Resonant solitons solutions to the time M-fractional Schrödinger equation
Subject Areas : Mathematical Engineering
موسی ایلی
1
(گروه ریاضی، واحد رشت، دانشگاه آزاد اسلامی، رشت، ایران)
علی خوش کنار
2
(گروه ریاضی، واحد رشت، دانشگاه آزاد اسلامی، رشت، ایران)
Keywords: Truncated M-fractional derivative, Kerr-law nonlinearity, Parabolic-law nonlinearity, Modified Kudryashov method, Sine-Gordon expansion approach, Soliton solutions, Resonant nonlinear Schrö dinger equation,
Abstract :
In this research the time M-fractional resonant nonlinear Schrödinger differential equation with different forms of nonlinearities, containing Kerr-law and parabolic-law has been studied. For this objective, the modified Kudryashov method and the sine-Gordon expansion approach have been implemented to retrieve a series of resonant solitons solutions for the abovementioned model. The prospective of the schemes in founding soliton solutions of nonlinear time-fractional equations in the truncated M-fractional derivative sense is confirmed.
Biswas, A. (2012). Soliton solutions of the perturbed resonant nonlinear dispersive Schrödinger’s equation with full nonlinearity by semi-inverse variational principle. Quantum Physics Letters, (1)2, 79–84.
Capelas de Oliveira, E., Tenreiro Machado, J. A. (2014). A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering, 2014 (238459).
Ekici, M., Zhou, Q. Sonmezoglu, A., Manafian, J., Mirzazadeh, M. (2017). The analytical study of solitons to the nonlinear Schrödinger equation with resonant nonlinearity, Optik, 130, 378-382, http://dx.doi.org/10.1016/j.ijleo.2016.10.098.
Eslami, M., Mirzazadeh, M, Biswas, A. (2013). Soliton solutions of the resonant nonlinear Schrödinger’s equation in optical fibers with time dependent coefficients by simplest equation approach. Journal of Modern Optics, (60)19, 1627–1636.
Figueiredo Camargo, R., Capelas de Oliveira, E. (2015). Fractional Calculus (In Portuguese). Editora Livraria da Fsica, So Paulo.
Goreno, R., Kilbas, A. A., Mainardi, F., Rogosin, S. V. (2014). Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin.
Herrmann, R. (2011). Fractional calculus: An Introduction for Physicists, World Scientific Publishing Company, Singapore.
Hosseini, K., Ansari, R. (2017). New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method, Wave in Random and Complex media, http://dx.doi.org/10.1080/17455030.2017.1296983.
Hosseini, K., Kumar, D., Kaplan, M., Yazdani Bejarbaneh, E. (2017). New Exact Traveling Wave Solutions of the Unstable Nonlinear Schrödinger Equations. Communications in Theoretical Physics, 68, 761–767.
Hosseini, K., Manafian, J., Samadani, F., Foroutan, M., Mirzazadeh, M., Zhou, Q. Resonant optical solitons with perturbation terms and fractional temporal evaluation using improved -expansion method and exp function approach. Optik, https://doi.org/10.1016/j.ijleo.2017.12.139.
Ilie, M., Biazar, J., Ayati, Z. (2018). Analytical study of exact traveling wave solutions for time-fractional nonlinear Schrödinger equations. Optical and Quantum Electronics, (50) 12, https://doi.org/10.1007/s11082-018-1682-y.
Ilie, M., Biazar, J., Ayati, Z. (2018). Resonant solitons to the nonlinear Schrödinger equation with different forms of nonlinearities. Optik, 164 (2018) 201-209.
Ilie, M., Biazar, J., Ayati, Z. (2018). The first integral method for solving some conformable fractional differential equations. Optical and Quantum Electronics, (50) 2, https://doi.org/10.1007/s11082-017-1307-x.
Inc, M., Yusufi, A., Aliyu, A. I. (2017). Dark optical and other soliton solutions for the three different nonlinear Schrödinger equations. Optical and Quantum Electronics, (2017), DOI 10.1007/s11082-017-1187-0.
Inc, M., Yusufi, A., Aliyu, A. I., Baleanu, D. (2018). Soliton structures to some time-fractional nonlinear differential equations with conformable derivative. Optical and Quantum Electronics (2018), https://doi.org/10.1007/s11082-017-1287-x.
Katugampola, U. N. (2014). A new fractional derivative with classical properties, arXiv:1410.6535v2.
Katugampola, U. N. (2016). New fractional integral unifying six existing fractional integrals. arxiv.org/abs/1612.08596, (2016).
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, (2014) 65-70.
Kilbas, A. A., Srivastava, H. M., Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier, Amsterdam, Vol. 207.
Kilbas, A. A., Srivastava, H. M., Trujillo, J. J. (2006). Theory and Applications of the Fractional Differential Equations. Vol. 204, Elsevier, Amsterdam, 2006.
Kudryashov, N. A. (2005). Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons and Fractals, 24 (2005), 1217–123.
Kudryashov, N. A. (2012). One method for finding exact solutions of nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 2248–2253.
Kudryashov, N. A. (2013). Polynomials in logistic function and solitary waves of nonlinear differential equations. Applied Mathematics and Computation, 219 (2013), 9245–9253.
Mirzazadeh, M., Eslami, M., Milovic, D., Biswas, A. (2014). Topological solitons of resonant nonlinear Schrödinger’s equation with dual-power law nonlinearity using G′/G-expansion technique. Optik, (125) 19, 5480–5489.
Mirzazadeh, M., Eslami, M., Fathi Vajargah, B., Biswas, A. (2014). Optical solitons and optical rogons of generalized resonant dispersive nonlinear Schrödinger’s equation with power law nonlinearity. Optik, (125) 9, 4246–4256.
Podlubny, I. (1999). Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Vol. 198.
Triki, H., Biswas, A., Babatin, M. M., Zhou, Q. (2018). Chirped dark solitons in optical metamaterials. Optik, 158, 312-315.
Triki, H., Yildirim, A., Hayat, T., Aldossary, O. M., Biswas, A. (2012). 1-soliton solution of the generalized resonant nonlinear dispersive Schrödinger’s equation with time-dependent coefficients. Advanced Science Letters, 16, 309–312.
Triki, H., Yildirim, A., Hayat, T., Aldossary, O. M., Biswas, A. (2012). Bright and dark solitons for the resonant nonlinear Schrödinger’s equation with time- dependent coefficients. Optics & Laser Technology, 44, 2223–2231.
Vanterler da C. Sousa, J., Capelas de Oliveira, E. (2017). M-fractional derivative with classical properties. arXiv:1704.08187.
Vanterler da C. Sousa, J., Capelas de Oliveira, E. (2018). A New Truncated M-Fractional Derivative Type Unifying Some Fractional Derivative Types with Classical Properties. International Journal of Analysis and Applications, (16) 1, 83-96.
Zhou, Q., Wei, C., Zhang, H., Lu, J., Yu, H., Yao, P., Zhu, Q. (2016). Exact solutions to the resonant nonlinear Schrödinger equation with both spatio-temporal and inter-modal dispersions. Proceedings of the Romanian Academy, Series A, (17) 4, 307-313.
Zhou, Q., Zhu, Q. (2015). Optical solitons in medium with parabolic law nonlinearity and higher order dispersion. Waves in Random and Complex Media, (25)1, 52-59.