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A compact split step Padé scheme for higher-order nonlinear Schrödinger equation (HNLS) with power law nonlinearity and fourth order dispersion. (English) Zbl 1217.65177
Summary: We propose a compact split step Padé scheme (CSSPS) to solve the scalar higher-order nonlinear Schrödinger equation (HNLS) with higher-order linear and nonlinear effects such as the third and fourth order dispersion effects, Kerr dispersion, stimulated Raman scattering and power law nonlinearity. The stability of this method is proved. It is shown as well that the CSSPS method gives the same results as classical numerical methods like the split step Fourier method and Crank-Nicolson method but it presents many advantages over theme. It is more efficient. This proposed scheme is well suited to higher-order dispersion effects and readily generalized for nonlinear and dispersion managed fibers. We tested this scheme for the case of the quintic nonlinearity and confirmed that this effect has no significant role on the propagation of single solitons.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
Full Text: DOI
[1] Agrawal, G.P., Nonlinear fiber optics, (2007), Academic Press US, pp. 177-220
[2] Hasegawa, A.; Kodama, Y., Solitons in optical communications, (1995), Clarendon Press Oxford · Zbl 0840.35092
[3] Hasegawa, A., Theory of information transfer in optical fibers: a tutorial review, Optical fiber technology, 10, 50-170, (2004)
[4] Lele, Sanjiva K., Compact finite difference schemes with spectral-like resolution, Journal of computational physics, 103, 16-42, (1992) · Zbl 0759.65006
[5] Konar, S.; Mishra, M.; Jana, S., Effect of quintic nonlinearity on soliton collisions in fibers, Physica D, 195, 123-140, (2004)
[6] Sonesone, J.; Peleg, A., The effect of quintic nonlinearity on the propagation characteristics of dispersion managed optical solitons, Chaos, solitons and fractals, 29, 823-828, (2006) · Zbl 1142.78318
[7] Latas, S.C.V.; Ferreira, M.F.S., Soliton propagation in the presence of intrapulse Raman scattering and nonlinear gain, Optics communications, 251, 415-422, (2005)
[8] Porsezian, K.; Hasegawa, A.; Serkin, V.N.; Belyaeva, T.L.; Ganapathy, R., Dispersion and nonlinear management for femtosecond optical solitons, Physics letters A, 361, 504-508, (2006)
[9] Reinsch, Matthias W., A simple expression for the terms in the Baker-Campbell-Hausdorff series, (February 4, 2008)
[10] Wazwas, A.M., Exact solutions for the fourth order nonlinear Schrödinger equations with cubic and power law nonlinearities, Mathematical computer modelling, 43, 7-8, 802-808, (2006) · Zbl 1136.35458
[11] Biswas, A.; Milovic, D., Optical solitons in power law media with fourth order dispersion, Communications in nonlinear science and numerical simulation, 14, 1834-1837, (2009) · Zbl 1221.78034
[12] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Mathematics and computers in simulation, 71, 16-30, (2006) · Zbl 1089.65085
[13] Dehghan, M.; Mirzaei, D., Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method, International journal for numerical methods in engineering, 76, 501-520, (2008) · Zbl 1195.81007
[14] Dehghan, M.; Taleei, A., A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Computer physics communications, 181, 1, 43-51, (2010) · Zbl 1206.65207
[15] Dehghan, M.; Mirzaei, D., The meshless local Petrov-Galerkin (MLPG) method for the generalized two-dimensional non-linear schrodinger equation, Engineering analysis with boundary elements, 32, 747-756, (2008) · Zbl 1244.65139
[16] Dehghan, M.; Shokri, A., A numerical method for two-dimensional schrodinger equation using collocation and radial basis functions, Computers and mathematics with applications, 54, 136-146, (2007) · Zbl 1126.65092
[17] Kohl, R.; Biswas, A.; Milovic, D.; Zerrad, E., Optical soliton perturbation in a non-Kerr law media, Optics & laser technology, 40, 4, 647-662, (2008)
[18] Biswas, A.; Konar, S., Introduction to non-Kerr law optical solitons, (2006), CRC Press Boca Raton, FL, USA · Zbl 1156.78001
[19] Dehghan, M.; Shakourifara, M.; Hamidi, A., The solution of linear and nonlinear systems of Volterra functional equations using Adomian-pade technique, Chaos, solitons and fractals, 39, 2509-2521, (2009) · Zbl 1197.65223
[20] Dehghan, M.; Hamidi, A.; Shakourifara, M., The solution of coupled Burgers equations using Adomian-pade technique, Applied mathematics and computation, 189, 1034-1047, (2007) · Zbl 1122.65388
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