Optical Solution Parameter Dynamics By  Variational Principle: Parabolic And Dual–Power Laws (Super–Gaussian And Super–Sech Pulses)

MohamedE.M. Alngar; Reham. M. A.  Shohib; Ahmed H.  Arnous; Anjan  Biswas; Yakup  Yıldırım; Anwar Jaafar Mohamad  Jawad; Ali Saleh  Alshomrani

doi:10.6180/jase.202504_28(4).0014

Optical Solution Parameter Dynamics By Variational Principle: Parabolic And Dual–Power Laws (Super–Gaussian And Super–Sech Pulses)

MohamedE.M.Alngar¹, Reham. M. A. Shohib², Ahmed H. Arnous³, Anjan Biswas^4,5,6,7This email address is being protected from spambots. You need JavaScript enabled to view it., Yakup Yıldırım^8,9,10, Anwar Jaafar Mohamad Jawad¹¹, and Ali Saleh Alshomrani⁵

¹Basic Science Department, Faculty of Computers and Artificial Intelligence, Modern University for Technology & Information, Cairo–11585, Egypt.

²Basic Science Department, Higher Institute of Foreign Trade & Management Sciences, New Cairo Academy, Cairo–379, Egypt.

³Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El Shoruk Academy, Cairo, Egypt.

⁴Department of Mathematics and Physics Grambling State University, Grambling, LA 71245–2715, USA.

⁵Mathematical Modeling and Applied Computation (MMAC) Research Group, Center of Modern Mathematical Sciences and their Applications (CMMSA), Department of Mathematics, King Abdulaziz University, Jeddah–21589, Saudi Arabia.

⁶Department of Applied Sciences, Cross–Border Faculty of Humanities, Economics and Engineering, Dunarea de Jos University of Galati, 111 Domneasca Street, Galati–800201, Romania.

⁷Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa–0204, South Africa.

⁸Department of Computer Engineering, Biruni University, Istanbul–34010, Turkey.

⁹Mathematics Research Center, Near East University, 99138 Nicosia, Cyprus.

¹⁰Faculty of Arts and Sciences, University of Kyrenia, 99320 Kyrenia, Cyprus.

¹¹Department of Computer Technical Engineering, Al Rafidain University College, 10064 Baghdad, Iraq.

Received: January 20, 2024
Accepted: March 17, 2024
Publication Date: June 20, 2024

Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Download Citation: ||https://doi.org/10.6180/jase.202504_28(4).0014

The current paper retrieves the optical soliton parameter dynamics that is considered with parabolic and dual—power laws of self—phase modulation structures. With linear chromatic dispersion and linear temporal evolution, the variational principle recovered the dynamical system of soliton parameters. Two specific forms of optical solitons are addressed in the paper which are super—Gaussian and super—sech pulses. These typically model RZ and NRZ types of pulses considered in telecommunications engineering. The special cases are naturally revealed when the parameter dictating the generalized nonlinearity is set to unity. The issue of soliton radiation has been tacitly disregarded to keep mathematics simple. The perturbation terms are also taken into account and the extended version of the Euler—Lagrange’s equation displays the extended dynamical system of these soliton parameters. The results naturally involved a range of special functions.

Keywords: Euler–Lagrange; perturbation; variational principle; solitons

[1] S. G. Ali, B. Talukdar, and S. K. Roy, (2007) “Bright solitons in asymmetrically trapped Bose-Einstein conden sates" ACTA PHYSICA POLONICAA111(3): 289–297.
[2] A.M.Ayela, G. Edah, C. Elloh, A. Biswas, M. Ekici, A. K. Alzahrani, and M. R. Belic, (2021) “Chirped super-Gaussian and super-sech pulse perturbation of non linear Schrodinger’s equation with quadratic-cubic non linearity by variational principle" PHYSICS LETTERS A396: DOI: 10.1016/j.physleta.2021.127231.
[3] A.M.Ayela,G.Edah,A.Biswas,Q.Zhou,Y.Yildirim, S. Khan, A. K. Alzahrani, and M. R. Belic, (2022) “Dynamical system of optical soliton parameters for anti cubic and generalized anti-cubic nonlinearities with super Gaussian and super-sech pulses" OPTICA APPLICATA 52(1): 117–128. DOI: 10.37190/oa220109.
[4] A.Biswas, (2002) “Dispersion-managed solitons in op tical fibres" JOURNAL OF OPTICS A-PURE AND APPLIED OPTICS 4(1): 84–97. DOI: 10.1088/1464 4258/4/1/315.
[5] Y. CHEN, (1991) “VARIATIONAL PRINCIPLE FOR VECTOR SPATIAL SOLITONS AND NONLINEAR MODES" OPTICS COMMUNICATIONS 84(5-6): 355–358. DOI: 10.1016/0030-4018(91)90102-J.
[6] F. K. Diakonos and P. Schmelcher, (2019) “Super Lagrangian and variational principle for generalized continuity equations" JOURNAL OF PHYSICS A MATHEMATICAL AND THEORETICAL 52(15): DOI: 10.1088/1751-8121/ab082f.
[7] M.F.S.Ferreira, (2018) “Variational approach to station ary and pulsating dissipative optical solitons" IET OP TOELECTRONICS12(3): 122–125. DOI: 10.1049/iet opt.2017.0121.
[8] P. Green, D. Milovic, A. K. Sarma, D. A. Lott, and A. Biswas, (2010) “DYNAMICS OF SUPER-SECH SOLITONS IN OPTICAL FIBERS" JOURNAL OF NONLINEAROPTICALPHYSICS&MATERIALS 19(2): 339–370. DOI: 10.1142/S0218863510005145.
[9] T. Hirooka and S. Wabnitz, (2000) “Nonlinear gain control of dispersion-managed soliton amplitude and colli sions" OPTICAL FIBER TECHNOLOGY6(2):109 121. DOI: 10.1006/ofte.1999.0325.
[10] S. C. V. Latas and M. F. S. Ferreira, (2010) “Soliton explosion control by higher-order effects" OPTICS LET TERS35(11): 1771–1773. DOI: 10.1364/OL.35.001771.
[11] S. C. Mancas and S. R. Choudhury, (2007) “A novel variational approach to pulsating solitons in the cubic quintic Ginzburg-Landau equation" THEORETICAL AND MATHEMATICAL PHYSICS 152(2): 1160 1172. DOI: 10.1007/s11232-007-0099-8.
[12] D. Pal, S. G. Ali, and B. Talukdar, (2008) “Embedded soliton solutions: A variational study" ACTA PHYS ICAPOLONICAA113(2):707–712. DOI: 10.12693/APhysPolA.113.707.
[13] V. Skarka and N. B. Aleksic, (2007) “Dissipative opti cal solitons" ACTA PHYSICA POLONICA A 112(5): 791–798. DOI: 10.12693/APhysPolA.112.791.
[14] E. M. E. Zayed, M. El-Horbaty, M. E. M. Alngar, R. M. A. Shohib, A. Biswas, Y. Yildirim, L. Moraru, C. Iticescu, D. Bibicu, P. Lucian Georgescu, and A. Asiri, (2023) “Dynamical system of optical soliton param eters by variational principle (super-Gaussian and super sech pulses)" JOURNAL OF THEEUROPEANOPTI CALSOCIETY-RAPIDPUBLICATIONS19(2): DOI: 10.1051/jeos/2023035.
[15] J. Zhang, J. Yu, and N. Pan, (2005) “Variational prin ciples for nonlinear fiber optics" CHAOS SOLITONS &FRACTALS24(1): 309–311. DOI: 10.1016/j.chaos.2004.09.032.
[16] A. J. M. Jawad and M. J. Abu-AlShaeer, (2023) “Highly dispersive optical solitons with cubic law and cubic-quinticseptic law nonlinearities by two methods" Al-Rafidain J. Eng. Sci. 1(1): 1–8.
[17] S. Wang, (2023) “Novel soliton solutions of CNLSEs with Hirota bilinear method" JOURNAL OF OPTICS INDIA: DOI: 10.1007/s12596-022-01065-x.
[18] B. Kopcasiz and E. Yasar, (2023) “The investigation of unique optical soliton solutions for dual-mode nonlinear Schrodinger’s equation with new mechanisms" JOUR NALOFOPTICS-INDIA52(3): 1513–1527. DOI: 10.1007/s12596-022-00998-7.
[19] L. Tang, (2023) “Bifurcations and optical solitons for the coupled nonlinear Schrodinger equation in optical fiber Bragg gratings" JOURNALOFOPTICS-INDIA52(3): 1388–1398. DOI: 10.1007/s12596-022-00963-4.
[20] T. N. Thi and L. C. Van, (2023) “Supercontinuum gen eration based on suspended core fiber infiltrated with bu tanol" JOURNAL OF OPTICS-INDIA 52(4): 2296 2305. DOI: 10.1007/s12596-023-01323-6.
[21] Z. Li and E. Zhu, (2023) “Optical soliton solutions of stochastic Schrodinger-Hirota equation in birefringent f ibers with spatiotemporal dispersion and parabolic law nonlinearity" JOURNAL OF OPTICS-INDIA: DOI: 10.1007/s12596-023-01287-7.
[22] T. Han, Z. Li, C. Li, and L. Zhao, (2023) “Bifurca tions, stationary optical solitons and exact solutions for complex Ginzburg-Landau equation with nonlinear chro matic dispersion in non-Kerr law media" JOURNAL OF OPTICS-INDIA52(2):831–844. DOI:10.1007/s12596022-01041-5.
[23] S. Nandy and V. Lakshminarayanan, (2015) “Ado mian decomposition of scalar and coupled nonlinear Schrodinger equations and dark and bright solitary wave solutions" JOURNAL OF OPTICS-INDIA44(4): 397–404. DOI: 10.1007/s12596-015-0270-9.
[24] W. Chen, M. Shen, Q. Kong, and Q. Wang, (2015) “The interaction of dark solitons with competing nonlocal cubic nonlinearities" JOURNAL OF OPTICS-INDIA 44(3): 271–280. DOI: 10.1007/s12596-015-0255-8.
[25] S.-L. Xu, N. Petrovic, and M. R. Belic, (2015) “Two dimensional dark solitons in diffusive nonlocal nonlinear media" JOURNAL OF OPTICS-INDIA 44(2): 172 177. DOI: 10.1007/s12596-015-0243-z.
[26] A.-M. Wazwaz, (2023) “Painleve integrability and lump solutions for two extended (3+1)- and (2+1)-dimensional Kadomtsev-Petviashvili equations" NONLINEAR DY NAMICS 111(4): 3623–3632. DOI: 10.1007/s11071022-08074-2.
[27] A.-M. Wazwaz, (2022) “Integrable (3+1)-dimensional Ito equation: variety of lump solutions and multiple soliton solutions" NONLINEAR DYNAMICS 109(3): 1929–1934. DOI: 10.1007/s11071-022-07517-0.
[28] A.-M. Wazwaz, (2022) “Multi-soliton solutions for inte grable (3+1)-dimensional modified seventh-order Ito and seventh-order Ito equations" NONLINEAR DYNAM ICS 110(4): 3713–3720. DOI: 10.1007/s11071-02207818-4.
[29] Y.-L. Ma, A.-M. Wazwaz, and B.-Q. Li, (2021) “New extended Kadomtsev–Petviashvili equation: multiple soli ton solutions, breather, lump and interaction solutions." NONLINEARDYNAMICS104(2):1581–1594. DOI: 10.1007/s11071-021-06357-8.
[30] A.-M. Wazwaz and L. Kaur, (2019) “New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions" NONLINEAR DYNAM ICS 97(1): 83–94. DOI: 10.1007/s11071-019-04955-1.
[31] G.-Q. Xu and A.-M. Wazwaz, (2020) “Bidirectional solitons and interaction solutions for a new integrable f ifth-order nonlinear equation with temporal and spatial dispersion" NONLINEAR DYNAMICS101(1): 581 595. DOI: 10.1007/s11071-020-05740-1.
[32] A.-M. Wazwaz and G.-Q. Xu, (2020) “Kadomtsev Petviashvili hierarchy: two integrable equations with time dependent coefficients" NONLINEAR DYNAMICS 100(4): 3711–3716. DOI: 10.1007/s11071-020-05708-1.
[33] D. Anderson, M. Lisak, and A. Berntson, (2001) “A variational approach to nonlinear evolution equations in optics" PRAMANA-JOURNALOFPHYSICS57(5 6): 917–936. DOI: 10.1007/s12043-001-0006-z.
[34] P.Zhang,W.Hu,Z.Wang,andZ.Qiao,(2023)“Multi Symplectic Simulation on Soliton-Collision for Nonlinear Perturbed Schrodinger Equation" JOURNAL OF NON LINEARMATHEMATICALPHYSICS30(4):1467 1482. DOI: 10.1007/s44198-023-00137-1.
[35] W.Hu,Z.Deng, and T. Yin, (2017) “Almost structure preserving analysis for weakly linear damping nonlinear Schrodinger equation with periodic perturbation" COM MUNICATIONSINNONLINEARSCIENCEAND NUMERICALSIMULATION42:298–312. DOI:10.1016/j.cnsns.2016.05.024.
[36] W.Hu,X.Xi, Z. Song, C. Zhang, and Z. Deng, (2023) “Coupling dynamic behaviors of axially moving cracked cantilevered beam subjected to transverse harmonic load" MECHANICAL SYSTEMS AND SIGNAL PRO CESSING204: DOI: 10.1016/j.ymssp.2023.110757.
[37] W. Hu, Z. Han, T. J. Bridges, and Z. Qiao, (2023) “Multi-symplectic simulations of W/M-shape-peaks soli tons and cuspons for FORQ equation" APPLIED MATHEMATICS LETTERS 145: DOI: 10.1016/j.aml.2023.108772.
[38] Y. Huai, W. Hu, W. Song, Y. Zheng, and Z. Deng, (2023) “Magnetic-field-responsive property of Fe₃O₄/polyaniline solvent free nanofluid" PHYSICS OF FLUIDS 35(1): DOI: 10.1063/5.0130588.
[39] W.Hu,M.Xu,F. Zhang, C. Xiao, and Z. Deng, (2022) “Dynamic analysis on flexible hub-beam with step-variable cross-section" MECHANICAL SYSTEMS AND SIG NALPROCESSING180: DOI:10.1016/j.ymssp.2022.109423.
[40] G. Akram, M. Sadaf, S. Arshed, and U. Ejaz, (2023) “Travelling wave solutions and modulation instability anal ysis of the nonlinear Manakov-system" JOURNAL OF TAIBAHUNIVERSITYFORSCIENCE17(1): DOI: 10.1080/16583655.2023.2201967.
[41] M.Sadaf, G. Akram, S. Arshed, and K. Farooq, (2023) “A study of fractional complex Ginzburg-Landau model with three kinds of fractional operators" CHAOS SOLI TONS&FRACTALS166: DOI:10.1016/j.chaos.2022.112976.
[42] S. Arshed, M. Sadaf, G. Akram, and M. M. Yasin, (2022) “Analysis of Sasa-Satsuma equa tion with beta fractional derivative using extended (G’/G²)-expansion technique and (exp(-()))-expansion technique" OPTIK 271: DOI: 10.1016/j.ijleo.2022.170087.
[43] M.Sadaf, S. Arshed, G. Akram, and Iqra, (2022) “Ex act soliton and solitary wave solutions to the Fokas system sing two variables G’ G, 1G-expansion technique and gen eralized projective Riccati equation method." OPTIK 268: DOI: 10.1016/j.ijleo.2022.169713.
[44] M.Sadaf, G. Akram, and M. Dawood, (2022) “An in vestigation of fractional complex Ginzburg-Landau equa tion with Kerr law nonlinearity in the sense of con formable, beta and M-truncated derivatives" OPTICAL ANDQUANTUMELECTRONICS54(4): DOI: 10.1007/s11082-022-03570-6.
[45] G. Akram, S. Arshed, and M. Sadaf, (2023) “Soliton solutions of generalized time-fractional Boussinesq-like equation via three techniques" CHAOS SOLITONS & FRACTALS173: DOI: 10.1016/j.chaos.2023.113653.
[46] S. Arshed, G. Akram, M. Sadaf, and K. Saeed, (2022) “Construction of new solutions of Korteweg-de Vries Caudrey-Dodd-Gibbon equation using two efficient in tegration methods" PLOS ONE 17(9): DOI: 10.1371/journal.pone.0275118.
[47] G.Akram,S.Arshed,M.Sadaf, andF. Sameen, (2022) “The generalized projective Riccati equations method for solving quadratic-cubic conformable time-fractional Klien Fock-Gordon equation" Ain Shams Engineering Jour nal 13(4): 101658. DOI: 10.1016/j.asej.2021.101658.
[48] H. Tariq, M. Sadaf, G. Akram, H. Rezazadeh, J. Baili, Y.-P. Lv, and H. Ahmad, (2021) “Computational study for the conformable nonlinear Schrodinger equation with cubic-quintic-septic nonlinearities" RESULTS IN PHYSICS30: DOI: 10.1016/j.rinp.2021.104839.
[49] G. Akram, M. Sadaf, M. Abbas, I. Zainab, and S. R. Gillani, (2022) “Efficient techniques for traveling wave so lutions of time-fractional Zakharov-Kuznetsov equation" MATHEMATICSANDCOMPUTERSINSIMULA TION193: 607–622. DOI: 10.1016/j.matcom.2021.11.004.
[50] G. Akram, M. Sadaf, and M. A. U. Khan, (2022) “Soli ton Dynamics of the Generalized Shallow Water Like Equation in Nonlinear Phenomenon" FRONTIERS IN PHYSICS10: DOI: 10.3389/fphy.2022.822042.
[51] A. M.Elsherbeny, A. Bekir, A. H. Arnous, M. Sadaf, and G. Akram, (2023) “Solitons to the time-fractional Radhakrishnan-Kundu-Lakshmanan equation with β and M-truncated fractional derivatives: a comparative analy sis" OPTICAL ANDQUANTUMELECTRONICS 55(12): DOI: 10.1007/s11082-023-05414-3.
[52] M.Sadaf,S.Arshed,G.Akram,M.R.Ali,andI.Bano, (2023) “Analytical investigation and graphical simula tions for the solitary wave behavior of Chaffee-Infante equation" RESULTS IN PHYSICS 54: DOI: 10.1016/j.rinp.2023.107097.
[53] S. Arshed, M. Sadaf, G. Akram, and K. Farooq, (2024) “Bright Solitons, dark solitons and periodic wave solutions of Chen-Lee-Liu model" INTERNATIONAL JOUR NALOFGEOMETRICMETHODSINMODERN PHYSICS21(04): DOI: 10.1142/S0219887824500737.
[54] G. Akram, M. Sadaf, S. Arshed, and U. Ejaz, (2023) “Travelling wave solutions and modulation instability anal ysis of the nonlinear Manakov-system" JOURNAL OF TAIBAHUNIVERSITYFORSCIENCE17(1): DOI: 10.1080/16583655.2023.2201967.