## Abstract

Optical solitons can be transmitted through a low-loss fiber to a distance exceeding 6 × 10^{3} without a conventional repeater if they are amplified periodically at appropriate distances. An example of a numerical simulation for a 2-gigabit/sec system is presented with solitons having peak power of 11.2 mW, a width of 34.2 psec in a monomode fiber with a cross-sectional area of 20 *μ*m^{2}, a loss rate of 0.2 dB/km at λ = 1.5 *μ*m, and amplifiers with 2-dB power gain spaced at every 10 km.

© 1982 Optical Society of America

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### Equations (23)

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(1)
$$\Gamma \ll {\left|q\right|}^{2}.$$
(2)
$$i\frac{\partial q}{\partial \xi}+\frac{1}{2}\frac{{\partial}^{2}q}{\partial {\tau}^{2}}+{\left|q\right|}^{2}q=-i\Gamma q+i\beta \frac{{\partial}^{3}q}{\partial {\tau}^{3}},$$
(3)
$$\xi ={10}^{-9}x/\lambda ,$$
(4)
$$\tau =\frac{{10}^{-4.5}}{{\left(-\lambda {k}^{\u2033}\right)}^{1/2}}\left(t-\frac{x}{{\upsilon}_{g}}\right),$$
(5)
$$q={10}^{4.5}{\left(\pi {n}_{2}\right)}^{1/2}\varphi ,$$
(6)
$$\Gamma ={10}^{9}\lambda \gamma ,$$
(7)
$$\beta =\frac{1}{6}\frac{{k}^{\u2034}}{{k}^{\u2033}}\frac{{10}^{-4.5}}{{\left(-\lambda {k}^{\u2033}\right)}^{1/2}}.$$
(8)
$${k}^{\prime}=\frac{\partial k}{\partial \omega}=\frac{n}{c}\left(1-\frac{\lambda}{n}\frac{\partial n}{\partial \lambda}\right),$$
(9)
$${k}^{\u2033}=\frac{{\partial}^{2}k}{\partial {\omega}^{2}}=\frac{\lambda}{2\pi {c}^{2}}\left({\lambda}^{2}\frac{{\partial}^{2}n}{\partial {\lambda}^{2}}\right),$$
(10)
$${k}^{\u2034}=\frac{{\partial}^{3}k}{\partial {\omega}^{3}}=-\frac{{\lambda}^{2}}{4{\pi}^{2}{c}^{3}}\frac{\partial}{\partial \lambda}\left({\lambda}^{3}\frac{{\partial}^{2}n}{\partial {\lambda}^{2}}\right),$$
(11)
$$\begin{array}{ll}q\left(\tau ,\xi \right)=\hfill & \stackrel{\u02c6}{q}\left(\tau ,\xi \right)(1+\beta \{2{\eta}^{2}\tau -3\eta \hfill \\ \hfill & \times \text{tanh}\left[\eta \left(\tau -2\beta \sigma \right)\right]\}+\frac{i}{2}\Gamma {\tau}^{2})\phantom{\rule{0.2em}{0ex}}\text{exp}\left(i\sigma \right)\hfill \\ \hfill & \begin{array}{ll}+O\left({\Gamma}^{2},{\beta}^{2},\Gamma \beta \right)\hfill & \left|\tau \right|\ll O\left({\Gamma}^{-1/2}\right),\hfill \end{array}\hfill \end{array}$$
(12)
$$\begin{array}{c}\stackrel{\u02c6}{q}\left(\tau ,\xi \right)=\eta \phantom{\rule{0.2em}{0ex}}\text{sech}\left[\eta \left(\tau -2\beta \sigma \right)\right],\\ \eta ={q}_{0}\phantom{\rule{0.2em}{0ex}}\text{exp}\left(-2\Gamma \xi \right),\end{array}$$
(13)
$$\sigma =\frac{{{q}_{0}}^{2}}{8\Gamma}\left[1-\text{exp}\left(-4\Gamma \xi \right)\right].$$
(16)
$$S\left(\xi ,\tau \right)=i\left[\text{exp}\left(\Gamma {\xi}_{0}\right)-1\right]{\displaystyle \sum _{n=1}^{N}\delta \left(\xi -n{\xi}_{0}\right)q\left(\xi ,\tau \right)}.$$
(17)
$$\begin{array}{ll}S\left(\xi ,\tau \right)\hfill & \simeq \phantom{\rule{0.2em}{0ex}}i\Gamma {\xi}_{0}{\displaystyle \sum _{n=1}^{N}\delta \left(\xi -n{\xi}_{0}\right)q\left(\xi ,\tau \right)}\hfill \\ \hfill & \simeq \phantom{\rule{0.2em}{0ex}}i\Gamma {\displaystyle \int \delta \left(x-\xi \right)q\left(\xi ,\tau \right)\mathrm{d}x\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}{\xi}_{0}\to 0}\hfill \\ \hfill & =i\Gamma q\left(\xi ,\tau \right).\hfill \end{array}$$
(18)
$$\eta \phantom{\rule{0.2em}{0ex}}\text{sech}\phantom{\rule{0.2em}{0ex}}\eta \tau \to a\eta \phantom{\rule{0.2em}{0ex}}\text{sech}\phantom{\rule{0.2em}{0ex}}\eta \tau ,$$
(19)
$$A=\left(2a-1\right)\eta .$$
(20)
$$\begin{array}{ll}\Delta E\hfill & ={\displaystyle {\int}_{-\infty}^{\infty}{a}^{2}{\eta}^{2}\phantom{\rule{0.2em}{0ex}}{\text{sech}}^{2}\phantom{\rule{0.2em}{0ex}}\eta \tau \mathrm{d}\tau}-{\displaystyle {\int}_{-\infty}^{\infty}{A}^{2}\phantom{\rule{0.2em}{0ex}}{\text{sech}}^{2}\phantom{\rule{0.2em}{0ex}}A\tau \mathrm{d}\tau}\hfill \\ \hfill & =2\eta {\left(a-1\right)}^{2}.\hfill \end{array}$$
(21)
$${\xi}_{0}<{{\tau}_{0}}^{2}={{q}_{0}}^{-2},$$
(22)
$${x}_{0}<{{t}_{0}}^{2}/{k}^{\u2033},$$
(23)
$$a=\text{exp}\left(T{\xi}_{0}\right).$$