{l \in \Lambda } {{b_{kjl}}} + \sum\limits_{l \in \Lambda } {b_{kjl}^*} + \sum\limits_{l \in \Lambda } {{b_l}} } \right)}}\\ \;\;\;\;\;\;\;k = 1,2, \cdots ,m \end{array} $

此解为方程组(1)的N-孤子解，其中Λ={1, 2, …, N}是指标集，$\sum\limits_{l \in \mathit{\Lambda }} {{X_l}} $表示从集合{X_l|l∈Λ}中选取部分适当的元素相加得到的累加和。N=1时，称为1-孤子解，又称单孤子解。

3.1 单孤子解

取常数N=1，即：

$ {g_{k1}} = {a_{k1}}\mathit{exp}\left( {{\mu _{k1}} + i{\nu _{k1}}} \right) $

(11)

其中a_k1是任意非零常数，b_k1=μ_k1+iν_k1是待求复函数。

将(11)代入系数方程组(9)，可得f的幂级数展开系数，根据双线性微分算子的性质(4)得：

$ \begin{array}{*{20}{l}} {{g_{kj}} = 0,k = 1,2, \cdots ,m,j = 3,5, \cdots }\\ {{f_j} = 0,j = 4,6, \cdots } \end{array} $

进而求得方程组(1)的单孤子解：

$ \begin{array}{*{20}{l}} {{u_k} = \frac{{{a_{k1}}\mathit{exp}\left( {{\mu _{k1}} + i{\nu _{k1}}} \right)}}{{1 + \mathit{exp}\left( {2{\mu _{k1}} + {b_0}} \right)}}}\\ {k = 1,2, \cdots ,m} \end{array} $

(12)

其中：

$ \begin{array}{l} \exp \left( {{b_0}} \right) = \frac{{\sum\limits_{i = 1}^n {a_{i1}^2} }}{{4C_1^2}}\\ {\mu _{k1}} = \sum\limits_{j = 1}^n {\left( {{C_1}{x_j} - 2{C_1}\int r \left( {\int \psi dt} \right)dt} \right)} \\ {\nu _{k1}} = \sum\limits_{j = 1}^n {\left( {\int {\left( {rC_1^2 + {F_j}(t) - r{{\left( {\int \psi dt} \right)}^2}} \right)dt} } \right.} \\ \left. { + {x_j}\int \psi dt} \right) \end{array} $

3.2 实例

取定n=1, m=1, r=1/2, F=f(t)时，变系数聚焦Manakov方程组(1)即为：

$ i{u_t} + \frac{1}{2}{u_{xx}} + {\left| u \right|^2}u + (f(t) + x \cdot \psi (t))u = 0 $

(13)

其中u=u(x, t)，f(t)是任意实函数。

方程(13)又描述准一维Bose-Einstein凝聚孤子动力学的具有线性势的单分量非线性Schrödinger方程^[7]。该方程的单孤子解为：

$ u = \frac{{{e^{{C_1}x - 2{C_1}}}\int {\frac{1}{2}} \left( {\int \psi dt} \right)dt + i\left( {\int {\left( {\frac{1}{2}C_1^2 + f(t) - \frac{1}{2}{{\left( {\int \psi dt} \right)}^2}} \right)dt} + x\int \psi dt} \right)}}{{1 + {e^{2\left( {{C_1}x - 2{C_1}\int {\frac{1}{2}} \left( {\int \psi dt} \right)dt} \right) + \ln \frac{1}{{4C_1^2}}}}}} $

(14)

(a)若ψ(t)=0，此时方程(13)就是我们平常所讨论的非线性Schrödinger方程。取定常数$ {C_1} = \frac{1}{2}$，则：

$ |u{|^2} = \frac{{{e^{x + 1}}}}{{{{\left( {1 + {e^{x + 1}}} \right)}^2}}} $

(b)若ψ(t)=2(1+cost)，取定常数C₁=12，则：

$ |u{|^2} = \frac{{{e^{2\cos t - {t^2} + x + 1}}}}{{{{\left( {1 + {e^{2\cos t - {t^2} + x + 1}}} \right)}^2}}} $

取定n=1, m=1, r=12·t, F=f(t)时，变系数聚焦Manakov方程组(1)变为：

$ i{u_t} + \frac{1}{2}t{u_{xx}} + |u{|^2}u + (f(t) + x \cdot \psi (t))u = 0 $

(15)

其单孤子解为：

$ u = \frac{{{e^{{C_1}x - 2{C_1}\int {\frac{1}{2}} t\left( {\int \psi dt} \right)dt + i\left( {\int {\left( {\frac{1}{2}tC_1^2 + f(t) - \frac{1}{2}t{{\left( {\int \psi dt} \right)}^2}} \right)} dt + x\int \psi dt} \right)}}}}{{1 + {e^{2\left( {{C_1}x - 2{C_1}\int {\frac{1}{2}} t\left( {\int \psi dt} \right)dt} \right) + \ln \frac{1}{{4C_1^2}}}}}} $

(c)若ψ(t)=3t²+2t+1，取定常数$ {C_1} = \frac{1}{2}$，则：

$ {\left| u \right|^2} = \frac{{{e^{x - \left( {\frac{1}{5}{t^5} + \frac{1}{4}{t^4} + \frac{1}{3}{t^3} + \frac{1}{2}{t^2}} \right)}}}}{{{{\left( {1 + {e^{x - \left( {\frac{1}{5}{t^5} + \frac{1}{4}{t^4}{t^4} + {{\frac{1}{3}}^3}{t^2} + {{\frac{1}{2}}^2}} \right)}}} \right)}^2}}} $

3.3 图像


4 结语

本文利用双线性形式和摄动方法研究了更一般的变系数聚焦Manakov方程组，给出了方程组的N-孤子精确解的求解方法，最后又画出了单孤子解的图像，研究了解其动力学行为。该方程组在物理学中的更广泛应用还有待进一步研究。