Abstract
In this paper, the generalized nonlinear Schrödinger equation with variable coefficients (gvcNLSE) arising in optical fiber is solved by using two different techniques the trail equation method and direct integration method. Many different new types of wave solutions like Jacobi, periodic and soliton wave solutions are obtained. From this study we have concluded that the direct integration method is more easy and straightforward than the trail equation method. As an application in optic fibers the propagation of the frequency modulated optical soliton is discussed and we have deduced that it's propagation shape is affected with the different values of both the amplification increment and the group velocity (GVD).
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This research was funded by Majmaah University.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] X. Lü, W. X. Ma, Y. Zhou, and C. M. Khalique, “Rational solutions to an extended Kadomtsev-Petviashvili-like equation with sym-bolic computation,” Comput. Math. with Appl., vol. 71, pp. 1560–1567, 2016, https://doi.org/10.1016/j.camwa.2016.02.017.Search in Google Scholar
[2] L. N. Gao, X. Y. Zhao, Y. Y. Zi, J. Yu, and X. Lü, “Resonant behavior of multiple wave solutions to a Hirota bilinear equation,” Comput. Math. with Appl., vol. 72, pp. 1225–1229, 2016, https://doi.org/10.1016/j.camwa.2016.06.008.Search in Google Scholar
[3] X. Lü, W. X. Ma, S. T. Chen, and C. M. Khalique, “A note on rational solutions to a Hirota-Satsuma-like equation,” Appl. Math. Lett., vol. 58, pp. 13–18, 2016, https://doi.org/10.1016/j.aml.2015.12.019.Search in Google Scholar
[4] X. Lü, S. T. Chen, and W. X. Ma, “Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation,” Nonlinear Dyn., vol. 86, pp. 523–534, 2016, https://doi.org/10.1007/s11071-016-2905-z.Search in Google Scholar
[5] L. N. Gao, Y. Y. Zi, Y. H. Yin, W. X. Ma, and X. Lü, “Bäcklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation,” Nonlinear Dyn., vol. 89, pp. 2233–2240, 2017, https://doi.org/10.1007/s11071-017-3581-3.Search in Google Scholar
[6] G. M. Wei, Y. L. Lu, Y. Q. Xie, and W. X. Zheng, “Lie symmetry analysis and conservation law of variable-coefficient Davey-Stewartson equation,” Comput. Math. with Appl., vol. 75, pp. 3420–3430, 2018, https://doi.org/10.1016/j.camwa.2018.02.008.664444.Search in Google Scholar
[7] H. Rezazadeh, H. Tariq, M. Eslami, M. Mirzazadeh, and Q. Zhou, “New exact solutions of nonlinear conformable time-fractional Phi-4 equation,” Chinese J. Phys., vol. 56, pp. 2805–2816, 2018, https://doi.org/10.1016/J.CJPH.2018.08.001.Search in Google Scholar
[8] A. Biswas, M. O. Al-Amr, H. Rezazadeh, et al., “Resonant optical solitons with dual-power 10 law nonlinearity and fractional temporal evolution,” Optik (Stuttg), vol. 165, pp. 233–239, 2018, https://doi.org/10.1016/J.IJLEO.2018.03.123.Search in Google Scholar
[9] N. Raza, U. Afzal, A. R. Butt, and H. Rezazadeh, “Optical solitons in nematic liquid crystals with Kerr and parabolic law nonlinearities,” Opt. Quantum Electron., vol. 51, p. 107, 2019, https://doi.org/10.1007/s11082-019-1813-0.Search in Google Scholar
[10] H. Rezazadeh, A. Korkmaz, M. Eslami, and S. M. Mirhosseini-Alizamini, “A large family of optical solutions to Kundu-Eckhaus model by a new auxiliary equation method,” Opt. Quantum Electron., vol. 51, p. 84, 2019, https://doi.org/10.1007/s11082-019-1801-4.Search in Google Scholar
[11] J. G. Liu, M. Eslami, H. Rezazadeh, and M. Mirzazadeh, “Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev-Petviashvili equation,” Nonlinear Dyn., vol. 95, pp. 1027–1033, 2019, https://doi.org/10.1007/s11071-018-4612-4.Search in Google Scholar
[12] H. Rezazadeh, “New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity,” Optik (Stuttg), vol. 167, pp. 218–227, 2018, https://doi.org/10.1016/J.IJLEO.2018.04.026.Search in Google Scholar
[13] A. Biswas, H. Rezazadeh, M. Mirzazadeh, et al., “Optical solitons having weak non-local nonlinearity by two integration schemes,” Optik (Stuttg), vol. 164, pp. 380–384, 2018, https://doi.org/10.1016/J.IJLEO.2018.03.026.Search in Google Scholar
[14] H. Rezazadeh, M. Mirzazadeh, S. M. Mirhosseini-Alizamini, A. Neirameh,M. Eslami, and Q. Zhou, “Optical solitons of Lakshmanan-Porsezian-Daniel model with a couple of nonlinearities,” Optik (Stuttg), vol. 164, pp. 414–423, 2018, https://doi.org/10.1016/J.IJLEO.2018.03.039.Search in Google Scholar
[15] Q. Zhou, D. Kumar, M. Mirzazadeh, M. Eslami, and H. Rezazadeh, “Opti-cal soliton in nonlocal nonlinear medium with cubic-quintic nonlinearities and spatio-temporal dispersion,” ACTA Phys. Pol. A, vol. 134, 2018, https://doi.org/10.12693/APhysPolA.134.1204.Search in Google Scholar
[16] K. U. Tariq, M. Younis, H. Rezazadeh, S. T. R. Rizvi, and M. S. Os-man, “Optical solitons with quadratic-cubic nonlinearity and fractional temporal evolution,” Mod. Phys. Lett. B, vol. 32, 2018, Art no. 1850317, https://doi.org/10.1142/S0217984918503177.Search in Google Scholar
[17] H. Rezazadeh, A. Korkmaz, M. Eslami, J. Vahidi, and R. Asghari, “Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method,” Opt. Quantum Electron., vol. 50, p. 150, 2018, https://doi.org/10.1007/s11082-018-1416-1.Search in Google Scholar
[18] M. S. Osman, H. Rezazadeh, M. Eslami, A. Neirameh, and M. Mirzazadeh, “Analtyical study of solitons to Benjamin-Bona-Mahony-Peregrine equation with power law nonlinearity by 11 using three methods,” U.P.B. Sci. Bull., Ser. A, vol. 80, 2018.https://www.scienti.cbulletin.upb.ro/rev_docs_arhiva/fullf37_570579.pdf [accessed: August 2, 2019].Search in Google Scholar
[19] H. Yépez-Martínez, H. Rezazadeh, A. Souleymanou, et al., “The extended modified method applied to optical solitons solutions in birefringent fibers with weak nonlocal nonlinearity and four wave mixing,” Chinese J. Phys., vol. 58, pp. 137–150, 2019, https://doi.org/10.1016/j.cjph.2019.02.002.Search in Google Scholar
[20] Y. H. Yin, Ma, W. X., Liu, J. G., Lü, X., “Diversity of Exact Solutions to a (3+1)-Dimensional Nonlinear Evolution Equation and Its Reduction,” Computers & mathematics with applications, vol. 76, pp. 1275–1283, 2018. https://doi.org/10.1016/j.camwa.2018.06.020.Search in Google Scholar
[21] H. N. Xu, W. Y. Ruan, Y. Zhang, and X. Lü, “Multi-exponential wave solutions to two extended Jimbo-Miwa equations and the resonance behavior,” Appl. Math. Lett., vol. 99, 2020, Art no. 105976, https://doi.org/10.1016/J.AML.2019.07.007.Search in Google Scholar
[22] Y. F. Hua, B. L. Guo, W. X. Ma, and X. Lü, “Interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation for nonlinear waves,” Appl. Math. Model., vol. 74, pp. 184–198, 2019, https://doi.org/10.1016/J.APM.2019.04.044.Search in Google Scholar
[23] X. Lü, F. Lin, and F. Qi, “Analytical study on a two-dimensional Korteweg-de Vries model with bilinear representation, Bäcklund transformation and soliton solutions,” Appl. Math. Model., vol. 39, pp. 3221–3226, 2015, https://doi.org/10.1016/j.apm.2014.10.046.Search in Google Scholar
[24] S. J. Chen, Y. H. Yin, W. X. Ma, and X. Lü, “Abundant exact solutions and interaction phenomena of the (2 + 1)-dimensional YTSF equation,” Anal. Math. Phys., vol. 9, pp. 2329–2344, 2019, https://doi.org/10.1007/s13324-019-00338-2.Search in Google Scholar
[25] R. M. El-Shiekh and A. G. Al-Nowehy, “Integral methods to solve the variable coefficient nonlinear Schrödinger equation,” Zeitschrift Fur Naturforsch. - Sect. C J. Biosci., vol. 68A, 2013, https://doi.org/10.5560/ZNA.2012-0108.Search in Google Scholar
[26] R. M. El-Shiekh, “Classes of new exact solutions for nonlinear Schrödinger equations with variable coefficients arising in optical fiber,” Results Phys., vol. 13, 2019, Art no. 102214. https://doi.org/10.1016/j.rinp.2019.102214.Search in Google Scholar
[27] H. Rezazadeh, S. M. Mirhosseini-Alizamini, M. Eslami, M. Rezazadeh, M. Mirzazadeh, and S. Abbagari, “New optical solitons of nonlinear conformable fractional Schrödinger-Hirota equation,” Optik (Stuttg), vol. 172, pp. 545–553, 2018, https://doi.org/10.1016/J.IJLEO.2018.06.111.Search in Google Scholar
[28] X. Lü and M. Peng, “Painlevé-integrability and explicit solutions of the general two-coupled nonlinear Schrödinger system in the optical fiber communications”, Nonlinear Dyn., vol. 73, pp. 405–410, 2013, https://doi.org/10.1007/s11071-013-0795-x.Search in Google Scholar
[29] X. Lü, “Madelung fluid description on a generalized mixed nonlinear Schrödinger equation,” Nonlinear Dyn., vol. 81, pp. 239–247, 2015, https://doi.org/10.1007/s11071-015-1985-5.Search in Google Scholar
[30] R. M. El-Shiekh and M. Gaballah-, “Solitary wave solutions for the variable- coefficient coupled nonlinear Schrödinger equations and Davey-Stewartson system using modified sine-Gordon equation method,” J. Ocean Eng. Sci., vol. 14, no. 1, pp. 783–789, 2019. https://doi.org/10.1016/j.joes.2019.10.003.Search in Google Scholar
[31] A. V. Zhukov, I. O. Zolotovskii, O. G. Okhotnikov, D. I. Sementsov, A. A. Sysolyatin, and I. O. Yavtushenko, “Dynamics of frequency-modulated soliton-like pulses in a longitudinally inhomogeneous active optical waveguide,” Opt. Spec-trosc., vol. 113, pp. 75–80, 2012, https://doi.org/10.1134/S0030400X12060239.Search in Google Scholar
[32] I. O. Zolotovskii, S. G. Novikov, O. G. Okhotnikov, D. I. Sementsov, I. O. Yavtushenko, and M. S. Yavtushenko, “Amplification of frequency-modulated soliton like pulses in inhomogeneous optical waveguides with normal dispersion,” Opt.Spectrosc., vol. 112, pp. 893–897, 2012, https://doi.org/10.1134/S0030400X12060252.Search in Google Scholar
[33] R. Hao, L. Li, Z. Li, W. Xue, and G. Zhou, “A new approach to ex-act soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients,” Opt. Commun., vol. 236, pp. 79–86, 2004, https://doi.org/10.1016/j.optcom.2004.03.005.Search in Google Scholar
[34] B. Q. Li and Y. L. Ma, “Periodic and N-kink-like optical solitons for a generalized Schrödinger equation with variable coefficients in an inhomogeneous fiber system,” Optik (Stuttg), vol. 179, pp. 854–860, 2019, https://doi.org/10.1016/j.ijleo.2018.11.008.Search in Google Scholar
[35] X. Zhang, H. Guo, G. Ma, W. Feng, X. Zhang, and W. Liu, “Anti-dark soliton interactions for the variable-coefficient nonlinear Schrödinger equation,” Optik (Stuttg), vol. 161 (2018), pp. 217–220, 2018. https://www.sciencedirect.com/science/article/pii/S003040261830202X [accessed: February 14, 2019].10.1016/j.ijleo.2018.02.040Search in Google Scholar
[36] M. H. M. Moussa and R. M. El Shikh, “Similarity Reduction and similarity solutions of Zabolotskay-Khoklov equation with a dissipative term via symmetry method,” Phys. A Stat. Mech. Its Appl., vol. 371, 2006, https://doi.org/10.1016/j.physa.2006.04.044.Search in Google Scholar
[37] M. H. M. Moussa and R. M. El Shikh, “Auto-Bäcklund transformation and similarity reductions to the variable coefficients variant Boussinesq system,” Phys. Lett. Sect. A Gen. At. Solid State Phys., vol. 372, 2008, https://doi.org/10.1016/j.physleta.2007.09.056.Search in Google Scholar
[38] R. M. El-Shiekh, “Periodic and solitary wave solutions for a generalized variable-coefficient Boiti-Leon-Pempinlli system,” Comput. Math. with Appl., vol. 73, 2017, https://doi.org/10.1016/j.camwa.2017.01.008.13.Search in Google Scholar
[39] R. M. El-Shiekh, “Jacobi elliptic wave solutions for two variable coefficients cylindrical Korteweg-de Vries models arising in dusty plasmas by using direct reduction method,” Comput. Math. with Appl., vol. 75, pp. 1676–1684, 2018, https://doi.org/10.1016/j.camwa.2017.11.031.Search in Google Scholar
[40] R. M. El-Shiekh, “New similarity solutions for the generalized variable coefficients KdV equation by using symmetry group method,” Arab J. Basic. Appl. Sci., vol. 25, pp. 66–70, 2018, https://doi.org/10.1080/25765299.2018.1449343.Search in Google Scholar
[41] R. M. El-Shiekh, “Painlevé test, Bäcklund transformation and consistent Riccati expansion solvability for two generalised cylindrical Korteweg-deries equations with variable coefficients,” Z. Naturforsch., vol. 73, pp. 207–213, 2018. https://doi.org/10.1515/zna-2017-0349.Search in Google Scholar
[42] M. H. M. Moussa and R. M. El-Shiekh, “Similarity solutions for generalized variable coefficients zakharov-kuznetso equation under some integrability conditions,” Commun. Theor. Phys., vol. 54, 2010, https://doi.org/10.1088/0253-6102/54/4/04.Search in Google Scholar
[43] M. H. M. Moussa and R. M. El-Shiekh, “Direct reduction and exact solutions for generalized variable coefficients 2D KdV equation under some integrability conditions,” Commun. Theor. Phys., vol. 55, 2011, https://doi.org/10.1088/02536102/55/4/03.Search in Google Scholar
[44] M. H. M. Moussa, R. A. K. Omar, R. M. El-Shiekh, and H. R. El-Melegy, “Non-equivalent similarity reductions and exact solutions for coupled burgers-type equations,” Commun. Theor. Phys., vol. 57, 2012, https://doi.org/10.1088/0253-6102/57/1/01.Search in Google Scholar
[45] R. M. El-Shiekh, “New exact solutions for the variable coefficient modified KdV equation using direct reduction method,” Math. Methods Appl. Sci., vol. 36, 2013, https://doi.org/10.1002/mma.2561.Search in Google Scholar
[46] G. M. Moatimid, R. M. El-Shiekh, and A. G. Al-Nowehy, “Exact solutions for Calogero-Bogoyavlenskii-Schiff equation using symmetry method,” Appl. Math. Comput., vol. 220, 2013, https://doi.org/10.1016/j.amc.2013.06.034.Search in Google Scholar
[47] M. F. El-Sayed, G. M. Moatimid, M. H. M. Moussa, R. M. El-Shiekh, and A. A. El-Satar, “Symmetry group analysis and similarity solutions for the (2+1)-dimensional coupled Burger’s system,” Math. Methods Appl. Sci., vol. 37, 2014, https://doi.org/10.1002/mma.2870.Search in Google Scholar
[48] M. F. El-Sayed, G. M. Moatimid, M. H. M. Moussa, R. M. El-Shiekh, F. A. H. El-Shiekh, and A.A. El-Satar, “A study of integrability and symmetry for the (p +1)th Boltzmann equation via Painlevé analysis and Lie-group method,” Math.Methods Appl. Sci., vol. 38, 2015, https://doi.org/10.1002/mma.3307.Search in Google Scholar
[49] R. M. El-Shiekh, “Direct similarity reduction and new exact solutions for the variable-coefficient kadomtsev-petviashvili equation,” Zeitschrift Fur Natur-forsch. - Sect. A J. Phys. Sci., vol. 70, 2015, https://doi.org/10.1515/zna-2015-0057.Search in Google Scholar
[50] M. Mirzazadeh, “Soliton solutions of Davey-Stewartson equation by trial equation method and ansatz approach,” Nonlinear Dyn., vol. 82, pp. 1775–1780, 2015, https://doi.org/10.1007/s11071-015-2276-x.Search in Google Scholar
[51] Q. Zhou, M. Ekici, A. Sonmezoglu, M. Mirzazadeh, and M. Eslami, “Optical solitons with Biswas-Milovic equation by extended trial equation method,” Nonlinear Dyn., vol. 84, pp. 1883–1900, 2016, https://doi.org/10.1007/s11071-016-2613-8.Search in Google Scholar
[52] M. El-Shiekh Rehab and Gaballah, Mahmoud, “Novel solitons and periodic wave solutions for Davey–Stewartson system with variable coefficients,” Journal of Taibah University for Science vol. 14, no. 1, pp. 783–789, 2020. https://doi.org/10.1080/16583655.2020.1774975.Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
