Optical Solutions of the Kundu-Eckhaus Equation via Two Different Methods

Yıl 2021, Cilt: 11 Sayı: 1, 126 - 135, 30.06.2021

Melike Kaplan

https://doi.org/10.37094/adyujsci.838536

Cited By: 1

Öz

This work is devoted to obtaining new optical solutions to the Kundu-Eckhaus (KE) equation which is believed to play a crucial part in the area of nonlinear optics. Two different methods, the exp(−𝜑 (ε)) method with the exponential rational function approach have been utilized. Both methods are efficient in finding the analytical solutions of many nonlinear partial differential equations and fractional differential equations. Results obtained in this research are dissimilar to the ones in the literature and the solutions are controlled by relocating them back to the primary equation. Finally, it can be stated that optical solutions have a promising future.

Anahtar Kelimeler

Nonlinear equation, Symbolic Computation, Optical solutions

Kaynakça

• [[1] Kaplan M., Ozer, M.N., Auto-Bäcklund transformations and solitary wave solutions for the nonlinear evolution equation, Optical and Quantum Electronics, 50(1), 33, 2018.
• [2] Akter, J., Akbar, M.A., Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method, Results in Physics, 5, 125-130, 2015.
• [3] Ma, W.X., Lee, J.H., A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation, Chaos, Solitons and Fractals, 42, 1356-1363, 2009.
• [4] Mirzazadeh, M., Arnous, A.H., Mahmood, M.F. Zerrad, E. Biswas, A., Soliton solutions to resonant nonlinear Schrödinger’s equation with time-dependent coefficients by trial solution approach, Nonlinear Dynamics, 81, 277-282, 2015.
• [5] Bulut, H., Akturk, T., Gurefe, Y., An application of the new function method to the generalized double sinh-Gordon equation, AIP Conference Proceedings 1648, 370014, 2015.
• [6] Islam, M.S., Khan, K., Akbar, M. A., An analytical method for finding exact solutions of modified Korteweg-de Vries equation, Results in Physics, 5, 131-135, 2015.
• [7] Inan, I.E., Ugurlu, Y., Inc, M., New Applications of the (G’/G,1/G)-Expansion Method, Acta Physica Polonica A, 128, 245-251, 2015.
• [8] Abdou, M.A., A generalized auxiliary equation method and its applications, Nonlinear Dynamics, 52, 95-102, 2008.
• [9] Adem, A.R., Khalique, C.M., Conserved quantities and solutions of a (2+1)- dimensional Haragus-Courcelle-Il’ichev model, Computers and Mathematics with Applications, 71, 1129-1136, 2016.
• [10] He, J.H., Abdou, M.A., New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos, Solitons and Fractals, 34, 1421-1429, 2007.
• [11] Abdou, M.A., Further improved F-expansion and new exact solutions for non- linear evolution equations, Nonlinear Dynamics, 52, 277-288, 2008.
• [12] Mirzazadeh, M., Eslami, M., Zerrad, E., Mahmood, M.F., Biswas, A., Belic, M., Optical solitons in nonlinear directional couplers by sine-cosine function method and Bernoulli’s equation approach, Nonlinear Dynamics, 81, 1933-1949, 2015.
• [13] Younis, M., Ali, S. , Mahmood, S.A., Solitons for compound KdV-Burgers equation with variable coefficients and power law nonlinearity, Nonlinear Dynamics, 81, 1191-1196, 2015.
• [14] Durur, H., Kurt, A., Tasbozan, O., New Travelling Wave Solutions for KdV6 Equation Using Sub Equation Method, Applied Mathematics and Nonlinear Sciences, 5(1), 455-460, 2020.
• [15] Yusufoğlu, E., New solitonary soltions for the MBBM equatios using Exp-function Method, Physic Letters A, 372, 442-446, 2008.
• [16] Biswas, A., Khalique, C.M., Stationary solutions for nonlinear dispersive Schrödinger’s equation, Nonlinear Dynamics, 63, 623-626, 2011.
• [17] Wazwaz, A.M., Multiple-soliton solutions for the Boussinesq equation, Applied Mathematics and Computation, 192, 479-486, 2007.
• [18] Lü, X., Tian, B., Zhang, H.Q., Xu, T., Li, H., Generalized (2+1)-dimensional Gardner model: bilinear equations, Bäcklund transformation, Lax representation and interaction mechanisms, Nonlinear Dynamics, 67, 2279-2290, 2012.
• [19] Ma, W.X., Abdeljabbar, A., Asaad, M.G., Wronskian and Grammian solutions to a (3+1)-dimensional generalized KP equation, Applied Mathematics and Computation, 217, 10016-10023, 2011. [20] Wang, M.L., Solitary wave solutions for variant Boussinesq equations, Physics Letters A, 199, 169-172, 1995.
• [21] Ablowitz, M.J., Segur, H., Solitons and Inverse Scattering Transformation, SIAM, Philadelphia, 1981.
• [22] Thabet, H., Kendre, S., Peters, J., Kaplan, M., Solitary wave solutions and traveling wave solutions for systems of time-fractional nonlinear wave equations via an analytical approach, Computational and Applied Mathematics, 39, 2020, 144 .
• [23] Zayed, E.M.E., Alngar, M.E.M., Al-Nowehy, A.G., On solving the nonlinear Schrödinger Equation with an anti-cubic nonlinearity in presence of Hamiltonian perturbation terms,Optik - International Journal for Light and Electron Optics, 178, 488-508, 2019.
• [24] Biswas, A., Jawad, A.J.M. and Zhou, Q., Resonant optical solitons with anti-cubic nonlinearity, Optik, 157, 525-531, 2018.
• [25] Biswas, A., Optical soliton perturbation with Radhakrishnan-Kundu-Lakshmanan equation by traveling wave hypothesis, Optik, 171, 217-220, 2018.
• [26] Kaplan, M., Application of two reliable methods for solving a nonlinear conformable time-fractional equation, Optical and Quantum Electronics, 49, 312, 2017.
• [27] Roshid, H.O., Kabir, M.R., Bhowmik, R.C., Datta, B.K., Investigation of Solitary wave Solutions for Vakhnenko-Parkes equation via exp-function and exp(−φ(ξ))-expansion method. SpringerPlus, 3, 692, 2014.
• [28] Kundu, A., Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations, Journal of Mathematical Physics, 25, 3433-3438, 1984.
• [29] Eckhaus, W., The long-time behaviour for perturbed wave-equations and related problems, Preprint no. 404, Department of Mathematics, University of Utrecht, 1985.
• [30] Mirzazadeh, M., Yıldırım, Y., Yas, E., Triki, H., Zhoud, Q., Moshokoa, S.P., Ullah, M.Z., Seadawy, A.R., Biswas, A., Belic, M., Optical solitons and conservation law of Kundu-Eckhaus equation, Optik, 154, 551-557, 2018.