×

Optical solitons for Chen-Lee-Liu equation with two spectral collocation approaches. (English) Zbl 1478.78053

Summary: This paper revisits the study of optical solitons that is governed by one of the three forms of derivative nonlinear Schrödinger’s equation that is also known as Chen-Lee-Liu model. This model is investigated by the aid of fully shifted Jacobi’s collocation method with two independent approaches. The first is discretization of the spatial variable, while the other is discretization of the temporal variable. It is concluded that the method of the current paper is far more efficient and reliable for the considered model. Numerical results illustrate the performance efficiency of the algorithm. The results also point out that the scheme can lead to spectral accuracy of the studied model.

MSC:

78A60 Lasers, masers, optical bistability, nonlinear optics
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
35C08 Soliton solutions
78M22 Spectral, collocation and related methods applied to problems in optics and electromagnetic theory
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Wazwaz, A.-M.; Kaur, L., Optical solitons and Peregrine solitons for nonlinear Schrödinger equation by the variational iteration method, Optik, 179, 804-809 (2019)
[2] Triki, H.; Wazwaz, A.-M., Combined optical solitary waves of the Fokas-Lennels equation, Waves Random Complex Media, 27, 587-593 (2017)
[3] Wazwaz, A.-M.; Kaur, L., Complex simplified Hirota’s forms and Lie symmetry analysis for multiple real and complex soliton solutions of the modified KdV-sine-Gordon equation, Nonlinear Dyn., 95, 2209-2215 (2019) · Zbl 1432.37093
[4] Wazwaz, A.-M., Two-mode fifth-order KdV equations: Necessary conditions for multiple-soliton solutions to exist, Nonlinear Dyn., 87, 1685-1691 (2017)
[5] Wazwaz, A.-M., Multiple soliton solutions and multiple complex soliton solutions for two distinct Boussinesq equations, Nonlinear Dyn., 85, 731-737 (2016)
[6] Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; Van Gorder, R. A., Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations, J. Comput. Phys., 261, 244-255 (2014) · Zbl 1349.65511
[7] Wang, H., Numerical studies on the split step finite difference method for the nonlinear Schrödinger equations, Appl. Math. Comput., 170, 17-35 (2005) · Zbl 1082.65570
[8] Dehghan, M.; Taleei, A., Numerical solution of nonlinear Schrödinger equation by using time-space pseudo-spectral method, Numer. Methods Partial Differ. Equations, 26, 979-992 (2010) · Zbl 1195.65137
[9] Dehghan, M.; Taleei, A., A Chebyshev pseudo-spectral multi-domain method for the soliton solution of coupled nonlinear Schrödinger equations, Comput. Phys. Commun., 182, 2519-2529 (2011) · Zbl 1261.65103
[10] Kaur, L.; Wazwaz, A.-M., Bright-dark optical solitons for Schrödinger-Hirota equation with variable coefficients, Optik, 179, 479-484 (2019)
[11] Chen, H. H.; Lee, Y. C.; Liu, C. S., Integrability of nonlinear Hamiltonian systems by inverse scattering method, Phys. Scr., 20, 490-492 (1979) · Zbl 1063.37559
[12] Fan, E. G., Integrable systems of derivative nonlinear Schrödinger type and their multi-Hamiltonian structure, J. Phys. A: Math. Gen., 34, 513-519 (2001) · Zbl 0970.37054
[13] Zhang, J.; Liu, W.; Qiu, D.; Zhang, Y.; Porsezian, K.; He, J., Rogue wave solutions of a higher-order Chen-Lee-Liu equation, Phys. Scr., 90, 055207 (2015)
[14] Jawad, A. J. M.; Biswas, A.; Zhou, Q.; Alfiras, M.; Moshokoa, S. P.; Belic, M., “Chirped singular and combo optical solitons for Chen-Lee-Liu equation with three forms of integration architecture,” Optik—Int, J. Light Electron Opt. Optics, 178, 172-177 (2019)
[15] Biswas, A.; Ekici, M.; Sonmezoglu, A.; Alshomrani, A. S.; Zhou, Q.; Moshokoa, S. P.; Belic, M., Chirped optical solitons of Chen-Lee-Liu equation by extended trial equation scheme, Optik, 156, 999-1006 (2018)
[16] Yildirim, Y., Optical solitons to Chen-Lee-Liu model with modified simple equation approach, Optik—Int. J. Light Electron Opt., 183, 792-796 (2019)
[17] Yildirim, Y., Optical solitons to Chen-Lee-Liu model in birefringent fibers with trial equation approach, Optik—Int. J. Light Electron Opt., 183, 881-886 (2019)
[18] Triki, H.; Hamaizi, Y.; Zhou, Q.; Biswas, A.; Ullah, M. Z.; Moshokoa, S. P.; Belic, M., Chirped singular solitons for Chen-Lee-Liu equation in optical fibers and PCF, Optik, 157, 156-160 (2018)
[19] Biswas, A., Chirp-free bright optical soliton perturbation with Chen-Lee-Liu equation by traveling wave hypothesis and semi-inverse variational principle, Optik—Int. J. Light Electron Opt., 172, 772-776 (2018)
[20] Mohammed, A. S. H. F.; Bakodah, H. O.; Banaja, M. A.; Alshaery, A. A.; Zhou, Q.; Biswas, A.; Moshokoa, S. P.; Belic, M. R., Bright optical solitons of Chen-Lee-Liu equation with improved Adomian decomposition method, Optik—Int. J. Light Electron Opt., 181, 964-970 (2019)
[21] Yang, B.; Zhang, W.-G.; Zhang, H.-Q.; Pei, S.-B., Generalized Darboux transformation and rational soliton solutions for Chen-Lee-Liu equation, Appl. Math. Comput., 242, 863-876 (2014) · Zbl 1334.35327
[22] Sousa, F. S.; Lages, C. F.; Ansoni, J. L.; Castelo, A.; Simao, A., A finite difference method with meshless interpolation for incompressible flows in non-graded tree-based grids, J. Comput. Phys., 396, 848-866 (2019) · Zbl 1452.76163
[23] Mbroh, N. A.; Munyakazi, J. B., A fitted operator finite difference method of lines for singularly perturbed parabolic convection-diffusion problems, Math. Comput. Simul., 165, 156-171 (2019) · Zbl 07316742
[24] Patil, H. M.; Maniyeri, R., Finite difference method based analysis of bio-heat transfer in human breast cyst, Thermal Sci. Eng. Prog., 10, 42-47 (2019)
[25] Li, P.-W.; Fu, Z.-J.; Gu, Y.; Song, L., The generalized finite difference method for the inverse Cauchy problem in two-dimensional isotropic linear elasticity, Int. J. Solids Struct., 174-175, 69-84 (2019)
[26] Ray, S.; Degweker, S. B.; Kannan, U., A hybrid method for reactor core simulations employing finite difference and polynomial expansion with improved treatment of transverse leakage, Ann. Nuclear Energy, 131, 102-111 (2019)
[27] Shu, Y.; Li, J.; Zhang, C., A local and parallel Uzawa finite element method for the generalized Navier-Stokes equations, Appl. Math. Comput., 387, 124671 (2020) · Zbl 1472.65148
[28] Wang, C.; Wang, J., Primal-dual weak Galerkin finite element methods for elliptic Cauchy problems, Comput. Math. Appl., 79, 746-763 (2020) · Zbl 1443.65374
[29] E. Burman, P. Hansbo, M. G. Larson, A. Massing, and S. Zahedi, “A stabilized cut streamline diffusion finite element method for convection-diffusion problems on surfaces,” Comput. Methods Appl. Mech. Eng. 358 (2020). arXiv:1807.01480. · Zbl 1441.76055
[30] Xiao, X.; Dai, Z.; Feng, X., A positivity preserving characteristic finite element method for solving the transport and convection-diffusion-reaction equations on general surfaces, Comput. Phys. Commun., 247, 106941 (2020)
[31] Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62, 2364-2373 (2011) · Zbl 1231.65126
[32] Doha, E. H.; Abdelkawy, M. A.; Amin, A. Z. M.; Lopes, A. M., Shifted Jacobi-Gauss-collocation with convergence analysis for fractional integro-differential equations, Commun. Nonlinear Sci. Numer. Simul., 72, 342-359 (2019) · Zbl 1464.65078
[33] Doha, E. H.; Abdelkawy, M. A.; Amin, A. Z. M.; Baleanu, D., Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations, Nonlinear Anal.: Model. Control, 24, 332-352 (2019) · Zbl 1480.65373
[34] Bhrawy, A. H.; Taha, T. M.; Tenreiro Machado, J. A., A review of operational matrices and spectral techniques for fractional calculus, Nonlinear Dyn., 81, 1023-105 (2015) · Zbl 1348.65106
[35] Zaky, M. A.; Ameen, I. G.; Abdelkawy, M. A., A new operational matrix based on Jacobi wavelets for a class of variable-order fractional differential equations, Proc. Rom. Acad. Ser. A, 18, 315-322 (2017)
[36] Bhrawy, A. H.; Zaky, M. A., A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, J. Comput. Phys., 281, 876-895 (2015) · Zbl 1352.65386
[37] Doha, E. H.; Bhrawy, A. H.; Baleanu, D.; Hafez, R. M., A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations, Appl. Numer. Math., 77, 43-54 (2014) · Zbl 1302.65175
[38] Kolmanovskii, V.; Myshkis, A., Applied Theory of Functional Differential Equations (1992), Amsterdam: Kluwer Academic, Amsterdam · Zbl 0917.34001
[39] Bhrawy, A. H.; Abdelkawy, M. A., Fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, J. Comput. Phys., 294, 462-483 (2015) · Zbl 1349.65503
[40] Bhrawy, A. H.; Doha, E. H.; Baleanu, D.; Ezz-eldein, S. S., A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys., 293, 142-156 (2015) · Zbl 1349.65504
[41] Doha, E. H.; Hafez, R. M.; Youssri, Y. H., Shifted Jacobi spectral-Galerkin method for solving hyperbolic partial differential equations, Comput. Math. Appl., 78, 889-904 (2019) · Zbl 1442.65290
[42] Doha, E. H.; Abd-Elhameed, W. M., Efficient spectral ultraspherical-dual-Petrov-Galerkin algorithms for the direct solution of (2n+1)th-order linear differential equations, Math. Comput. Simul., 79, 3221-3242 (2009) · Zbl 1169.65326
[43] Triki, H.; Babatin, M. M.; Biswas, A., Chirped bright solitons for Chen-Lee-Liu equation in optical fibers and PCF, Optik, 149, 300-303 (2017)
[44] Triki, H.; Zhou, Q.; Moshokoa, S. P.; Ullah, M. Z.; Biswas, A.; Belic, M., Chirped W-shaped optical solitons of Chen-Lee-Liu equation, Optik, 55, 208-212 (2018)
[45] Triki, H.; Hamaizi, Y.; Zhou, Q.; Biswas, A.; Ullah, M. Z.; Moshokoa, S. P.; Belic, M., Chirped dark and gray optical solitons for Chen-Lee-Liu equation in optical fibers and PCF, Optik, 155, 329-333 (2018)
[46] Kara, A. H.; Biswas, A.; Zhou, Q.; Moraru, L.; Moshokoa, S. P.; Belic, M., Conservation laws for optical solitons with Chen-Lee-Liu equation, Optik, 174, 195-198 (2018)
[47] Gonzalez-Gaxiola, O.; Biswas, A., W-shaped optical solitons of Chen-Lee-Liu equation by Laplace-Adomian decomposition method, Opt. Quantum Electron., 50, 1-11 (2020)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.