Certain hybrid prototypes of dispersive optical solitons that we are looking for can correspond to new or future behaviors, observable or not, developed or will be developed by optical media that present the cubic-quintic-septic law coupled, with strong dispersions. The equation considered for this purpose is that of non-linear Schr?dinger. The solutions are obtained using the Bogning-Djeumen Tchaho-Kofané method extended to the new implicit Bogning’ functions. Some of the obtained solutions show that their existence is due only to the Kerr law nonlinearity presence. Graphical representations plotted have confirmed the hybrid and multi-form character of the obtained dispersive optical solitons. We believe that a good understanding of the hybrid dispersive optical solitons highlighted in the context of this work allows to grasp the physical description of systems whose dynamics are governed by nonlinear Schr?dinger equation as studied in this work, allowing thereby a relevant improvement of complex problems encountered in particular in nonliear optaics and in optical fibers.
References
[1]
Hirota, R. (1971) Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons. Physical Review Letters, 27, 1192-1194. https://doi.org/10.1103/PhysRevLett.27.1192
[2]
Lee, J.-H., Lin, C.-K. and Pashaev, O.K. (2004) Shock Waves, Chiral Solitons and Semiclassical Limit of One-Dimensional Anyons. Chaos, Solitons and Fractals, 19, 109-128. https://doi.org/10.1016/S0960-0779(03)00084-5
[3]
Tariq, K.U., Younis, M., Rezazadeh, H., Rizvi, S.T.R. and Osman, M.S. (2018) Optical Solitons with Quadratic-Cubic Nonlinearity and Fractional Temporal Evolution. Modern Physics Letters B, 32, 1850317-185049. https://doi.org/10.1142/S0217984918503177
[4]
Seadawy, A.R., Arshad, M. and Lu, D. (2017) Stability Analysis of New Exact Traveling-Wave Solutions of New Coupled KdV and New Coupled Zakharov-Kuznetsov Systems. European Physical Journal Plus, 132, 162-180. https://doi.org/10.1140/epjp/i2017-11437-5
[5]
Ngouo Tchinda, C. and Bogning, J.R. (2020) Solitary Waves and Property Management of Nonlinear Dispersive and Flattened Optical Fiber. American Journal Optics and Photonics, 8, 27-32. https://doi.org/10.11648/j.ajop.20200801.13
[6]
Mahak, N. and Akram, G. (2019) Novel Approaches to Extract Soliton Solutions of the (1+1) Dimensional Fokas-Lenells Equation by Means of the Complex Transformation. Optik, 192, 162912-162919. https://doi.org/10.1016/j.ijleo.2019.06.012
[7]
Bhrawy, A.H., Alshaery, A.A., Hilal, E.M., Manrakhan, W.N., Savescu, M. and Biswas, A. (2014) Dispersive Optical Solitons with Schrödinger-Hirota Equation. Journal of Nonlinear Optical Physics & Materials, 23, 1450014-1450034. https://doi.org/10.1142/S0218863514500143
[8]
Ebaid, A., El-Zahar, E.R., Aljohani, A.F., Salah, B., Krid, M. and Machado, J.T. (2019) Exact Solutions of the Generalized Nonlinear Fokas-Lennells Equation. Results in Physics, 14, 102472-102476. https://doi.org/10.1016/j.rinp.2019.102472
[9]
Osman, M.S., Lu, D. and Khater, M.M.A. (2019) A Study of Optical Wave Propagation in the Nonautonomous Schrödinger-Hirota Equation with Power-Law Nonlinearity. Results in Physics, 13, 102157-102160. https://doi.org/10.1016/j.rinp.2019.102157
[10]
Biswas, A. (2004) Stochastic Perturbation of Optical Solitons in Schrödinger-Hirota Equation. Optics Communications, 239, 461-466. https://doi.org/10.1016/j.optcom.2004.06.047
[11]
Zhang, Z.-Y. (2015) Jacobi Elliptic Function Expansion Method for the Modified Korteweg-de Vries-Zakharov Kuznetsov and the Hirota Equations. Romanian Journal of Physics, 60, 1384-1394.
[12]
Biswas, A., Yildrim, Y., Yasar, E., Zhou, Q., Alshomrani, A.S. and Belic, M. (2019) Optical Soliton Perturbation in Parabolic Law Medium Having Weak Non-Local Nonlinearity by a Couple of Strategic Integration Architectures. Results in Physics, 13, 102334-102346. https://doi.org/10.1016/j.rinp.2019.102334
[13]
Abdou, M.A. and Elhanbaly, A. (2007) Construction of Periodic and Solitary Wave Solutions by the Extended Jacobi Elliptic Function Expansion Method. Communications in Nonlinear Science and Numerical Simulation, 12, 1229-1241. https://doi.org/10.1016/j.cnsns.2006.01.013
[14]
Kudryashov, N.A. (2019) First Integrals and General Solution of the Traveling Wave Reduction for Schrödinger Equation with Anti-Cubic Nonlinearity. Optik, 185, 665-671. https://doi.org/10.1016/j.ijleo.2019.03.167
[15]
Yildirim, Y. (2019) Optical Soliton Molecules of Manakov Model by Trial Equation Technique. Optik, 185, 1146-1151. https://doi.org/10.1016/j.ijleo.2019.04.041
[16]
Biswas, A., Ekici, M., Sonmezoglu, A. and Belic, M.R. (2019) Highly Dispersive Optical Solitons with Cubic-Quintic-Septic Law by Extended Jacobi’s Elliptic Function Expansion. Optik, 183, 571-578. https://doi.org/10.1016/j.ijleo.2019.02.127
[17]
Biswas, A., Ekici, M., Sonmezoglu, A. and Belic, M.R. (2019) Highly Dispersive Optical Solitons with Cubic-Quintic-Septic Law by Exp-Expansion. Optik, 186, 321-325. https://doi.org/10.1016/j.ijleo.2019.04.085
[18]
Biswas, A., Ekici, M., Sonmezoglu, A. and Belic, M.R. (2019) Highly Dispersive Optical Solitons with Cubic-Quintic-Septic Law by F-Expansion. Optik, 182, 897-906. https://doi.org/10.1016/j.ijleo.2019.01.058
[19]
Biswas, A., Vega-Guzman, J., Mahmood, M.F., Khan, S., Ekici, M., Zhou, Q., Moshoko, S.P. and Belic, M.R. (2019) Highly Dispersive Optical Solitons with Undetermined Coefficients. Optik, 182, 890-896. https://doi.org/10.1016/j.ijleo.2019.01.087
[20]
Ur Rehman, H., Ullah, N. and Imran, M.A. (2019) Highly Dispersive Optical Solitons Using Kudryashov’s Method. Optik, 199, 163349-163355. https://doi.org/10.1016/j.ijleo.2019.163349
[21]
Djeumen Tchaho, C.T., Bogning, J.R. and Kofané, T.C. (2010) Construction of the Analytical Solitary Wave Solutions of Modified Kuramoto-Sivashinsky Equation by the Method of Identification of Coefficients of the Hyperbolic Functions. Far East Journal of Dynamical Systems, 14, 17-34.
[22]
Djeumen Tchaho, C.T., Bogning, J.R. and Kofané, T.C. (2011) Multi-Soliton Solutions of the Modified Kuramoto-Sivashinsky Equation by the BDK Method. Far East Journal of Dynamical Systems, 15, 83-98.
[23]
Bogning, J.R., Djeumen Tchaho, C.T. and Kofané, T.C. (2012) Construction of the Soliton Solutions of the Ginzburg-Landau Equations by the New Bogning-Djeumen Tchaho-Kofané Method. Physica Scripta, 85, 25013-25017. https://doi.org/10.1088/0031-8949/85/02/025013
[24]
Djeumen Tchaho, C.T. (2015) New Method of Construction of the Solitary Wave Solutions of Some Physical Nonlinear Partial Differential Equations Doctorat. Ph.D. Thesis, University of Yaounde I, Yaounde.
[25]
Bogning, J.R. (2019) Mathematics for Nonlinear Physics: Solitary Waves in the Center of Resolutions of Dispersive Nonlinear Partial Differential Equations. Dorrance Publishing Co., Pittsburgh.
[26]
Rezazadeh, H., Tariq, K.U., Sabi’u, J. and Bekir, A. (2020) Abundant Traveling Wave Solutions to the Resonant Nonlinear Schrödinger’s Equation with Variable Coefficients. Modern Physics Letters B, 34, 2050118-2050128. https://doi.org/10.1142/S0217984920501183
[27]
Inc, M., Aliyu, A.I. and Yusuf, A. (2017) Dark Optical, Singular Solitons and Conservation Laws to the Nonlinear Schrödinger’s Equation with Spatio-Temporal Dispersion. Modern Physics Letters B, 31, Article ID: 1750163. https://doi.org/10.1142/S0217984917501639
[28]
Das, A., Biswas, A., Ekici, M., Zhou, Q., Alshomrani, A.S. and Belic, M.R. (2019) Optical Solitons with Complex Ginzburg-Landau Equation for Two Nonlinear Forms Using F-Expansion. Chinese Journal of Physics, 61, 255-261. https://doi.org/10.1016/j.cjph.2019.08.009
[29]
Biswas, A. (2003) Optical Solitons: Quasistationarity versus Lie Transform. Optical and Quantum Electronics, 35, 979-998. https://doi.org/10.1023/A:1025121931885
[30]
Bogning, J.R. (2019) Mathematics for Nonlinear Physics: The Implicit Bogning Functions and Applications. Lambert Academic Publishing, Saarbrücken.
[31]
Bogning, J.R. (2020) Elements of Analytical Mechanics and Quantum Physics. Lambert Academic Publishing, Saarbrücken.
[32]
Djeumen Tchaho, C.T., Bogning, J.R. and Kofané, T.C. (2012) Modulated Soliton Solution of the Modified Kuramoto-Sivashinsky’s Equation. American Journal of Computational and Applied Mathematics, 2, 218-224. https://doi.org/10.5923/j.ajcam.20120205.03
[33]
Bogning, J.R., Djeumen Tchaho, C.T. and Kofané, T.C. (2012) Generalization of the Bogning-Djeumen Tchaho-Kofané Method for the Construction of the Solitary Waves and the Survey of the Instabilities. Far East Journal of Dynamical Systems, 20, 101-119. https://doi.org/10.5923/j.ajcam.20120205.03
[34]
Djeumen Tchaho, C.T., Omanda, H.M. and Belobo Belobo, D. (2018) Hybrid Solitary Waves for the Generalized Kuramoto-Sivashinsky Equation. The European Physical Journal Plus, 133, 387-395. https://doi.org/10.1140/epjp/i2018-12218-4
[35]
Djeumen Tchaho, C.T., Omanda, H.M., N’tchayi Mbourou, G., Bogning, J.R. and Koafné, T.C. (2019) Multi-Form Solitary Waves Solutions of the KdV-Burgers-Ku- ramoto Equation. Journal of Physics Communications, 3, 105013-105023. https://doi.org/10.1088/2399-6528/ab4ba1
[36]
Njikue, R., Bogning, J.R. and Kofané, T.C. (2018) Exact Bright and Dark Solitary Wave Solutions of the Generalized Higher-Order Nonlinear Schrödinger Equation Describing the Propagation of Ultra-Short Pulse in Optical Fibers. Journal of Physics Communications, 2, 25030-25039. https://doi.org/10.1088/2399-6528/aaaf3b
[37]
Bogning, J.R. (2018) Exact Solitary Wave Solutions of the (3+1)-Modified B-Type Kadomtsev-Petviashvili Family Equations. American Journal of Computational and Applied Mathematics, 8, 85-92.
[38]
Bogning, J.R., Djeumen Tchaho, C.T. and Omanda, H.M. (2016) Combined Solitary Wave Solutions in Higher-order Effects Optical Fibers. British Journal of Mathematics & Computer Science, 13, 1-12. https://doi.org/10.9734/BJMCS/2016/10620
[39]
Bogning, J.R., Fautso Kuiaté, G., Omanda, H.M. and Djeumen Tchaho, C.T. (2015) Combined Peakons and Multiple-Peak Solutions of the Camassa-Holm and Modified KdV Equations and Their Conditions of Obtention. Physics Journal, 1, 367-374.
[40]
Bogning, J.R., Porsezian, K., Fautso Kuiaté, G. and Omanda, H.M. (2015) Gap Solitary Pulses Induced by the Modulational Instability and Discrete Effects in Array of Inhomogeneous Optical Fibers. Physics Journal, 1, 216-224.
[41]
Tiague Takongmo, G. and Bogning, J.R. (2018) Construction of Solutions in the Shape (Pulse; Pulse) and (Kink; Kink) of a Set of Two Equations Modeled in a Nonlinear Inductive Electrical Line with Crosslink Capacitor. American Journal of Circuits, Systems and Signal Processing, 4, 28-35.
[42]
Tiague Takongmo, G. and Bogning, J.R. (2018) Construction of Breather Soliton Solutions of a Modeled Equation in a Discrete Nonlinear Electrical Line and the Survey of Modulationnal Instability. Journal of Physics Communications, 2, 115007-115019. https://doi.org/10.1088/2399-6528/aaeaa1
[43]
Tiague Takongmo, G. and Bogning, J.R. (2018) Coupled Soliton Solutions of Modeled Equations in a Noguchi Electrical Line with Crosslink Capacitor. Journal of Physics Communications, 2, 105016-105026. https://doi.org/10.1088/2399-6528/aae7e6
[44]
Biswas, A., Milovic, D. and Milic, D. (2011) Solitons in Alpha-Helix Proteins by He’s Variational Principle. International Journal of Biomathematics, 4, 423-429. https://doi.org/10.1142/S1793524511001325
[45]
Biswas, A., Kara, A.H., Savscu, M., Bokhari, A.H. and Zaman, F.D. (2013) Solitons and Conservation Laws in Neurosciences. International Journal of Biomathematics, 6, Article ID: 1350017. https://doi.org/10.1142/S1793524513500174
[46]
Biswas, A., Milovic, D., Savescu, M., Mahmood, M.F., Khan, K.R. and Kohl, R. (2012) Optical Soliton Perturbation in Nanofibers with Improved Nonlinear Schrödinger’s Equation by Semi-Inverse Variational Principle. Journal of Nonlinear Optical Physics & Materials, 21, Article ID: 1250054. https://doi.org/10.1142/S0218863512500543
[47]
Kudryashov, A. (2020) Highly Dispersive Solitary Wave Solutions of Perturbed Nonlinear Schrödinger Equations. Applied Mathematics and Computation, 371, Article ID: 124972. https://doi.org/10.1016/j.amc.2019.124972