The investigation of several soliton solutions to the complex Ginzburg-Landau model with Kerr law nonlinearity

Year 2022, , 147 - 163, 30.09.2022

Muhammad Abubakar Isah , Asıf Yokus

https://doi.org/10.53391/mmnsa.2022.012

Cited By: 7

Abstract

This work investigates the complex Ginzburg-Landau equation (CGLE) with Kerr law in nonlinear optics, which represents soliton propagation in the presence of a detuning factor. The $\varphi^{6}$-model expansion approach is used to find optical solitons such as dark, bright, singular, and periodic as well as the combined soliton solutions to the model. The results presented in this study are intended to improve the CGLE's nonlinear dynamical characteristics, it might also assist in comprehending some of the physical implications of various nonlinear physics models. The hyperbolic sine, for example, appears in the calculation of the Roche limit and gravitational potential of a cylinder, while the hyperbolic cotangent appears in the Langevin function for magnetic polarization. The current research is frequently used to report a variety of fascinating physical phenomena, such as the Kerr law of non-linearity, which results from the fact that an external electric field causes non-harmonic motion of electrons bound in molecules, which causes nonlinear responses in a light wave in an optical fiber. The obtained solutions' 2-dimensional, 3-dimensional, and contour plots are shown.

Keywords

$\varphi^{6}$-model expansion method, complex Ginzburg-Landau equation, traveling wave solution, Kerr law nonlinearity

References

• Isah, M.A., & Külahcı, M.A. A study on null cartan curve in Minkowski 3-space. Applied Mathematics and Nonlinear Sciences, 5(1), 413-424, (2020).
• Isah, M.A., & Külahcı, M.A. Involute Curves in 4-dimensional Galilean space G4. In Conference Proceedings of Science and Technology, 2(2), 134-141, (2019).
• Isah, M.A., Isah, I., Hassan, T.L., & Usman, M. Some characterization of osculating curves according to darboux frame in three dimensional euclidean space. International Journal of Advanced Academic Research, 7(12), 47-56, (2021).
• Isah, M.A., & Külahçı, M.A. Special curves according to bishop frame in minkowski 3-space. Applied Mathematics and Nonlinear Sciences, 5(1), 237-248, (2020).
• Aydin, M.E., Mihai, A., & Yokus, A. Applications of fractional calculus in equiaffine geometry: plane curves with fractional order. Mathematical Methods in the Applied Sciences, 44(17), 13659-13669, (2021).
• Myint-U, T., & Debnath, L. Linear partial differential equations for scientists and engineers. Springer Science & Business Media, Boston, (2007).
• Duran, S. Breaking theory of solitary waves for the Riemann wave equation in fluid dynamics. International Journal of Modern Physics B, 35(09), 2150130, (2021).
• Hasegawa, A. Optical solitons in fibers. In Optical solitons in fibers (pp. 1-74). Springer, Berlin, Heidelberg, (1989).
• Yokuş, A., Durur, H., & Duran, S. Simulation and refraction event of complex hyperbolic type solitary wave in plasma and optical fiber for the perturbed Chen-Lee-Liu equation. Optical and Quantum Electronics, 53(7), 1-17, (2021).
• Kivshar, Y.S., & Agrawal, G.P. Optical solitons: from fibers to photonic crystals. Academic Press, (2003).
• Hasegawa, A., & Kodama, Y. Signal transmission by optical solitons in monomode fiber. Proceedings of the IEEE, 69(9), 1145-1150, (1981).
• Kaya, D., & Yokus, A. A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations. Mathematics and Computers in Simulation, 60(6), 507-512, (2002).
• Durur, H. Energy-carrying wave simulation of the Lonngren-wave equation in semiconductor materials. International Journal of Modern Physics B, 35(21), 2150213, (2021).
• Gao, W., Silambarasan, R., Baskonus, H.M., Anand, R.V., & Rezazadeh, H. Periodic waves of the non dissipative double dispersive micro strain wave in the micro structured solids. Physica A: Statistical Mechanics and its Applications, 545, 123772, (2020).
• Yokuş, A., Durur, H., & Abro, K.A. Symbolic computation of Caudrey–Dodd–Gibbon equation subject to periodic trigonometric and hyperbolic symmetries. The European Physical Journal Plus, 136(4), 1-16, (2021).
• Yokuş, A. Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 24-31, (2021).
• Tarla, S., Ali, K.K., Yilmazer, R., & Osman, M.S. New optical solitons based on the perturbed Chen-Lee-Liu model through Jacobi elliptic function method. Optical and Quantum Electronics, 54(2), 1-12, (2022).
• Tarla, S., Ali, K.K., Sun, T.C., Yilmazer, R., & Osman, M.S. Nonlinear pulse propagation for novel optical solitons modeled by Fokas system in monomode optical fibers. Results in Physics, 36, 105381, (2022).
• Duran, S., & Karabulut, B. Nematicons in liquid crystals with Kerr Law by sub-equation method. Alexandria Engineering Journal, 61(2), 1695-1700, (2022).
• Yokus, A., & Tuz, M. An application of a new version of (G0/G)-expansion method. In AIP Conference Proceedings 1798(1), 020165. AIP Publishing LLC, (2017).
• Kaya, D., Yokuş, A., & Demiroğlu, U. Comparison of exact and numerical solutions for the Sharma-Tasso-Olver equation. In Numerical Solutions of Realistic Nonlinear Phenomena (pp. 53-65). Springer, Cham (2020).
• Baskonus, H.M., Gao, W., Rezazadeh, H., Mirhosseini-Alizamini, S.M., Baili, J., Ahmad, H., & Gia, T.N. New classifications of nonlinear Schrödinger model with group velocity dispersion via new extended method. Results in Physics, 31, 104910, (2021).
• Yokus, A., & Isah, M.A. Stability analysis and solutions of (2+1)-Kadomtsev–Petviashvili equation by homoclinic technique based on Hirota bilinear form. Nonlinear Dynamics, 1-12, (2022).
• Khan, A., Khan, A., & Sinan, M. Ion temperature gradient modes driven soliton and shock by reduction perturbation method for electron-ion magneto-plasma. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 1-12, (2022).
• Yokus, A., & Isah, M.A. Investigation of internal dynamics of soliton with the help of traveling wave soliton solution of Hamilton amplitude equation. Optical and Quantum Electronics, 54(8), 1-21, (2022).
• Zhou, Q., Xiong, X., Zhu, Q., Liu, Y., Yu, H., Yao, P., ... & Belicd, M. Optical solitons with nonlinear dispersion in polynomial law medium. Journal of Optoelectronics and Advanced Materials, 17(1-2), 82-86, (2015).
• Zayed, E.M., & Al-Nowehy, A.G. Many new exact solutions to the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms using three different techniques. Optik, 143, 84-103, (2017).
• Zayed, E.M., Al-Nowehy, A.G., & Elshater, M.E. New-model expansion method and its applications to the resonant nonlinear Schrödinger equation with parabolic law nonlinearity. The European Physical Journal Plus, 133(10), 417, (2018).
• Mehdizadeh Khalsaraei, M., Shokri, A., Noeiaghdam, S., & Molayi, M. Nonstandard Finite Difference Schemes for an SIR Epidemic Model. Mathematics, 9(23), 3082, (2021).
• Zarin, R., Ahmed, I., Kumam, P., Zeb, A., & Din, A. Fractional modeling and optimal control analysis of rabies virus under the convex incidence rate. Results in Physics, 28, 104665, (2021).
• Khan, A., Zarin, R., Khan, S., Saeed, A., Gul, T., & Humphries, U.W. Fractional dynamics and stability analysis of COVID-19 pandemic model under the harmonic mean type incidence rate. Computer Methods in Biomechanics and Biomedical Engineering, 25(6), 619-640, (2022).
• Zarin, R., Ahmed, I., Kumam, P., Zeb, A., & Din, A. Fractional modeling and optimal control analysis of rabies virus under the convex incidence rate. Results in Physics, 28, 104665, (2021).
• Chu, Y., Shallal, M.A., Mirhosseini-Alizamini, S.M., Rezazadeh, H., Javeed, S., & Baleanu, D. Application of modified extended Tanh technique for solving complex Ginzburg-Landau equation considering Kerr law nonlinearity. CMC-Computers Materials & Continua, 66(2), 1369-1378, (2021).
• Arnous, A.H., Seadawy, A.R., Alqahtani, R.T., & Biswas, A. Optical solitons with complex Ginzburg–Landau equation by modified simple equation method. Optik, 144, 475-480, (2017).
• Liu, W., Yu, W., Yang, C., Liu, M., Zhang, Y., & Lei, M. Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers. Nonlinear Dynamics, 89(4), 2933-2939, (2017).
• Kudryashov, N.A. First integrals and general solution of the complex Ginzburg-Landau equation. Applied Mathematics and Computation, 386, 125407, (2020).
• Rezazadeh, H. New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity. Optik, 167, 218-227, (2018).
• Mirzazadeh, M., Ekici, M., Sonmezoglu, A., Eslami, M., Zhou, Q., Kara, A.H., ... & Belić, M. Optical solitons with complex Ginzburg–Landau equation. Nonlinear Dynamics, 85(3), 1979-2016, (2016).
• Osman, M.S., Ghanbari, B., & Machado, J.A.T. New complex waves in nonlinear optics based on the complex Ginzburg-Landau equation with Kerr law nonlinearity. The European Physical Journal Plus, 134(1), 1-10, (2019).
• Ahmed, I., Seadawy, A.R., & Lu, D. Combined multi-waves rational solutions for complex Ginzburg–Landau equation with Kerr law of nonlinearity. Modern Physics Letters A, 34(03), 1950019, (2019).
• Sajid, N., & Akram, G. Novel solutions of Biswas-Arshed equation by newly φ6-model expansion method. Optik, 211, 164564, (2020).