×

zbMATH — the first resource for mathematics

Semi-implicit operator splitting Padé method for higher-order nonlinear Schrödinger equations. (English) Zbl 1102.65093
Summary: We propose a semi-implicit finite difference operator splitting Padé (OSPD) method for solving the higher-order nonlinear Schrödinger equation which describes the optical soliton wave propagation in fibers. The method achieves fourth order of accuracy in space and is proven to be stable by linear stability analysis. Numerical experiments and comparisons are investigated to show the advantages and effectivity of the OSPD method. Some interesting collision behaviors are also observed.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hasegawa, A., Optical solitons in fibers, (1989), Springer Heidelberg
[2] Porsezian, K.; Nakkeeran, K., Optical solitons in presence of Kerr dispersion and self-frequency shift, Phys. rev. lett., 76, 3955-3958, (1996)
[3] Agrawal, G.P., Nonlinear fiber optics, (2001), Academic Press San Diego
[4] Pathria, D.; Morris, J.L., Pseudo-spectral solution of nonlinear Schrödinger equations, J. comput. phys., 87, 108-125, (1990) · Zbl 0691.65090
[5] Chen, J.B.; Qin, M.Z.; Tang, Y.F., Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation, Comp. math. appl., 43, 1095-1106, (2002) · Zbl 1050.65127
[6] Gardner, L.; Gardner, G.; Zaki, S.; Sahrawi, Z.E., B-spline finite element studies of the non-linear Schrödinger equation, Comput. methods appl. mech. eng., 108, 303-318, (1993) · Zbl 0842.65083
[7] Taha, T.R.; Ablowitz, M.J., Analytical and numerical aspects of certain nonlinear evolution equations, II: numerical, nonlinear Schrödinger equation, J. comput. phys., 55, 203-230, (1984) · Zbl 0541.65082
[8] Chang, Q.S.; Jia, E.H.; Sun, W., Difference schemes for solving the generalized nonlinear Schrödinger equation, J. comp. phys., 148, 397-415, (1999) · Zbl 0923.65059
[9] McLachlan, R.I.; Quispel, G.R.W., Splitting methods, Acta numer., 11, 341-434, (2002) · Zbl 1105.65341
[10] Yoshida, H., Construction of higher order symplectic integrators, Phys. lett. A, 150, 262-268, (1990)
[11] Holden, H.; Karlsen, K.H.; Risebroz, N.H., Operator splitting methods for generalized Korteweg-de Vries equations, J. comp. phys., 153, 203-222, (1999) · Zbl 0947.65102
[12] Bao, W.Z.; Jin, S.; Markowich, P.A., On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. comp. phys., 175, 487-524, (2002) · Zbl 1006.65112
[13] Bao, W.Z.; Jin, S.; Markowich, P.A., Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semi-classical regimes, SIAM J. sci. comp., 25, 27-64, (2003) · Zbl 1038.65099
[14] Blow, K.J., System analysis using the split operator method, (), 127-140
[15] Wang, H.Q., Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations, Appl. math. comput., 170, 17-35, (2005) · Zbl 1082.65570
[16] Lele, S.K., Compact finite difference schemes with spectral-like resolution, J. comput. phys., 103, 16-42, (1992) · Zbl 0759.65006
[17] M.W. Reinsch, A simple expression for the terms in the Baker-Campbell-Hausdorff series, preprint, 13 January 2000. http://arxiv.org/abs/math-ph/9905012/. · Zbl 0974.22015
[18] Sanders, B.F.; Katopodes, N.D.; Boyd, J.P., Spectral modeling of nonlinear dispersive waves, J. hydraul. eng., 124, 2-12, (1998)
[19] Muslu, G.M.; Erbay, H.A., Higher-order split-step Fourier schemes for the generalized Schrödinger equation, Math. comput. simul., 67, 581-595, (2005) · Zbl 1064.65117
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.