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Specific optical solitons solutions to the coupled Radhakrishnan–Kundu–Lakshmanan model and modulation instability gain spectra in birefringent fibers

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Abstract

In this work, we examine optical solitons to the Radhakrishnan–Kundu–Lakshmanan equation (RKL) equation which describes the optical pulses in birefringent fiber (Raza and Javid in J Appl Anal Comput 10:1375–1395, 2020; Seadawy et al. in Opt Quant Electron 53:324, 2021) by using the New Generalized Auxiliary Equation Method (NGAEM). After some mathematical transformations, owing some constraint relations it is obtained two categories of soliton solutions. The first includes bright and dark optical solitons, while in the second class it is divulged the combined bright-dark and bright-bright optical solitons. Taking some suitable parameters of the model and the NGAEM, it is put up W-shaped optical solitons and diverse other solutions. Thereafter, we use the continuous waves as solutions of the model with small perturbations, to show the effects of the TOD, ellipticity angle and XPM on the Modulation Instability (MI) gain in normal and anomalous dispersion regime. It has been indicated that the third-order dispersion in normal/aanomalous dispersion regime can generate MI growth rate. At the same time, the ellipticity angle and such others parameters of the model play an important role during the MI growth rate (gain). Compared the obtained appropriate results in terms of analytical results and dynamics of the MI to Refs. (Raza and Javid 2020; Seadawy et al. 2021; Yepez-Martinez et al. in Chin J Phys 58:137–150, 2019; Drummond et al. in Opt Commun 78:137–142, 1990; Li et al. in Commun Theor Phys 65:231–236, 2016), they are new in our knowledge.

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Authors and Affiliations

  1. Department of Basic Science, Faculty of Mines and Petroleum Industries, University of Maroua, P.O Box 08, Kaélé, Cameroon

    Souleymanou Abbagari

  2. Laboratory of Mechanics, Materials and Structures, Department of Physics, Faculty of Science, University of Yaounde I, P.O Box 812, Yaoundé, Cameroon

    Souleymanou Abbagari

  3. Department of Marine Engineering, Limbe Nautical Arts and Fisheries Institute, P.O Box 485, Limbe, Cameroon

    Alphonse Houwe

  4. Department of Physics, Faculty of Science, University of Maroua, P.O Box 814, Maroua, Cameroon

    Alphonse Houwe

  5. Department of Physics, Faculty of Science, University of Ngaoundere, P.O. Box 454, Ngaoundere, Cameroon

    Serge Y. Doka

  6. Department of Computer Engineering, Biruni University, Istanbul, Turkey

    Mustafa Inc

  7. Department of Mathematics, Science Faculty, Firat University, Elazig, 23119, Turkey

    Mustafa Inc

  8. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

    Mustafa Inc

  9. National Advanced School of Engineering, University of Yaounde I, Yaoundé, Cameroon

    Thomas B. Bouetou

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  1. Souleymanou Abbagari
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  2. Alphonse Houwe
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  3. Serge Y. Doka
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  4. Mustafa Inc
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  5. Thomas B. Bouetou
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Appendix

Appendix

$$\begin{aligned}&L_{0}=-{\frac{ \left( {f}^{8}+4\,{f}^{6}+6\,{f}^{4}+4\,{f}^{2}+1 \right) {\Omega }^{4}}{ \left( {f}^{2}+1 \right) ^{4}}} &\quad -{\frac{ \left( -3\,{f}^{8}-6\,{f}^{6}-3\,{f}^{4} \right) {\Omega }^{2}{P}^{2}{b_{{1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( -3\,{f }^{4}-6\,{f}^{2}-3 \right) {\Omega }^{2}{P}^{2}{b_{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{9{f}^{4}{b_{{2}}}^{2}{P}^{4}{b_{{1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}+{\frac{4{f}^{4}c_{{2}}c_{{1}}b_{{2}}{P}^{4}b_{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}},\nonumber \\&L_{{1}}=-{\frac{ \left( 4\,{f}^{8}+12\,{f}^{6}+12\,{f}^{4}+4\,{f}^{2}\right) {\Omega }^{3}P\lambda _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( 4\,{f}^{6}+12\,{f}^{4}+12\,{f}^{2}+4 \right) {\Omega }^{3}P\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -12\,{f}^{6}-12\,{f}^{4} \right) \Omega \,{P}^{3}{b_{{1}}}^{2}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 4\,{f}^{6}+4\,{f}^{4} \right) \Omega \,{P}^{3}b_{{1}}c_{{1}}\gamma _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( -12\,{f}^{4}-12\,{f}^{2} \right) \Omega \,{P}^{3}{b_{{2}}}^{2}\lambda _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( 4\,{f}^{4}+4\,{f}^{2} \right) \Omega \,{P}^{3}b_{{2}}c_{{2}}\gamma _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}},\nonumber \\&L_{{2}}=-{\frac{ \left( -4\,{f}^{6}-8\,{f}^{4}-4\,{f}^{2} \right) {\Omega }^{2}{P}^{2}\gamma _{{1}}\gamma _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( 3\,{f}^{8}+6\,{f}^{6}+3\,{f}^{4} \right) {\Omega }^{2}{P}^{2}{\lambda _{{1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 16\,{f}^{6}+32\,{f}^{4}+16\,{f}^{2} \right) {\Omega }^{2}{P}^{2}\lambda _{{1}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 3\,{f}^{4}+6\,{f}^{2}+3 \right) {\Omega }^{2}{P}^{2}{\lambda _{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( 4\,{f}^{8}+12\,{f}^{6}+12\,{f}^{4}+4\,{f}^{2} \right) {\Omega }^{2}Pa_{{1}}b_{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 4\,{f}^{6}+12\,{f}^{4}+12\,{f}^{2}+4 \right) {\Omega }^{2}Pa_{{2}}b_{{2}}}{\left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&+{\frac{9{f}^{4}{\lambda _{{2}}}^{2}{P}^{4}{b_{{1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{4{f}^{4}\lambda _{{2}}\gamma _{{2}}c_{{1}}{P}^{4}b_{{1}}}{ \left( {f}^{2}+1\right) ^{4}}}+{\frac{9{f}^{4}{\lambda _{{1}}}^{2}{P}^{4}{b_{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{4{f}^{4}\lambda _{{1}}\gamma _{{1}}c_{{2}}{P}^{4}b_{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -12\,{f}^{4}-12\,{f}^{2} \right) {P}^{3}a_{{1}}b_{{1}}{b_{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 4\,{f}^{4}+4\,{f}^{2} \right) {P}^{3}a_{{1}}b_{{2}}c_{{1}}c_{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( -12\,{f}^{6}-12\,{f}^{4} \right) {P}^{3}a_{{2}}{b_{{1}}}^{2}b_{{2}}}{ \left( {f}^{2}+1\right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 4\,{f}^{6}+4\,{f}^{4} \right) {P}^{3}a_{{2}}b_{{1}}c_{{1}}c_{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}},\nonumber \\&L_{{3}}=-{\frac{ \left( 2\,{f}^{8}+8\,{f}^{6}+12\,{f}^{4}+8\,{f}^{2}+2 \right) {\Omega }^{3}\beta _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( 2\,{f}^{8}+8\,{f}^{6}+12\,{f}^{4}+8\,{f}^{2}+2 \right) {\Omega }^{3}\beta _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{\left( -4\,{f}^{6}-4\,{f}^{4} \right) \Omega \,{P}^{3}\gamma _{{1}}\gamma _{{2}}\lambda _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{\left( -4\,{f}^{4}-4\,{f}^{2} \right) \Omega \,{P}^{3}\gamma _{{1}}\gamma _{{2}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{\left( 12\,{f}^{6}+12\,{f}^{4} \right) \Omega \,{P}^{3}{\lambda _{{1}}}^{2}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 12\,{f}^{4}+12\,{f}^{2} \right) \Omega \,{P}^{3}\lambda _{{1}}{\lambda _{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 16\,{f}^{6}+32\,{f}^{4}+16\,{f}^{2} \right) \Omega \,{P}^{2}a_{{1}}b_{{1}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( -4\,{f}^{6}-8\,{f}^{4}-4\,{f}^{2} \right) \Omega \,{P}^{2}a_{{1}}c_{{1}}\gamma _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 16\,{f}^{6}+32\,{f}^{4}+16\,{f}^{2} \right) \Omega \,{P}^{2}a_{{2}}b_{{2}}\lambda _{{1}}}{\left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -4\,{f}^{6}-8\,{f}^{4}-4\,{f}^{2} \right) \Omega \,{P}^{2}a_{{2}}c_{{2}}\gamma _{{1}}}{\left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( -6\,{f}^{8}-12\,{f}^{6}-6\,{f}^{4} \right) \Omega \,{P}^{2}{b_{{1}}}^{2}\beta _{{2}}}{\left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -6\,{f}^{4}-12\,{f}^{ 2}-6 \right) \Omega \,{P}^{2}{b_{{2}}}^{2}\beta _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}},\nonumber \\&L_{{4}}=-{\frac{ \left( 4\,{f}^{8}+12\,{f}^{6}+12\,{f}^{4}+4\,{f}^{2} \right) {\Omega }^{2}P\beta _{{1}}\lambda _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 8\,{f}^{6}+24\,{f}^{4}+24\,{f}^{2}+8 \right) {\Omega }^{2}P\beta _{{1}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 8\,{f}^{8}+24\,{f}^{6}+24\,{f}^{4}+8\, {f}^{2} \right) {\Omega }^{2}P\beta _{{2}}\lambda _{{1}}}{ \left( {f}^{2} +1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 4\,{f}^{6}+12\,{f}^{4}+12\,{f}^{2}+4 \right) {\Omega }^{2}P\beta _{{2}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\nonumber -{\frac{ \left( -{f}^{8}-4\,{f}^{6}-6\,{f}^{4}-4\,{f}^ {2}-1 \right) {\Omega }^{2}{a_{{1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4 }}}\nonumber \\&\quad -{\frac{ \left( -{f}^{8}-4\,{f}^{6}-6\,{f}^{4}-4\,{f}^{2}-1 \right) {\Omega }^{2}{a_{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad +{\frac{4 {f}^{4}\lambda _{{2}}\lambda _{{1}}\gamma _{{2}}{P}^{4}\gamma _{ {1}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{9{f}^{4}{\lambda _{{1} }}^{2}{\lambda _{{2}}}^{2}{P}^{4}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 12\,{f}^{4}+12\,{f}^{2} \right) {P}^{3}a_{{1}}b_{{1}}{ \lambda _{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( - 4\,{f}^{4}-4\,{f}^{2} \right) {P}^{3}a_{{1}}c_{{1}}\gamma _{{2}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 12\,{f}^{6}+ 12\,{f}^{4} \right) {P}^{3}a_{{2}}b_{{2}}{\lambda _{{1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\nonumber -{\frac{ \left( -4\,{f}^{6}-4\,{f}^{4} \right) {P}^{3}a_{{2}}c_{{2}}\gamma _{{1}}\lambda _{{1}}}{ \left( {f}^{ 2}+1 \right) ^{4}}}-{\frac{ \left( -12\,{f}^{6}-12\,{f}^{4} \right) { P}^{3}{b_{{1}}}^{2}\beta _{{2}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 4\,{f}^{6}+4\,{f}^{4} \right) {P}^{3}b _{{1}}\beta _{{2}}c_{{1}}\gamma _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -12\,{f}^{4}-12\,{f}^{2} \right) {P}^{3}{b_{{2}}}^{2} \beta _{{1}}\lambda _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( 4\,{f}^{4}+4\,{f}^{2} \right) {P}^{3}b_{{2}}\beta _{{1}}c_{{2}} \gamma _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 3\,{f}^ {4}+6\,{f}^{2}+3 \right) {P}^{2}{a_{{1}}}^{2}{b_{{2}}}^{2}}{ \left( {f }^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 16\,{f}^{6}+32\,{f}^{4}+16\,{f}^{2} \right) {P}^{2}a_{{1}}a_{{2}}b_{{1}}b_{{2}}}{ \left( {f}^{2}+1\right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -4\,{f}^{6}-8\,{f}^{4}-4\,{f}^{2}\right) {P}^{2}a_{{1}}a_{{2}}c_{{1}}c_{{2}}}{ \left( {f}^{2}+1\right) ^{4}}}-{\frac{ \left( 3\,{f}^{8}+6\,{f}^{6}+3\,{f}^{4}\right) {P}^{2}{a_{{2}}}^{2}{b_{{1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}},\end{aligned}$$
(37)
$$\begin{aligned}&L_{{5}}=-{\frac{ \left( -4\,{f}^{6}-8\,{f}^{4}-4\,{f}^{2} \right) \Omega \,{P}^{2}\beta _{{1}}\gamma _{{1}}\gamma _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 16\,{f}^{6}+32\,{f}^{4}+16\,{f}^{2} \right) \Omega \,{P}^{2}\beta _{{1}}\lambda _{{1}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 6\,{f}^{4}+12\,{f}^{2 }+6 \right) \Omega \,{P}^{2}\beta _{{1}}{\lambda _{{2}}}^{2}}{ \left( {f} ^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -4\,{f}^{6}-8\,{f}^{4}-4\,{f}^{2 } \right) \Omega \,{P}^{2}\beta _{{2}}\gamma _{{1}}\gamma _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( 6\,{f}^{8}+12\,{f}^{6}+6\,{f} ^{4} \right) \Omega \,{P}^{2}\beta _{{2}}{\lambda _{{1}}}^{2}}{ \left( {f }^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 16\,{f}^{6}+32\,{f}^{4}+16\,{f} ^{2} \right) \Omega \,{P}^{2}\beta _{{2}}\lambda _{{1}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -4\,{f}^{6}-12\,{f}^{ 4}-12\,{f}^{2}-4 \right) \Omega \,P{a_{{1}}}^{2}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( 8\,{f}^{8}+24\,{f}^{6}+24\,{f }^{4}+8\,{f}^{2} \right) \Omega \,Pa_{{1}}b_{{1}}\beta _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -4\,{f}^{8}-12\,{f}^{6}-12\,{f }^{4}-4\,{f}^{2} \right) \Omega \,P{a_{{2}}}^{2}\lambda _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( 8\,{f}^{6}+24\,{f}^{4}+24\,{f }^{2}+8 \right) \Omega \,Pa_{{2}}b_{{2}}\beta _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}},\nonumber \\&L_{{6}}=-{\frac{ \left( {f}^{8}+4\,{f}^{6}+6\,{f}^{4}+4\,{f}^{2}+1 \right) {\Omega }^{2}{\beta _{{1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4} }}\nonumber \\&\quad -{\frac{ \left( 4\,{f}^{8}+16\,{f}^{6}+24\,{f}^{4}+16\,{f}^{2}+4 \right) {\Omega }^{2}\beta _{{1}}\beta _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -4\,{f}^{4}-4\,{f}^{2} \right) {P}^{3}\beta _{{1 }}\gamma _{{1}}\gamma _{{2}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4 }}}-{\frac{ \left( 12\,{f}^{4}+12\,{f}^{2} \right) {P}^{3}\beta _{{1}} \lambda _{{1}}{\lambda _{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{ \frac{ \left( -4\,{f}^{6}-4\,{f}^{4} \right) {P}^{3}\beta _{{2}}\gamma _{{1}}\gamma _{{2}}\lambda _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{ \frac{ \left( 12\,{f}^{6}+12\,{f}^{4} \right) {P}^{3}\beta _{{2}}{ \lambda _{{1}}}^{2}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{ \frac{ \left( -3\,{f}^{4}-6\,{f}^{2}-3 \right) {P}^{2}{a_{{1}}}^{2}{ \lambda _{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 16\,{f}^{6}+32\,{f}^{4}+16\,{f}^{2} \right) {P}^{2}a_{{1}}b_{{1}}\beta _{{2}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -4\,{f}^{6}-8\,{f}^{4}-4\,{f}^{2} \right) {P}^{2}a_{{1}}\beta _{{2}}c_{ {1}}\gamma _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -3 \,{f}^{8}-6\,{f}^{6}-3\,{f}^{4} \right) {P}^{2}{a_{{2}}}^{2}{\lambda _{ {1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 16\,{f}^{6 }+32\,{f}^{4}+16\,{f}^{2} \right) {P}^{2}a_{{2}}b_{{2}}\beta _{{1}} \lambda _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -4\,{f }^{6}-8\,{f}^{4}-4\,{f}^{2} \right) {P}^{2}a_{{2}}\beta _{{1}}c_{{2}} \gamma _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}-{\frac{ \left( -3\,{f} ^{8}-6\,{f}^{6}-3\,{f}^{4} \right) {P}^{2}{b_{{1}}}^{2}{\beta _{{2}}}^{ 2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -3\,{f}^{4}-6\,{f }^{2}-3 \right) {P}^{2}{b_{{2}}}^{2}{\beta _{{1}}}^{2}}{ \left( {f}^{2} +1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -4\,{f}^{6}-12\,{f}^{4}-12\,{f}^{2}-4 \right) P{a_{{1}}}^{2}a_{{2}}b_{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -4\,{f}^{8}-12\,{f}^{6}-12\,{f}^{4}-4\,{f}^{2}\right) Pa_{{1}}{a_{{2}}}^{2}b_{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( {f}^{8}+4\,{f}^{6}+6\,{f}^{4}+4\,{f}^{2}+1 \right) {\Omega }^{2}{\beta _{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}},\nonumber \\&L_{{7}}=-{\frac{ \left( 4\,{f}^{6}+12\,{f}^{4}+12\,{f}^{2}+4 \right) \Omega \,P{\beta _{{1}}}^{2}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4 }}}\nonumber \\&\quad -{\frac{ \left( 8\,{f}^{8}+24\,{f}^{6}+24\,{f}^{4}+8\,{f}^{2} \right) \Omega \,P\beta _{{1}}\beta _{{2}}\lambda _{{1}}}{ \left( {f}^{2} +1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 8\,{f}^{6}+24\,{f}^{4}+24\,{f}^{2}+8 \right) \Omega \,P\beta _{{1}}\beta _{{2}}\lambda _{{2}}}{ \left( {f}^{2} +1 \right) ^{4}}}-{\frac{ \left( 4\,{f}^{8}+12\,{f}^{6}+12\,{f}^{4}+4 \,{f}^{2} \right) \Omega \,P{\beta _{{2}}}^{2}\lambda _{{1}}}{ \left( {f} ^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -2\,{f}^{8}-8\,{f}^{6}-12\,{f}^{ 4}-8\,{f}^{2}-2 \right) \Omega \,{a_{{1}}}^{2}\beta _{{2}}}{ \left( {f}^ {2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -2\,{f}^{8}-8\,{f}^{6}-12\,{f}^{4 }-8\,{f}^{2}-2 \right) \Omega \,{a_{{2}}}^{2}\beta _{{1}}}{ \left( {f}^{ 2}+1 \right) ^{4}}},\nonumber \\&L_{{8}}=-{\frac{ \left( 3\,{f}^{4}+6\,{f}^{2}+3 \right) {P}^{2}{\beta _{{1}}}^{2}{\lambda _{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{ \frac{ \left( -4\,{f}^{6}-8\,{f}^{4}-4\,{f}^{2} \right) {P}^{2}\beta _ {{1}}\beta _{{2}}\gamma _{{1}}\gamma _{{2}}}{ \left( {f}^{2}+1 \right) ^{ 4}}}\nonumber \\&\quad -{\frac{ \left( 16\,{f}^{6}+32\,{f}^{4}+16\,{f}^{2} \right) {P}^{ 2}\beta _{{1}}\beta _{{2}}\lambda _{{1}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 3\,{f}^{8}+6\,{f}^{6}+3\,{f}^{4} \right) {P}^{2}{\beta _{{2}}}^{2}{\lambda _{{1}}}^{2}}{ \left( {f}^{2}+ 1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -4\,{f}^{6}-12\,{f}^{4}-12\,{f}^{2}-4 \right) P{a_{{1}}}^{2}\beta _{{2}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 4\,{f}^{8}+12\,{f}^{6}+12\,{f}^{4}+4\, {f}^{2} \right) Pa_{{1}}b_{{1}}{\beta _{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -4\,{f}^{8}-12\,{f}^{6}-12\,{f}^{4}-4 \,{f}^{2} \right) P{a_{{2}}}^{2}\beta _{{1}}\lambda _{{1}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( 4\,{f}^{6}+12\,{f}^{4}+12\,{f}^{2 }+4 \right) Pa_{{2}}b_{{2}}{\beta _{{1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( {f}^{8}+4\,{f}^{6}+6\,{f}^{4}+4\,{f}^{ 2}+1 \right) {a_{{1}}}^{2}{a_{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}},\nonumber \\&L_{{9}}=-{\frac{ \left( 2\,{f}^{8}+8\,{f}^{6}+12\,{f}^{4}+8\,{f}^{2}+ 2 \right) \Omega \,{\beta _{{1}}}^{2}\beta _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 2\,{f}^{8}+8\,{f}^{6}+12\,{f}^{4}+8\,{ f}^{2}+2 \right) \Omega \,\beta _{{1}}{\beta _{{2}}}^{2}}{ \left( {f}^{2} +1 \right) ^{4}}},\nonumber \\&L_{{10}}=-{\frac{ \left( 4\,{f}^{6}+12\,{f}^{4}+12\,{f}^{2}+4 \right) P{\beta _{{1}}}^{2}\beta _{{2}}\lambda _{{2}}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( 4\,{f}^{8}+12\,{f}^{6}+12\,{f}^{4}+4\, {f}^{2} \right) P\beta _{{1}}{\beta _{{2}}}^{2}\lambda _{{1}}}{ \left( {f }^{2}+1 \right) ^{4}}}\nonumber \\&-{\frac{ \left( -{f}^{8}-4\,{f}^{6}-6\,{f}^{4}- 4\,{f}^{2}-1 \right) {a_{{1}}}^{2}{\beta _{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}\nonumber \\&\quad -{\frac{ \left( -{f}^{8}-4\,{f}^{6}-6\,{f}^{4}-4\,{f}^ {2}-1 \right) {a_{{2}}}^{2}{\beta _{{1}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}},\nonumber \\&L_{12}=-{\frac{ \left( {f}^{8}+4\,{f}^{6}+6\,{f}^{4}+4\,{ f}^{2}+1 \right) {\beta _{{1}}}^{2}{\beta _{{2}}}^{2}}{ \left( {f}^{2}+1 \right) ^{4}}}. \end{aligned}$$
(38)

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Abbagari, S., Houwe, A., Doka, S.Y. et al. Specific optical solitons solutions to the coupled Radhakrishnan–Kundu–Lakshmanan model and modulation instability gain spectra in birefringent fibers. Opt Quant Electron 54, 35 (2022). https://doi.org/10.1007/s11082-021-03359-z

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