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Role of nonlinear saturation on modulational instability in Kundu–Eckhaus equation with the presence of inter modal, XPM and SPM

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Abstract

With cross phase modulation (XPM), conventional saturable nonlinearity (CSNL), and self phase modulation (SPM) present, we analyze the effect of inter modal dispersion on the modulation instability (MI) of an optical rogue waves in the Kundu-Eckhaus (KE) equation. We take into account the two component KE model without the four wave mixing term and with intermodal dispersion along with CSNL. On the basis of the conventional linear stability analysis, the MI criterion for the KE equation is determined. Using the MI criterion, it is possible to determine how inter modal dispersion and CSNL affect the MI of an optical rogue waves in a birefringent fiber when XPM and SPM are present. Through a graphical illustration of MI gain, the effects of intermodal dispersion and CSNL on MI in optical fiber for the presence of SPM and XPM are explored.

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Acknowledgements

P. Mohanraj thanks the UGC for providing financial assistance under the RGNF programmed F1-17.1/2015-16/NFST-2015-17-STPON-2332. R. Sivakumar, one of the authors, thanks UGC for supporting this work in part through the major research project grant letter F. No. 37-312/2009 (SR) dated January 12, 2010, as well as DST for funding through the FIST programmed through the order SR/FST / PSII-021/2009 dated August 13, 2010, which will be used to build a basic computing cluster facility. This paper is a dedication in honor of Prof. K. Porsezian, one of the work's research guides, in honor of his love and support while he was alive.

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This research received no specific grant from any funding agency in the public, commercial or not for profit sectors.

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Authors and Affiliations

  1. Department of Physics, Vel Tech Rangarajan Dr, Sagunthala R&D Institute of Science and Technology, Chennai, 600062, India

    P. Mohanraj, Jayaprakash Kaliyamurthy & Rajesh Kumar

  2. Department of Physics, Pondicherry University, Puducherry, 605014, India

    R. Sivakumar

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  1. P. Mohanraj
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PM carried out the concept and design of the article. RS carried out data interpretation for the article. JK and RK carried out the draught manuscript. All authors read and approved the final manuscript.

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Appendix

Appendix

$$ \begin{aligned} A_{1} = & \;k^{2} a_{1} - 2kQa_{1} - Q^{2} a_{1} + iQr_{1} u_{0}^{2} - 2iQ\Gamma r_{1} u_{0}^{4} - k\alpha_{1} - Q\alpha_{1} + 3u_{0}^{4} v_{0}^{4} \eta_{1} - 5\Gamma u_{0}^{8} v_{0}^{4} \eta_{1} \\ & \; - 3\Gamma u_{0}^{4} v_{0}^{8} \eta_{1} + 5\Gamma^{2} u_{0}^{8} v_{0}^{8} \eta_{1} + 5u_{0}^{8} \xi_{1} - 9\Gamma u_{0}^{16} \xi_{1} + \omega_{1} + v_{0}^{8} \varsigma_{1} - \Gamma v_{0}^{16} \varsigma_{1} ; \\ \end{aligned} $$
$$ \begin{aligned} A_{2} = & \;k^{2} a_{2} - 2kQa_{2} - Q^{2} a_{2} + iQr_{2} v_{0}^{2} - 2iQ\Gamma r_{2} v_{0}^{4} - k\alpha_{2} - Q\alpha_{2} + 3u_{0}^{4} v_{0}^{4} \eta_{2} - 3\Gamma u_{0}^{8} v_{0}^{4} \eta_{2} \\ & \; - 5\Gamma u_{0}^{4} v_{0}^{8} \eta_{2} + 5\Gamma^{2} u_{0}^{8} v_{0}^{8} \eta_{2} + 5v_{0}^{8} \xi_{2} - 9\Gamma v_{0}^{16} \xi_{2} + \omega_{2} + u_{0}^{8} \varsigma_{2} - \Gamma u_{0}^{16} \varsigma_{2} ; \\ \end{aligned} $$
$$ \begin{aligned} A_{3} = & \;k^{2} a_{1} + 2kQa_{1} - Q^{2} a_{1} - iQr_{1} u_{0}^{2} + 2iQ\Gamma r_{1} u_{0}^{4} - k\alpha_{1} + Q\alpha_{1} + 3u_{0}^{4} v_{0}^{4} \eta_{1} - 5\Gamma u_{0}^{8} v_{0}^{4} \eta_{1} \\ & \; - 3\Gamma u_{0}^{4} v_{0}^{8} \eta_{1} + 5\Gamma^{2} u_{0}^{8} v_{0}^{8} \eta_{1} + 5u_{0}^{8} \xi_{1} - 9\Gamma u_{0}^{16} \xi_{1} + \omega_{1} + v_{0}^{8} \varsigma_{1} - \Gamma v_{0}^{16} \varsigma_{1} ; \\ \end{aligned} $$
$$ \begin{aligned} A_{4} = & \;k^{2} a_{2} + 2kQa_{2} - Q^{2} a_{2} - iQr_{2} v_{0}^{2} + 2iQ\Gamma r_{2} v_{0}^{4} - k\alpha_{2} + Q\alpha_{2} + 3u_{0}^{4} v_{0}^{4} \eta_{2} - 3\Gamma u_{0}^{8} v_{0}^{4} \eta_{2} \\ & \; - 5\Gamma u_{0}^{4} v_{0}^{8} \eta_{2} + 5\Gamma^{2} u_{0}^{8} v_{0}^{8} \eta_{2} + 5v_{0}^{8} \xi_{2} - 9\Gamma v_{0}^{16} \xi_{2} + \omega_{2} + u_{0}^{8} \varsigma_{2} - \Gamma u_{0}^{16} \varsigma_{2} ; \\ \end{aligned} $$
$$ B_{1} = - iQr_{1} u_{0}^{2} + 2iQ\Gamma r_{1} u_{0}^{4} + 2u_{0}^{4} v_{0}^{4} \eta_{1} - 4\Gamma u_{0}^{8} v_{0}^{4} \eta_{1} - 2\Gamma u_{0}^{4} v_{0}^{8} \eta_{1} + 4\Gamma^{2} u_{0}^{8} v_{0}^{8} \eta_{1} + 4u_{0}^{8} \xi_{1} - 8\Gamma u_{0}^{16} \xi_{1} ; $$
$$ B_{2} = iQs_{2} u_{0} v_{0} - 2iQ\Gamma s_{2} u_{0}^{3} v_{0} + 2u_{0}^{3} v_{0}^{5} \eta_{2} - 4\Gamma u_{0}^{7} v_{0}^{5} \eta_{2} - 2\Gamma u_{0}^{3} v_{0}^{9} \eta_{2} + 4\Gamma^{2} u_{0}^{7} v_{0}^{9} \eta_{2} + 4u_{0}^{7} v_{0} \varsigma_{2} - 8\Gamma u_{0}^{15} v_{0} \varsigma_{2} ; $$
$$ B_{3} = - iQs_{2} u_{0} v_{0} + 2iQ\Gamma s_{2} u_{0}^{3} v_{0} + 2u_{0}^{3} v_{0}^{5} \eta_{2} - 4\Gamma u_{0}^{7} v_{0}^{5} \eta_{2} - 2\Gamma u_{0}^{3} v_{0}^{9} \eta_{2} + 4\Gamma^{2} u_{0}^{7} v_{0}^{9} \eta_{2} + 4u_{0}^{7} v_{0} \varsigma_{2} - 8\Gamma u_{0}^{15} v_{0} \varsigma_{2} ; $$
$$ B_{4} = iQs_{1} u_{0} v_{0} - 2iQ\Gamma s_{1} u_{0} v_{0}^{3} + 2u_{0}^{5} v_{0}^{3} \eta_{1} - 2\Gamma u_{0}^{9} v_{0}^{3} \eta_{1} - 4\Gamma u_{0}^{5} v_{0}^{7} \eta_{1} + 4\Gamma^{2} u_{0}^{9} v_{0}^{7} \eta_{1} + 4u_{0} v_{0}^{7} \varsigma_{1} - 8\Gamma u_{0} v_{0}^{15} \varsigma_{1} ; $$
$$ C_{1} = - iQr_{2} v_{0}^{2} + 2iQ\Gamma r_{2} v_{0}^{4} + 2u_{0}^{4} v_{0}^{4} \eta_{2} - 2\Gamma u_{0}^{8} v_{0}^{4} \eta_{2} - 4\Gamma u_{0}^{4} v_{0}^{8} \eta_{2} + 4\Gamma^{2} u_{0}^{8} v_{0}^{8} \eta_{2} + 4v_{0}^{8} \xi_{2} - 8\Gamma v_{0}^{16} \xi_{2} ; $$
$$ C_{2} = - iQs_{1} u_{0} v_{0} + 2iQ\Gamma s_{1} u_{0} v_{0}^{3} + 2u_{0}^{5} v_{0}^{3} \eta_{1} - 2\Gamma u_{0}^{9} v_{0}^{3} \eta_{1} - 4\Gamma u_{0}^{5} v_{0}^{7} \eta_{1} + 4\Gamma^{2} u_{0}^{9} v_{0}^{7} \eta_{1} + 4u_{0} v_{0}^{7} \varsigma_{1} - 8\Gamma u_{0} v_{0}^{15} \varsigma_{1} ; $$
$$ C_{3} = iQs_{2} u_{0} v_{0} - 2iQ\Gamma s_{2} u_{0}^{3} v_{0} + 2u_{0}^{3} v_{0}^{5} \eta_{2} - 4\Gamma u_{0}^{7} v_{0}^{5} \eta_{2} - 2\Gamma u_{0}^{3} v_{0}^{9} \eta_{2} + 4\Gamma^{2} u_{0}^{7} v_{0}^{9} \eta_{2} + 4u_{0}^{7} v_{0} \varsigma_{2} - 8\Gamma u_{0}^{15} v_{0} \varsigma_{2} ; $$
$$ C_{4} = iQr_{1} u_{0}^{2} - 2iQ\Gamma r_{1} u_{0}^{4} + 2u_{0}^{4} v_{0}^{4} \eta_{1} - 4\Gamma u_{0}^{8} v_{0}^{4} \eta_{1} - 2\Gamma u_{0}^{4} v_{0}^{8} \eta_{1} + 4\Gamma^{2} u_{0}^{8} v_{0}^{8} \eta_{1} + 4u_{0}^{8} \xi_{1} - 8\Gamma u_{0}^{16} \xi_{1} ; $$
$$ D_{1} = - iQs_{2} u_{0} v_{0} + 2iQ\Gamma s_{2} u_{0}^{3} v_{0} + 2u_{0}^{3} v_{0}^{5} \eta_{2} - 4\Gamma u_{0}^{7} v_{0}^{5} \eta_{2} - 2\Gamma u_{0}^{3} v_{0}^{9} \eta_{2} + 4\Gamma^{2} u_{0}^{7} v_{0}^{9} \eta_{2} + 4u_{0}^{7} v_{0} \varsigma_{2} - 8\Gamma u_{0}^{15} v_{0} \varsigma_{2} ; $$
$$ D_{2} = iQs_{1} u_{0} v_{0} - 2iQ\Gamma s_{1} u_{0} v_{0}^{3} + 2u_{0}^{5} v_{0}^{3} \eta_{1} - 2\Gamma u_{0}^{9} v_{0}^{3} \eta_{1} - 4\Gamma u_{0}^{5} v_{0}^{7} \eta_{1} + 4\Gamma^{2} u_{0}^{9} v_{0}^{7} \eta_{1} + 4u_{0} v_{0}^{7} \varsigma_{1} - 8\Gamma u_{0} v_{0}^{15} \varsigma_{1} ; $$
$$ D_{3} = - iQs_{1} u_{0} v_{0} + 2iQ\Gamma s_{1} u_{0} v_{0}^{3} + 2u_{0}^{5} v_{0}^{3} \eta_{1} - 2\Gamma u_{0}^{9} v_{0}^{3} \eta_{1} - 4\Gamma u_{0}^{5} v_{0}^{7} \eta_{1} + 4\Gamma^{2} u_{0}^{9} v_{0}^{7} \eta_{1} + 4u_{0} v_{0}^{7} \varsigma_{1} - 8\Gamma u_{0} v_{0}^{15} \varsigma_{1} ; $$
$$ D_{4} = iQr_{2} v_{0}^{2} - 2iQ\Gamma r_{2} v_{0}^{4} + 2u_{0}^{4} v_{0}^{4} \eta_{2} - 2\Gamma u_{0}^{8} v_{0}^{4} \eta_{2} - 4\Gamma u_{0}^{4} v_{0}^{8} \eta_{2} + 4\Gamma^{2} u_{0}^{8} v_{0}^{8} \eta_{2} + 4v_{0}^{8} \xi_{2} - 8\Gamma v_{0}^{16} \xi_{2} ; $$

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Mohanraj, P., Sivakumar, R., Kaliyamurthy, J. et al. Role of nonlinear saturation on modulational instability in Kundu–Eckhaus equation with the presence of inter modal, XPM and SPM. Opt Quant Electron 55, 327 (2023). https://doi.org/10.1007/s11082-023-04588-0

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