⇦Chinese Physics Letters, 2023, Vol. 40, No. 10, Article code 100503 Effective Control of Three Soliton Interactions for the High-Order Nonlinear Schrödinger Equation Yanli Yao (姚延立)1, Houhui Yi (伊厚会)2, Xin Zhang (张鑫)1, and Guoli Ma (马国利)1* Affiliations 1Institute of Aeronautical Engineering, Binzhou University, Binzhou 256603, China 2School of Intelligent Manufacturing, Weifang University of Science and Technology, Weifang 262700, China Received 10 August 2023; accepted manuscript online 14 September 2023; published online 28 September 2023 *Corresponding author. Email: bz_mgl@163.com Citation Text: Yao Y L, Yi H H, Zhang X et al. 2023 Chin. Phys. Lett. 40 100503    Abstract We take the higher-order nonlinear Schrödinger equation as a mathematical model and employ the bilinear method to analytically study the evolution characteristics of femtosecond solitons in optical fibers under higher-order nonlinear effects and higher-order dispersion effects. The results show that the effects have a significant impact on the amplitude and interaction characteristics of optical solitons. The larger the higher-order nonlinear coefficient, the more intense the interaction between optical solitons, and the more unstable the transmission. At the same time, we discuss the influence of other free parameters on third-order soliton interactions. Effectively regulate the interaction of three optical solitons by controlling relevant parameters. These studies will lay a theoretical foundation for experiments and further practicality of optical soliton communications.
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 DOI:10.1088/0256-307X/40/10/100503 © 2023 Chinese Physics Society Article Text With the development of society and economy, the perfection of people's lives, the increasing need for communication, and the continuous development and maturity of fiber optic communication technology, fiber optic communication has gradually become one of the main transmission methods of some communication networks. With the development of fiber optic communication, as a carrier for transmitting information, how to generate femtosecond optical solitons with richer frequency components and narrower pulses has become one of the main considerations for researchers.[1-8] Therefore, generation and transmission of femtosecond optical solitons in optical communication systems have important scientific significance, and the relevant results will provide a certain theoretical reference for further obtaining ultra-short and ultra-high energy optical solitons.[9-14] For optical communication systems, the nonlinear Schrödinger (NLS) equation describes the transmission process of picosecond solitons in single-mode fibers.[15-17] Based on equations and some theoretical methods, researchers have studied the basic structure and characteristics of picosecond solitons.[18-20] Due to the fact that the equation only considers the group velocity dispersion and nonlinear Kerr effect and does not include the influence of nonlinear effects such as self-steepening and self-frequency shift induced by the nonlinear dispersion and stimulated Raman scattering, the NLS equation is no longer applicable to solitons with pulse widths in the sub-picosecond and femtosecond order. With a model of the higher-order NLS equation, we generally research the characteristics of femtosecond solitons propagating in optical fibers.[21-23] On the other hand, the interaction of optical solitons can be split into short-range interaction and long-range interaction from the perspective of interaction distance, and the interaction frequency can be divided into the interaction between optical solitons of the same frequency and the interaction between optical solitons of different frequencies.[24-27] In polarization multiplexing systems, there are also interactions between orthogonal polarization solitons and parallel polarization solitons, which are all nonlinear interactions from a physical perspective. Among them, the interaction of short-range solitons in systems, especially in dense wavelength division multiplexing and high-speed optical time division multiplexing systems, can cause temporary or permanent frequency shifts of optical solitons, and the magnitude of their forces is greatly influenced by the soliton spacing, relative phase, and relative amplitude.[28-30] This study starts from describing the propagation of femtosecond solitons in optical fibers with the higher-order NLS equation, obtains the analytical three soliton solution and its existence conditions through the bilinear method, and analyzes in detail the higher-order effects and effects of free parameters in optical fibers on the transmission and interaction characteristics of optical solitons. The form of the higher-order NLS equation is[31] \begin{align} &i u_{t}+u_{x x}+\alpha |u|^{2}u+i(\gamma_{1}u_{\rm x x x}+\gamma_{2}|u|^{2}u_{x})=0, \tag {1} \end{align} where $\alpha$ is the second order nonlinearity, $\gamma_{1}$ is the third-order-dispersion effect, and $\gamma_{2}$ is the higher-order nonlinear effect. The modulation instability for Eq. (1) was studied by the F-expansion method.[31] The transmission characteristics of two soliton interactions were investigated.[32] Some experimental studies have shown that for the system simulated by Eq. (1), when the power of optical solitons injected into the fiber is high, the optical solitons will split and interact to form multiple soliton pulses. Their existence will limit the generation of narrow pulse width and high energy optical solitons. However, there are only a few previous studies based on multi-soliton interactions to solve those limiting problems. In order to further improve the performance of optical communication systems, it is necessary to have a clear understanding of the physical mechanisms that produce those effects. This study firstly analyzes and investigates the high-order dispersion effect, high-order nonlinear effect and other effects on the nonlinear dynamic evolution process and characteristics of three-soliton interactions in optical communication systems analytically, providing a basis for exploring how to improve the pulse energy of optical solitons. Secondly, this study will employ the bilinear method, which can be used to solve the soliton solutions for integrable equations, to obtain the analytical three soliton solution of Eq. (1). Thirdly, the influence of some parameters on the interaction of three solitons based on the obtained solutions is analyzed. Finally, the conclusions will be presented. Three-Soliton Solutions. First, we make $u(x,t)=p(x,t)/f(x,t)$, here $p(x,t)$ and $f(x,t)$ read \begin{align} &p(x,t)=\epsilon p_{1}(x,t)+\epsilon^{3}p_{3}(x,t)+\epsilon^{5}p_{5}(x,t),\notag\\ &f(x,t) =1+\epsilon^{2}f_{2}(x,t)+\epsilon^{4}f_{4}(x,t)+\epsilon ^{6}f_{6}(x,t). \tag {2} \end{align} Substituting $u(x,t)$ into Eq. (1), we can reach the bilinear forms \begin{align} &\Big[iD_{t}+D_{x}^{2}+i\gamma_{1}D_{x}^{3}\Big] p \cdot f=0,\notag\\ &D_{x}^{2}f \cdot f-\gamma_{2}/(3\gamma_{1})|p|^{2}=0, \tag {3} \end{align} with $\alpha=\gamma_{2}/(3\gamma_{1})$. We assume \begin{align} &p_{1}(x,t)=A_{1}e^{\vartheta_{1}}+A_{2}e^{\vartheta_{2}}+A_{3}e^{\vartheta_{3}} \tag {4} \end{align} with \begin{align} \vartheta_{j}=\,&k_{j}x +\nu_{j}t +\eta_{j}\notag\\ =\,&(k_{j1}+ik_{j2})x +(\nu_{j1}t+i\nu_{j2})t +\eta_{j1}+i\eta_{j2},\notag\\ & (j=1,2,3). \tag {5} \end{align} Thus, we can obtain \begin{align} &\nu_{j1}=3\gamma_{1}k_{j1}k_{j2}^2-\gamma_{1}k_{j1}^3-2k_{j1}k_{j2},\notag\\ &\nu_{j2}=k_{j1}^2-k_{j2}^2-3\gamma_{1}k_{j1}^2 k_{j2}+\gamma_{1}k_{j2}^3. \tag {6} \end{align} We assume \begin{align} f_{2}(x,t)=\,&B_{1}e^{\vartheta_{1}+\vartheta_{1}^{*}}+B_{2}e^{\vartheta_{1}+\vartheta_{2}^{*}}+B_{3}e^{\vartheta_{1}+\vartheta_{3}^{*}} +B_{4}e^{\vartheta_{2}+\vartheta_{1}^{*}}\notag\\ &+B_{5}e^{\vartheta_{2}+\vartheta_{2}^{*}}+B_{6}e^{\vartheta_{2}+\vartheta_{3}^{*}}+B_{7}e^{\vartheta_{3} +\vartheta_{1}^{*}}\notag\\ &+B_{8}e^{\vartheta_{3}+\vartheta_{2}^{*}}+B_{9}e^{\vartheta_{3}+\vartheta_{3}^{*}}, \tag {7} \end{align} then we can derive the coefficients \begin{align} &B_{1}=\frac{|A_{1}|^{2}\gamma_{2}}{24\gamma_{1}k_{11}^2},~ B_{2}=\frac{A_{1}A_{2}^{*}\gamma_{2}}{6 \gamma_{1}(k_{1}+k_{2}^{*})^2},~ B_{3}=\frac{A_{1}A_{3}^{*}\gamma_{2}}{6 \gamma_{1}(k_{1}+k_{3}^{*})^2},\notag\\ &B_{4}=\frac{A_{2}A_{1}^{*}\gamma_{2}}{6 \gamma_{1}(k_{2}+k_{1}^{*})^2},~ B_{5}=\frac{|A_{2}|^{2}\gamma_{2}}{24\gamma_{1}k_{21}^2},~ B_{6}=\frac{A_{2}A_{3}^{*}\gamma_{2}}{6 \gamma_{1}(k_{2}+k_{3}^{*})^2},\notag\\ &B_{7}=\frac{A_{3}A_{1}^{*}\gamma_{2}}{6 \gamma_{1}(k_{3}+k_{1}^{*})^2},~ B_{8}=\frac{A_{3}A_{2}^{*}\gamma_{2}}{6 \gamma_{1}(k_{3}+k_{2}^{*})^2},\notag\\ &B_{9}=\frac{|A_{3}|^{2}\gamma_{2}}{24\gamma_{1}k_{31}^2}. \tag {8} \end{align} Also, we assume \begin{align} p_{3}(x,t)=\,&D_{1}e^{\vartheta_{1}+\vartheta_{2}+\vartheta_{1}^{*}}+D_{2}e^{\vartheta_{1}+\vartheta_{2}+\vartheta_{2}^{*}} +D_{3}e^{\vartheta_{1}+\vartheta_{2}+\vartheta_{3}^{*}}\notag\\ &+D_{4}e^{\vartheta_{1}+\vartheta_{3}+\vartheta_{1}^{*}}+D_{5}e^{\vartheta_{1}+\vartheta_{3}+\vartheta_{2}^{*}}\notag\\ &+D_{6}e^{\vartheta_{1}+\vartheta_{3}+\vartheta_{3}^{*}}+D_{7}e^{\vartheta_{2}+\vartheta_{3}+\vartheta_{1}^{*}}\notag\\ &+D_{8}e^{\vartheta_{2}+\vartheta_{3}+\vartheta_{2}^{*}}+D_{9}e^{\vartheta_{2}+\vartheta_{3}+\vartheta_{3}^{*}}, \tag {9} \end{align} then we can derive the coefficients \begin{align} &D_{1}=\frac{|A_{1}|^{2}A_{2}\gamma_{2}(k_{1}-k_{2})^2}{24\gamma_{1}k_{11}^2(k_{2}+k_{1}^{*})^2},\notag\\ &D_{2}=\frac{|A_{2}|^{2}A_{1}\gamma_{2}(k_{1}-k_{2})^2}{24\gamma_{1}k_{21}^2(k_{1}+k_{2}^{*})^2}, \notag\\ &D_{3}=\frac{A_{1}A_{2}A_{3}^{*}\gamma_{2}(k_{1}-k_{2})^2}{6\gamma_{1}(k_{1}+k_{3}^{*})^2(k_{2}+k_{3}^{*})^2},\notag\\ &D_{4}=\frac{|A_{1}|^{2}A_{3}\gamma_{2}(k_{1}-k_{3})^2}{24\gamma_{1}k_{11}^2(k_{3}+k_{1}^{*})^2}, \notag\\ &D_{5}=\frac{A_{1}A_{3}A_{2}^{*}\gamma_{2}(k_{1}-k_{3})^2}{6\gamma_{1}(k_{1}+k_{2}^{*})^2(k_{3}+k_{2}^{*})^2}, \notag\\ &D_{6}=\frac{|A_{3}|^{2}A_{1}\gamma_{2}(k_{1}-k_{3})^2}{24\gamma_{1}k_{31}^2(k_{1}+k_{3}^{*})^2},\notag\\ &D_{7}=\frac{A_{2}A_{3}A_{1}^{*}\gamma_{2}(k_{2}-k_{3})^2}{6\gamma_{1}(k_{2}+k_{1}^{*})^2(k_{3}+k_{1}^{*})^2}, \notag\\ &D_{8}=\frac{|A_{2}|^{2}A_{3}\gamma_{2}(k_{2}-k_{3})^2}{24\gamma_{1}k_{21}^2(k_{3}+k_{2}^{*})^2},\notag\\ &D_{9}=\frac{|A_{3}|^{2}A_{2}\gamma_{2}(k_{2}-k_{3})^2}{24\gamma_{1}k_{31}^2(k_{2}+k_{3}^{*})^2}. \tag {10} \end{align} Again, we assume \begin{align} f_{4}(x,t)=\,&G_{1}e^{\vartheta_{1}+\vartheta_{2}+\vartheta_{1}^{*}+\vartheta_{2}^{*}} +G_{2}e^{\vartheta_{1}+\vartheta_{2}+\vartheta_{1}^{*}+\vartheta_{3}^{*}}\notag\\ &+G_{3}e^{\vartheta_{1}+\vartheta_{2}+\vartheta_{2}^{*}+\vartheta_{3}^{*}} +G_{4}e^{\vartheta_{1}+\vartheta_{3}+\vartheta_{1}^{*}+\vartheta_{2}^{*}}\notag\\ &+G_{5}e^{\vartheta_{1}+\vartheta_{3}+\vartheta_{1}^{*}+\vartheta_{3}^{*}} +G_{6}e^{\vartheta_{1}+\vartheta_{3}+\vartheta_{2}^{*}+\vartheta_{3}^{*}}\notag\\ &+G_{7}e^{\vartheta_{2}+\vartheta_{3}+\vartheta_{1}^{*}+\vartheta_{2}^{*}} +G_{8}e^{\vartheta_{2}+\vartheta_{3}+\vartheta_{1}^{*}+\vartheta_{3}^{*}}\notag\\ &+G_{9}e^{\vartheta_{2}+\vartheta_{3}+\vartheta_{2}^{*}+\vartheta_{3}^{*}}, \tag {11} \end{align} then we can derive the coefficients \begin{align} &G_{1}=\frac{|A_{1}|^{2}|A_{2}|^{2}\gamma_{2}^{2}[(k_{11}-k_{21})^{2}+(k_{12}-k_{22})^{2}]^2} {576k_{11}^2k_{21}^2\gamma_{1}^{2}(k_{1}+k_{2}^{*})^{2}(k_{2}+k_{1}^{*})^{2}}, \notag\\ &G_{2}=\frac{|A_{1}|^{2}A_{2}A_{3}^{*}\gamma_{2}^{2}(k_{1}-k_{2})^{2}(k_{1}^{*}-k_{3}^{*})^{2}} {144k_{11}^2\gamma_{1}^{2}(k_{2}+k_{1}^{*})^{2}(k_{1}+k_{3}^{*})^{2}(k_{2}+k_{3}^{*})^{2}}, \notag\\ &G_{3}=\frac{|A_{2}|^{2}A_{1}A_{3}^{*}\gamma_{2}^{2}(k_{1}-k_{2})^{2}(k_{2}^{*}-k_{3}^{*})^{2}} {144k_{21}^2\gamma_{1}^{2}(k_{1}+k_{2}^{*})^{2}(k_{1}+k_{3}^{*})^{2}(k_{2}+k_{3}^{*})^{2}}, \notag\\ &G_{4}=\frac{|A_{1}|^{2}A_{3}A_{2}^{*}\gamma_{2}^{2}(k_{1}^{*}-k_{2}^{*})^{2}(k_{1}-k_{3})^{2}} {144k_{11}^2\gamma_{1}^{2}(k_{1}+k_{2}^{*})^{2}(k_{3}+k_{1}^{*})^{2}(k_{3}+k_{2}^{*})^{2}}, \notag\\ &G_{5}=\frac{|A_{1}|^{2}|A_{3}|^{2}\gamma_{2}^{2}[(k_{11}-k_{31})^{2}+(k_{12}-k_{32})^{2}]^2} {576k_{11}^2k_{31}^2\gamma_{1}^{2}(k_{1}+k_{3}^{*})^{2}(k_{3}+k_{1}^{*})^{2}}, \notag\\ &G_{6}=\frac{|A_{3}|^{2}A_{1}A_{2}^{*}\gamma_{2}^{2}(k_{1}-k_{3})^{2}(k_{2}^{*}-k_{3}^{*})^{2}} {144k_{31}^2\gamma_{1}^{2}(k_{1}+k_{2}^{*})^{2}(k_{1}+k_{3}^{*})^{2}(k_{3}+k_{2}^{*})^{2}},\notag\\ &G_{7}=\frac{|A_{2}|^{2}A_{3}A_{1}^{*}\gamma_{2}^{2}(k_{1}^{*}-k_{2}^{*})^{2}(k_{2}-k_{3})^{2}} {144k_{21}^2\gamma_{1}^{2}(k_{2}+k_{1}^{*})^{2}(k_{3}+k_{1}^{*})^{2}(k_{3}+k_{2}^{*})^{2}}, \notag\\ &G_{8}=\frac{|A_{3}|^{2}A_{2}A_{1}^{*}\gamma_{2}^{2}(k_{2}-k_{3})^{2}(k_{1}^{*}-k_{3}^{*})^{2}} {144k_{31}^2\gamma_{1}^{2}(k_{2}+k_{1}^{*})^{2}(k_{2}+k_{3}^{*})^{2}(k_{3}+k_{1}^{*})^{2}},\notag\\ &G_{9}=\frac{|A_{2}|^{2}|A_{3}|^{2}\gamma_{2}^{2}[(k_{21}-k_{31})^{2}+(k_{22}-k_{32})^{2}]^2} {576k_{21}^2k_{31}^2\gamma_{1}^{2}(k_{2}+k_{3}^{*})^{2}(k_{3}+k_{2}^{*})^{2}}. \tag {12} \end{align} We assume \begin{align} p_{5}(x,t)=\,&H_{1}e^{\vartheta_{1}+\vartheta_{2}+\vartheta_{3}+\vartheta_{1}^{*}+\vartheta_{2}^{*}} +H_{2}e^{\vartheta_{1}+\vartheta_{2}+\vartheta_{3}+\vartheta_{1}^{*}+\vartheta_{3}^{*}}\notag\\ &+H_{3}e^{\vartheta_{1}+\vartheta_{2}+\vartheta_{3}+\vartheta_{2}^{*}+\vartheta_{3}^{*}}, \tag {13} \end{align} then we can derive the coefficients \begin{align} &H_{1}=\frac{|A_{1}|^{2}|A_{2}|^{2}A_{3}\gamma_{2}^{2}[(k_{11}-k_{21})^{2}+(k_{12}-k_{22})^{2}]^2(k_{1}-k_{3})^{2}(k_{2}-k_{3})^{2}} {576k_{11}^2k_{21}^2\gamma_{1}^{2}(k_{1}+k_{2}^{*})^{2}(k_{2}+k_{1}^{*})^{2}(k_{3}+k_{1}^{*})^{2}(k_{3}+k_{2}^{*})^{2}},\notag\\ &H_{2}=\frac{|A_{1}|^{2}|A_{3}|^{2}A_{2}\gamma_{2}^{2}[(k_{11}-k_{31})^{2}+(k_{12}-k_{32})^{2}]^2(k_{1}-k_{2})^{2}(k_{2}-k_{3})^{2}} {576k_{11}^2k_{31}^2\gamma_{1}^{2}(k_{2}+k_{1}^{*})^{2}(k_{1}+k_{3}^{*})^{2}(k_{2}+k_{3}^{*})^{2}(k_{3}+k_{1}^{*})^{2}},\notag\\ &H_{3}=\frac{|A_{2}|^{2}|A_{3}|^{2}A_{1}\gamma_{2}^{2}[(k_{21}-k_{31})^{2}+(k_{22}-k_{32})^{2}]^2(k_{1}-k_{2})^{2}(k_{1}-k_{3})^{2}} {576k_{21}^2k_{31}^2\gamma_{1}^{2}(k_{1}+k_{2}^{*})^{2}(k_{1}+k_{3}^{*})^{2}(k_{2}+k_{3}^{*})^{2}(k_{3}+k_{2}^{*})^{2}}. \tag {14} \end{align} We assume \begin{align} f_{6}(x,t) =Le^{\vartheta_{1}+\vartheta_{2}+\vartheta_{3}+\vartheta_{1}^{*}+\vartheta_{2}^{*}+\vartheta_{3}^{*}}, \tag {15} \end{align} then we can derive the coefficients \begin{align} L=\frac{|A_{1}|^{2}|A_{2}|^{2}|A_{3}|^{2}\gamma_{2}^{3}[(k_{11}-k_{21})^{2}+(k_{12}-k_{22})^{2}]^2[(k_{11}-k_{31})^{2} +(k_{12}-k_{32})^{2}]^2[(k_{21}-k_{31})^{2}+(k_{22}-k_{32})^{2}]^2}{13824k_{11}^2k_{21}^2k_{31}^2\gamma_{1}^{3}(k_{1}+k_{2}^{*})^{2} (k_{2}+k_{1}^{*})^{2}(k_{1}+k_{3}^{*})^{2}(k_{3}+k_{1}^{*})^{2}(k_{2}+k_{3}^{*})^{2}(k_{3}+k_{2}^{*})^{2}}. \tag {16} \end{align} Thus, when $\epsilon=1$, the form of the three-soliton solution is \begin{align} &u(x,t)=\frac{p_{1}(x,t)+p_{3}(x,t)+p_{5}(x,t)}{1+f_{2}(x,t)+f_{4}(x,t)+f_{6}(x,t)}. \tag {17} \end{align} Discussions. In three-soliton solution (17), there are eleven parameters $\gamma_{1}$, $\gamma_{2}$, $k_{j}$, $\xi_{j}$, and $A_{j}$. Among them, $k_{j}$, $\xi_{j}$, and $A_{j}$ are free parameters. Here, we analyze their influences on the three-soliton interactions. Firstly, we set $k_{1}=1+2i$, $k_{2}=2+3i$, $k_{3}=1+i$, $\xi_{1}=1$, $\xi_{2}=3$, $\xi_{3}=2$, $A_{1}=1$, $A_{2}=2$, and $A_{3}=1$ in Fig. 1, and discuss the influences of $\gamma_{1}$ and $\gamma_{2}$ on the interactions of three solitons.
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Fig. 1. Influence of $\gamma_{1}$ and $\gamma_{2}$ on three-soliton interactions for parameters $k_{1}=1+2i$, $k_{2}=2+3i$, $k_{3}=1+i$, $\xi_{1}=1$, $\xi_{2}=3$, $\xi_{3}=2$, $A_{1}=1$, $A_{2}=2$, and $A_{3}=1$: (a) $\gamma_{1}=-0.06$, $\gamma_{2}=-1.8$; (b) $\gamma_{1}=0.34$, $\gamma_{2}=0.5$; (c) $\gamma_{1} =0.28$, $\gamma_{2}=0.56$; (d) $\gamma_{1}=0.56$, $\gamma_{2}=1.4$.
In Fig. 1(a), $\gamma_{1}=-0.06$ and $\gamma_{2}=-1.8$, we can see that the optical solitons with large amplitudes are more affected by optical solitons with small amplitudes. During the interaction between large amplitude optical solitons and small amplitude optical solitons, there will be a very large protrusion, and the amplitude change is very obvious. After the interaction, this large amplitude optical soliton will quickly split into two small amplitude optical solitons. By changing the values of $\gamma_{1}$ and $\gamma_{2}$, we can significantly change the amplitude of optical solitons, as shown in Figs. 1(b)–1(d). Due to differences in dispersion and efficient nonlinear effects, the initial states of optical solitons (such as incident phase, initial amplitude, and initial velocity) are also different, resulting in different interaction states among three optical solitons.
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Fig. 2. Influence of $k_{11}$, $k_{21}$, and $k_{31}$ on three-soliton interactions for parameters $k_{12}=1$, $k_{22}=0$, $k_{32}=3$, $\xi_{1}=1$, $\xi_{2}=3$, $\xi_{3}=2$, $\gamma_{1}=0.34$, $\gamma_{2}=0.5$, $A_{1}=1$, $A_{2}=2$, and $A_{3}=1$: (a) $k_{11}=-1.8$, $k_{21}=-1.7$, $k_{31}=-1.8$; (b) $k_{11}=1.4$, $k_{21}=1.2$, $k_{31}=-0.66$; (c) $k_{11}=1.3$, $k_{21}=-1.5$, $k_{31}=-1$; (d) $k_{11}=-1.4$, $k_{21}=-0.53$, $k_{31}=1.4$.
In Fig. 2, based on the relevant parameters in Fig. 1(b), we discuss their impact on the interactions of three optical solitons by changing the values of $k_{11}$, $k_{21}$, and $k_{31}$. In Fig. 2(a), $k_{11}=-1.8$, $k_{21}=-1.7$, and $k_{31}=-1.8$, the values of those three parameters are very similar, and the amplitudes of the three optical solitons are almost equal. They interact near $x=0$ and their interactions are not too intense. When the values of $k_{j1}$ differ significantly, the amplitude of optical solitons will undergo significant changes. In Fig. 2(b), the value of $k_{31}$ differs significantly from $k_{11}$ and $k_{21}$, the amplitude of one optical soliton is small, and the optical soliton is more intense in the interaction process, and there is obvious oscillation and modulation at the interaction point. By changing the values of $k_{11}$, $k_{21}$, and $k_{31}$, we can change the intensity of soliton interactions and the presence of modulation in Figs. 2(c) and 2(d).
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Fig. 3. Influence of $k_{12}$, $k_{22}$, and $k_{32}$ on three-soliton interactions for parameters $k_{11}=1.3$, $k_{21}=-1.5$, $k_{31}=-1$, $\xi_{1}=1$, $\xi_{2}=3$, $\xi_{3}=2$, $\gamma_{1}=0.34$, $\gamma_{2}=0.5$, $A_{1}=1$, $A_{2}=2$, and $A_{3}=1$: (a) $k_{12}=1.5$, $k_{22}=-0.31$, $k_{32}=1.3$; (b) $k_{12}=-0.13$, $k_{22}=-1.4$, $k_{32}= 1.7$; (c) $k_{12}=0.13$, $k_{22}=-0.63$, $k_{32}=1.7$; (d) $k_{12}=0.5$, $k_{22}=-0.34$, $k_{32}=-1$.
In Fig. 3, based on the relevant parameters in Fig. 2(c), we achieve the parallel transmission of optical solitons by adjusting the values of $k_{j2}$. In Fig. 3(a), $k_{12}=1.5$, $k_{22}=-0.31$, and $k_{32}=1.3$, two optical solitons exhibit parallel transmission states, and the other optical soliton will interact with these two optical solitons separately. The interaction process is more relaxed compared to Fig. 2. By changing the values of $k_{j2}$, the states of those two parallel propagating optical solitons can be adjusted. In Fig. 3(b), the parallel transmission state of two optical solitons is broken, and their amplitude changes sharply during the interaction process, and they exhibit strong modulation. In Fig. 3(c), the spacing between those two parallel propagating optical solitons is reduced, and after their interaction, the phase shift further reduces the spacing between the solitons, resulting in a significant impact between the parallel propagating optical solitons. In Fig. 3(d), those three optical solitons exhibit opposite interactions, and they do not exhibit significant changes in amplitude during the interaction process. Thus, we can adjusting the values of $k_{j2}$ to control the parallel transmission of optical solitons. In Fig. 4, we mainly depict the influence of $\xi_{j}$ on the interaction of three optical solitons. The parameters in Figs. 4(a) and 4(b) are based on Fig. 3(d). We can see that by changing the values of $\xi_{j}$, we can control the interaction point of optical solitons and the intensity of the interactions. The parameters in Figs. 4(c) and 4(d) are based on Fig. 3(a). Changing the values of $\xi_{j}$ can control the spacing of parallel transmission optical solitons, thereby regulating the parallel transmission characteristics of optical solitons and improving the communication quality and capacity of the system.
cpl-40-10-100503-fig4.png
Fig. 4. Influence of $\xi_{1}$, $\xi_{2}$, and $\xi_{3}$ on three-soliton interactions for parameters $\gamma_{1}=0.34$, $\gamma_{2}=0.5$, $A_{1}=1$, $A_{2}=2$, and $A_{3}=1$: (a) $k_{1}=1.3+0.5i$, $k_{2}=-1.5-0.34i$, $k_{3}=-1-i$, $\xi_{1}=-0.94$, $\xi_{2}=-0.41$, $\xi_{3}=-0.16$; (b) $k_{1}=1.3+0.5i$, $k_{2}=-1.5-0.34i$, $k_{3}= -1-i$, $\xi_{1}=0.31$, $\xi_{2}=0.41$, $\xi_{3}=-0.16$; (c) $k_{1}=1.3+1.5i$, $k_{2}=-1.5-0.31i$, $k_{3}=-1+1.3i$, $\xi_{1}=0.1$, $\xi_{2}=-0.13$, $\xi_{3}=-1.6$; (d) $k_{1}=1.3+1.5i$, $k_{2}=-1.5-0.31i$, $k_{3}=-1+1.3i$, $\xi_{1}=-0.47$, $\xi_{2}=-0.53$, $\xi_{3}=-0.31$.
In summary, based on the nonlinear Schrödinger equation (1), the interaction characteristics of three optical solitons have been studied analytically. Based on the bilinear method, we have obtained the three-soliton solution (17) of Eq. (1). With the help of the solution, we have analyzed the influences of dispersion effects, nonlinear effects and free parameters on the interactions of three optical solitons. The results show that by changing the values of $\gamma_{1}$ and $\gamma_{2}$, the amplitudes of optical solitons have been significantly changed due to the differences in dispersion and efficient nonlinear effects. The intensity of soliton interactions and the presence of modulation have been changed with the values of $k_{j1}$. The values of $k_{j2}$ have been used to control the parallel transmission of optical solitons. The interaction point of optical solitons and the intensity of the interactions have been controlled by changing the values of $\xi_{j}$. The conclusion is of guiding significance for how to weaken the interaction of multiple optical solitons and to improve communication capacity. Acknowledgements. Scientific Research Foundation of Weifang University of Science and Technology (Grant No. KJRC2022002), Shandong Province Higher Educational Science and Technology Program (Grant No. J18KB108), Research start-up fees for doctoral degree holders and senior professional title holders with master's degrees of Binzhou University (Grant No. 2022Y12). References Preparation of High Damage Threshold Device Based on Bi2 Se3 Film and Its Application in Fiber LasersTunable Dual-Wavelength Fiber Laser in a Novel High Entropy van der Waals MaterialMode conversions and molecular forms of breathers under parameter controlExact analysis and elastic interaction of multi-soliton for a two-dimensional Gross-Pitaevskii equation in the Bose-Einstein condensationBi4 Br4 -based saturable absorber with robustness at high power for ultrafast photonic deviceResearch progress of mode-locked pulsed fiber lasers with high damage threshold saturable absorberStable transmission characteristics of double-hump solitons for the coupled Manakov equations in fiber lasersNonlinear optical property and application of yttrium oxide in erbium-doped fiber lasersTungsten disulfide saturable absorbers for 67 fs mode-locked erbium-doped fiber lasersTungsten disulphide for ultrashort pulse generation in all-fiber lasersOptical properties and applications for MoS2 -Sb2 Te3 -MoS2 heterostructure materialsSupercontinuum generation in photonic crystal fiberTemporal solitons in optical microresonatorsThe Peregrine soliton in nonlinear fibre opticsSoliton Rectangular Pulses and Bound States in a Dissipative System Modeled by the Variable-Coefficients Complex Cubic-Quintic Ginzburg–Landau EquationDynamic Behavior of Optical Soliton Interactions in Optical Communication SystemsDynamics and spectral analysis of optical rogue waves for a coupled nonlinear Schrödinger equation applicable to pulse propagation in isotropic mediaSoliton fusion and fission for the high-order coupled nonlinear Schrödinger system in fiber lasersChirped Bright and Kink Solitons in Nonlinear Optical Fibers with Weak Nonlocality and Cubic-Quantic-Septic NonlinearityInfluence of Parameters of Optical Fibers on Optical Soliton InteractionsEvolution of periodic wave and dromion-like structure solutions in the variable coefficients coupled high-order complex Ginzburg–Landau systemSoliton molecules in the kink, antikink and oscillatory backgroundThe dynamic characteristics of pure-quartic solitons and soliton moleculesDromion-like soliton interactions for nonlinear Schrödinger equation with variable coefficients in inhomogeneous optical fibersAnalytic study on interactions between periodic solitons with controllable parametersOptical Spatial Solitons and Their Interactions: Universality and DiversityModulational instability, solitons and beam propagation in spatially nonlocal nonlinear mediaOptical solitons due to quadratic nonlinearities: from basic physics to futuristic applicationsLong-range interactions between optical solitonsSoliton interaction in a fiber ring laserDynamics of soliton solutions in optical fibers modelled by perturbed nonlinear Schrödinger equation and stability analysisHigh-order effect on the transmission of two optical solitons
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