Vector bright solitons and their interactions of the couple Fokas–Lenells system in a birefringent optical fiber

Abstract

In the fiber communication domain, people are facing the challenges due to the rapidly growing requirement on the capacity from new functions and services. Multi-hump solitons are therefore noticed and studied on the feasibility of improving the capacity of the optical fiber communication. In this paper, we study the vector bright solitons and their interaction properties of the coupled Fokas–Lenells system, which models the femtosecond optical pulses in a birefringent optical fiber. We derive the so-called degenerate and nondegenerate vector solitons associated with the one and two eigenvalues, respectively, and the latter admits the symmetric profile. Asymptotically and graphically, interaction patterns of such solitons are classified as follows: Interactions between the degenerate solitons can be elastic or inelastic, reflecting the intensity redistribution between the two components; Interactions between the degenerate and nondegenerate solitons are inelastic, which make the nondegenerate solitons maintaining or losing the profiles in the different situations; Interactions between the nondegenerate solitons do not cause the intensity redistribution, while their shapes change slightly or remain unchanged.

Appendices

Appendix A

Furthermore, $u_{\alpha }[N]$ can be rewritten as

$$\begin{aligned} u_{\alpha }[N]=2\sigma \frac{\text {det} \left( C_{\alpha , N}'\right) }{\text {det}\left( B_{N}'\right) } \mathrm{e}^{i\frac{\tau }{\epsilon }}=2\sigma \frac{\text {det} \left( \begin{array}{cc} B'_{N} &{}\quad Y_{2}'^{[\alpha ]\dag } \\ Y_{1}' &{}\quad 0 \\ \end{array} \right) }{\text {det}\left( B_{N}'\right) }\mathrm{e}^{i\frac{\tau }{\epsilon }}, \end{aligned}$$

where

$$\begin{aligned}&B'_{N}=\left[ \frac{2}{\lambda _{j}^{*2}-\lambda _{k}^{2}}\left( \lambda _{j}^{*}\mathrm{e}^{2\theta _{j}^{*}+2\theta _{k}} -\sigma \lambda _{k}\gamma _{j,k}\right) \right] _{N\times N}, \\&Y_{1}'=(\mathrm{e}^{2\theta _{1}},\mathrm{e}^{2\theta _{2}},\ldots ,\mathrm{e}^{2\theta _{N}}), \quad Y_{2}'=\left( \begin{array}{c} l_{1},l_{2},\ldots , l_{N} \\ m_{1},m_{2},\ldots ,m_{N} \\ \end{array} \right) . \end{aligned}$$

Supposing that $v_{1}<v_{2}<\cdots <v_{N}$ and $\theta _{1R},\theta _{2R},\ldots ,\theta _{k-1,R}\rightarrow -\infty ; \theta _{k+1,R},\theta _{k+2,R},\ldots ,\theta _{N,R}\rightarrow +\infty$ along the track $\tau -v_{k}\xi =$const as $\xi \rightarrow -\infty$. It follows that

$$\begin{aligned}&\text {det}\left( B'_{N}\right) \rightarrow \mathrm{e}^{4 (\theta _{k+1}+\theta _{k+2}+\cdots +\theta _{N})_{R} }\left[ \text {det}( {\widetilde{B}}_{k}^{-})\right] ,\\&\text {det}\left( C_{\alpha , N}'\right) \rightarrow \mathrm{e}^{4(\theta _{k+1}+\theta _{k+2}+\cdots +\theta _{N})_{R}} \left[ \text {det}( {\widetilde{C}}_{\alpha ,k}^{-})\right] ,\\&u_{\alpha }[N]\rightarrow 2\sigma \frac{\text {det}({\widetilde{C}}_{\alpha ,k}^{-})}{\text {det}({\widetilde{B}}_{k}^{-})}\mathrm{e}^{i\frac{\tau }{\epsilon }}, \end{aligned}$$

where

$$\begin{aligned}&{\widetilde{B}}_{k}^{-}=\left[ \begin{array}{ccccccc} -\frac{2\sigma \lambda _{1}\gamma _{1,1}}{\lambda _{1}^{*2}-\lambda _{1}^{2}} &{}\quad \cdots &{}\quad -\frac{2\sigma \lambda _{k-1}\gamma _{1,k-1}}{\lambda _{1}^{*2}-\lambda _{k-1}^{2}} &{}\quad -\frac{2\sigma \lambda _{k}\gamma _{1,k}}{\lambda _{1}^{*2}-\lambda _{k}^{2}} &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ -\frac{2\sigma \lambda _{1}\gamma _{k-1,1}}{\lambda _{k-1}^{*2}-\lambda _{1}^{2}} &{}\quad \cdots &{}\quad -\frac{2\sigma \lambda _{k-1}\gamma _{k-1,k-1}}{\lambda _{k-1}^{*2}-\lambda _{k-1}^{2}} &{}\quad -\frac{2\sigma \lambda _{k}\gamma _{k-1,k}}{\lambda _{k-1}^{*2}-\lambda _{k}^{2}} &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ -\frac{2\sigma \lambda _{1}\gamma _{k,1}}{\lambda _{k}^{*2}-\lambda _{1}^{2}} &{}\quad \cdots &{}\quad -\frac{2\sigma \lambda _{k-1}\gamma _{k,k-1}}{\lambda _{k}^{*2}-\lambda _{k-1}^{2}} &{}\quad \frac{2\left( \lambda _{k}^{*}\mathrm{e}^{2\theta _{k}^{*}+2\theta _{k}} -\sigma \lambda _{k}\gamma _{k,k}\right) }{\lambda _{k}^{*2}-\lambda _{k}^{2}} &{}\quad \frac{2\lambda _{k}^{*}\mathrm{e}^{2\theta _{k}^{*}}}{\lambda _{k}^{*2}-\lambda _{k+1}^{2}} &{}\quad \cdots &{}\quad \frac{2\lambda _{k}^{*}\mathrm{e}^{2\theta _{k}^{*}}}{\lambda _{k}^{*2}-\lambda _{N}^{2}} \\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad \frac{2\lambda _{k+1}^{*}\mathrm{e}^{2\theta _{k}}}{\lambda _{k+1}^{*2}-\lambda _{k}^{2}} &{}\quad \frac{2\lambda _{k+1}^{*}}{\lambda _{k+1}^{*2}-\lambda _{k+1}^{2}} &{}\quad \cdots &{}\quad \frac{2\lambda _{k+1}^{*}}{\lambda _{k+1}^{*2}-\lambda _{N}^{2}} \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad \frac{2\lambda _{N}^{*}\mathrm{e}^{2\theta _{k}}}{\lambda _{N}^{*2}-\lambda _{k}^{2}} &{}\quad \frac{2\lambda _{N}^{*}}{\lambda _{N}^{*2}-\lambda _{k+1}^{2}} &{}\quad \cdots &{}\quad \frac{2\lambda _{N}^{*}}{\lambda _{N}^{*2}-\lambda _{N}^{2}} \\ \end{array} \right] , \\&{\widetilde{C}}_{\alpha ,k}^{-}=\left( \begin{array}{cc} {\widetilde{B}}_{k}^{-} &{}\quad {\widetilde{Y}}_{2,k}^{-[\alpha ]\dag } \\ {\widetilde{Y}}_{1,k}^{-} &{}\quad 0 \\ \end{array} \right) ,\ {\widetilde{Y}}_{1,k}^{-}=(0,\ldots ,0,\mathrm{e}^{2\theta _{k}},1,\ldots ,1),\ {\widetilde{Y}}_{2,k}^{-}=\left( \begin{array}{c} l_{1},\ldots ,l_{k},0,\ldots ,0 \\ m_{1},\ldots ,m_{k},0,\ldots ,0 \\ \end{array} \right) . \end{aligned}$$

By the same thoughts above, we can derive the asymptotical behavior as $\xi \rightarrow +\infty$ along the trajectory $\tau -v_{k}\xi =\text {const}$ with $\theta _{1R},\theta _{2R},\ldots ,\theta _{k-1,R}\rightarrow +\infty ; \theta _{k+1,R},\theta _{k+2,R},\ldots ,\theta _{N,R}\rightarrow -\infty$. It follows that

$$\begin{aligned}&\text {det}\left( B'_{N}\right) \rightarrow \mathrm{e}^{4 (\theta _{1}+\theta _{2}+\cdots +\theta _{k-1})_{R} }\left[ \text {det}( {\widetilde{B}}_{k}^{+})\right] ,\\&\text {det}\left( C_{\alpha , N}'\right) \rightarrow \mathrm{e}^{4(\theta _{1}+\theta _{2}+\cdots +\theta _{k-1})_{R}} \left[ \text {det}( {\widetilde{C}}_{\alpha ,k}^{+})\right] ,\\&u_{\alpha }[N]\rightarrow 2\sigma \frac{\text {det}({\widetilde{C}}_{\alpha ,k}^{+})}{\text {det}{{\widetilde{B}}_{k}^{+}}}\mathrm{e}^{i\frac{\tau }{\epsilon }}, \end{aligned}$$

where

$$\begin{aligned}&{\widetilde{B}}_{k}^{+}=\left[ \begin{array}{ccccccc} \frac{2\lambda _{1}^{*}}{\lambda _{1}^{*2}-\lambda _{1}^{2}} &{}\quad \cdots &{}\quad \frac{2\lambda _{1}^{*}}{\lambda _{1}^{*2}-\lambda _{k-1}^{2}} &{}\quad \frac{2\lambda _{1}^{*}\mathrm{e}^{2\theta _{k}}}{\lambda _{1}^{*2}-\lambda _{k}^{2}} &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \frac{2\lambda _{k-1}^{*}}{\lambda _{k-1}^{*2}-\lambda _{1}^{2}} &{}\quad \cdots &{}\quad \frac{2\lambda _{k-1}^{*}}{\lambda _{k-1}^{*2}-\lambda _{k-1}^{2}} &{}\quad \frac{2\lambda _{k-1}^{*}\mathrm{e}^{2\theta _{k}}}{\lambda _{k-1}^{*2}-\lambda _{k}^{2}} &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \frac{2\lambda _{k}^{*}\mathrm{e}^{2\theta _{k}^{*}}}{\lambda _{k}^{*2}-\lambda _{1}^{2}} &{}\quad \cdots &{}\quad \frac{2\lambda _{k}^{*}\mathrm{e}^{2\theta _{k}^{*}}}{\lambda _{k}^{*2}-\lambda _{k-1}^{2}} &{}\quad \frac{2\left( \lambda _{k}^{*}\mathrm{e}^{2\theta _{k}^{*}+2\theta _{k}} -\sigma \lambda _{k}\gamma _{k,k}\right) }{\lambda _{k}^{*2}-\lambda _{k}^{2}} &{}\quad -\frac{2\sigma \lambda _{k+1}\gamma _{k,k+1}}{\lambda _{k}^{*2}-\lambda _{k+1}^{2}} &{}\quad \cdots &{}\quad -\frac{2\sigma \lambda _{N}\gamma _{k,N}}{\lambda _{k}^{*2}-\lambda _{N}^{2}} \\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad -\frac{2\sigma \lambda _{k}\gamma _{k+1,k}}{\lambda _{k+1}^{*2}-\lambda _{k}^{2}} &{}\quad -\frac{2\sigma \lambda _{k+1}\gamma _{k+1,k+1}}{\lambda _{k+1}^{*2}-\lambda _{k+1}^{2}} &{}\quad \cdots &{}\quad -\frac{2\sigma \lambda _{N}\gamma _{k+1,N}}{\lambda _{k+1}^{*2}-\lambda _{N}^{2}} \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad -\frac{2\sigma \lambda _{k}\gamma _{N,k}}{\lambda _{N}^{*2}-\lambda _{k}^{2}} &{}\quad -\frac{2\sigma \lambda _{k+1}\gamma _{N,k+1}}{\lambda _{N}^{*2}-\lambda _{k+1}^{2}} &{}\quad \cdots &{}\quad -\frac{2\sigma \lambda _{N}\gamma _{N,N}}{\lambda _{N}^{*2}-\lambda _{N}^{2}} \\ \end{array} \right] , \\&{\widetilde{C}}_{\alpha ,k}^{+}=\left( \begin{array}{cc} {\widetilde{B}}_{k}^{+} &{}\quad {\widetilde{Y}}_{2,k}^{+[\alpha ]\dag } \\ {\widetilde{Y}}_{1,k}^{+} &{}\quad 0 \\ \end{array} \right) ,\ {\widetilde{Y}}_{1,k}^{+}=(1,\ldots ,1,\mathrm{e}^{2\theta _{k}},0,\ldots ,0),\ {\widetilde{Y}}_{2,k}^{+}=\left( \begin{array}{c} 0,\ldots ,0, l_{k},\ldots ,l_{N} \\ 0,\ldots ,0,m_{k},\ldots ,m_{N} \\ \end{array} \right) . \end{aligned}$$

Appendix B

$$\begin{aligned} a_{1}^{+}= & {} \frac{(\lambda _{1}^{2}-\lambda _{2}^{2}) (\lambda _{2}^{2}-\lambda _{2}^{*2})(\lambda _{2}^{2}-\lambda _{3}^{*2}) \left[ |m_{3}|^{2}(\lambda _{2}^{*2}-\lambda _{3}^{2})+|l_{3}|^{2} (\lambda _{2}^{*2}-\lambda _{3}^{*2})\right] }{l_{2}\lambda _{2}(\lambda _{2}^{2}-\lambda _{1}^{*2}) \left( |m_{3}|^{2}|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{\rho _{1}^{+}}= & {} \frac{|m_{1}|^{2}\lambda _{1}(\lambda _{2}^{2}-\lambda _{2}^{*2}) (\lambda _{2}^{2}-\lambda _{3}^{*2})\left[ |m_{3}|^{2}|\lambda _{1}^{2} -\lambda _{3}^{2}|^{2} (\lambda _{3}^{2}-\lambda _{2}^{*2})+|l_{3}|^{2}|\lambda _{1}^{2}-\lambda _{3}^{*2}|^{2} (\lambda _{3}^{*2}-\lambda _{2}^{*2})\right] }{-\sigma l_{2}\lambda _{1}^{*}\lambda _{2}|\lambda _{1}^{2}-\lambda _{3}^{*2}|^{2} \left( |m_{3}|^{2}|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{s_{11}}= & {} \frac{l_{3}^{*}m_{1}^{*}m_{3}\lambda _{2}^{*} (\lambda _{1}^{*2}-\lambda _{1}^{2})(\lambda _{1}^{2}-\lambda _{2}^{2}) (\lambda _{2}^{*2}-\lambda _{3}^{2}) (\lambda _{2}^{2}-\lambda _{3}^{*2})(\lambda _{3}^{2}-\lambda _{3}^{*2})}{|l_{2}|^{2}\lambda _{1}^{*}\lambda _{2} (\lambda _{1}^{2}-\lambda _{2}^{*2}) (\lambda _{1}^{*2}-\lambda _{3}^{2})\left( |m_{3}|^{2}|\lambda _{2}^{*2} -\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{s_{12}}= & {} \frac{l_{3}^{*}m_{1}^{*}m_{3} (-\lambda _{1}^{*2}+\lambda _{1}^{2})(\lambda _{2}^{*2}-\lambda _{3}^{2}) (\lambda _{2}^{*2}-\lambda _{3}^{*2}) (\lambda _{3}^{2}-\lambda _{3}^{*2})}{\sigma \lambda _{1}^{*} (\lambda _{1}^{*2}-\lambda _{3}^{2})\left( |m_{3}|^{2}|\lambda _{2}^{*2} -\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ b_{1}^{+}= & {} \frac{m_{1}^{*}(-\lambda _{1}^{*2}+\lambda _{1}^{2})\left[ |m_{3}|^{2} (\lambda _{1}^{*2}-\lambda _{3}^{*2})|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}(\lambda _{1}^{*2}-\lambda _{3}^{2})|\lambda _{2}^{2} -\lambda _{3}^{2}|^{2}\right] }{\sigma \lambda _{1}^{*} (\lambda _{1}^{*2}-\lambda _{3}^{2})\left( |m_{3}|^{2}|\lambda _{2}^{*2} -\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{\rho _{2}^{+}}= & {} \frac{m_{1}^{*}\lambda _{2}^{*}(\lambda _{1}^{*2}-\lambda _{1}^{2}) (\lambda _{1}^{2}-\lambda _{2}^{2})|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2} \left[ |m_{3}|^{2} (\lambda _{1}^{*2}-\lambda _{3}^{*2})+ |l_{3}|^{2}(\lambda _{1}^{*2}-\lambda _{3}^{2})\right] }{\lambda _{1}^{*} (\lambda _{1}^{*2}-\lambda _{3}^{2})\left( |m_{3}|^{2}|\lambda _{2}^{*2} -\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{s_{21}}= & {} -\frac{l_{3}m_{3}^{*}|m_{1}|^{2}\lambda _{1} (\lambda _{2}^{2}-\lambda _{2}^{*2})(\lambda _{1}^{*2}-\lambda _{3}^{*2}) (\lambda _{2}^{2}-\lambda _{3}^{*2}) (\lambda _{3}^{*2}-\lambda _{3}^{2})}{\sigma l_{2}\lambda _{1}^{*}\lambda _{2} (\lambda _{1}^{2}-\lambda _{3}^{*2}) \left( |m_{3}|^{2}|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{s_{22}}= & {} \frac{l_{3}m_{3}^{*} (\lambda _{1}^{2}-\lambda _{1}^{2})(\lambda _{2}^{2}-\lambda _{2}^{*2}) (\lambda _{2}^{2}-\lambda _{3}^{*2}) (\lambda _{3}^{*2}-\lambda _{3}^{2})}{l_{2}\lambda _{2} (\lambda _{2}^{2}-\lambda _{1}^{*2}) \left( |m_{3}|^{2}|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{\eta _{1}^{+}}= & {} \frac{|m_{1}|^{2}\lambda _{1} \left[ |m_{3}|^{2}|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}|\lambda _{1}^{2} -\lambda _{3}^{2}|^{2} +|l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}|\lambda _{1}^{2}-\lambda _{3}^{*2}|^{2} \right] }{-\sigma \lambda _{1}^{*}|\lambda _{1}^{2}-\lambda _{3}^{*2}|^{2} \left( |m_{3}|^{2}|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{\eta _{2}^{+}}= & {} \frac{-\sigma (|l_{3}|^{2}+|m_{3}|^{2})\lambda _{2}^{*} |\lambda _{1}^{2}-\lambda _{2}^{2}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{*2}|^{2} }{|l_{2}|^{2}\lambda _{2} |\lambda _{1}^{2}-\lambda _{2}^{*2}|^{2} \left( |m_{3}|^{2}|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{\eta _{11}^{+}}= & {} \frac{|m_{1}|^{2}\lambda _{1}\lambda _{2}^{*} |\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2} \left[ |m_{3}|^{2}|\lambda _{1}^{2}-\lambda _{3}^{2}|^{2} +|l_{3}|^{2}|\lambda _{1}^{2}-\lambda _{3}^{*2}|^{2} \right] }{|l_{2}|^{2}\lambda _{1}^{*}\lambda _{2}|\lambda _{1}^{2}-\lambda _{3}^{*2}|^{2} \left( |m_{3}|^{2}|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{\eta _{12}}= & {} \frac{-l_{3}m_{1}m_{3}^{*}\lambda _{1} (\lambda _{1}^{2}-\lambda _{1}^{*2})(\lambda _{2}^{2}-\lambda _{2}^{*2}) (\lambda _{3}^{2}-\lambda _{3}^{*2})(\lambda _{2}^{2}-\lambda _{3}^{*2})}{l_{2}\lambda _{2}(\lambda _{1}^{2}-\lambda _{3}^{*2})(\lambda _{2}^{2}-\lambda _{1}^{*2}) \left( |m_{3}|^{2}|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) },\\ \mathrm{e}^{\eta _{21}}= & {} \frac{-l_{3}^{*}m_{3}m_{1}^{*}\lambda _{2}^{*} (\lambda _{1}^{2}-\lambda _{1}^{*2})(\lambda _{2}^{2}-\lambda _{2}^{*2}) (\lambda _{3}^{2}-\lambda _{3}^{*2})(\lambda _{2}^{*2}-\lambda _{3}^{2})}{l_{2}^{*}\lambda _{1}^{*}(\lambda _{1}^{2}-\lambda _{2}^{*2})(\lambda _{1}^{*2} -\lambda _{3}^{2}) \left( |m_{3}|^{2}|\lambda _{2}^{*2}-\lambda _{3}^{2}|^{2}+ |l_{3}|^{2}|\lambda _{2}^{2}-\lambda _{3}^{2}|^{2}\right) }. \end{aligned}$$