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Original Articles

A conservative exponential time differencing method for the nonlinear cubic Schrödinger equation

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Pages 230-251 | Received 05 May 2015, Accepted 15 Sep 2015, Published online: 29 Oct 2015

References

  • D. Anderson, Variational approach to nonlinear pulse propagation in optical fibers, Phys. Rev. A Gen. Phys. 3rd Ser. 27 (1983), pp. 3135–3145. doi: 10.1103/PhysRevA.27.3135
  • J. Argyris and M. Haase, An engineer's guide to soliton phenomena: Application of the finite element method, Comput. Methods Appl. Mech. Eng. 61 (1987), pp. 71–122. doi: 10.1016/0045-7825(87)90117-4
  • G. Beylkin, J.M. Keiser, and L. Vozovoi, A new class of time discretization schemes for the solution of nonlinear PDEs, J. Comput. Phys. 147 (1998), pp. 362–387. doi: 10.1006/jcph.1998.6093
  • H.P. Bhatt and A.Q.M. Khaliq, The locally extrapolated exponential time differencing LOD scheme for multidimensional reaction–diffusion systems, J. Comput. Appl. Math. 285 (2015), pp. 256–278. doi: 10.1016/j.cam.2015.02.017
  • A.G. Bratsos, A linearized finite-difference scheme for the numerical solution of the nonlinear cubic Schrödinger Equation, Korean J. Comput. Appl. Math. 8(3) (2001), pp. 459–467.
  • A.G. Bratsos, On the numerical solution of the nonlinear cubic Schrödinger equation, Int. J. Pure Appl. Math. Sci. 2(2) (2005), pp. 217–226.
  • A.G. Bratsos, A modified numerical scheme for the cubic Schrödinger equation, Numer. Methods Partial Differential Equations. 27 (2011), pp. 608–620. doi: 10.1002/num.20541
  • A.G. Bratsos, An improved second-order numerical method for the generalized Burgers–Fisher equation, ANZIAM J. 54(3) (2013), pp. 181–199. doi: 10.1017/S1446181113000138
  • A.G. Bratsos, M. Ehrhardt, and I.Th. Famelis, A discrete Adomian decomposition method for discrete nonlinear Schrödinger equations, Appl. Math. Comput. 197(1) (2008), pp. 190–205. doi: 10.1016/j.amc.2007.07.055
  • J.B. Chen and M.Z. Qin, Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation, Electron. Trans Numer. Anal. 12 (2001), pp. 193–204.
  • S.M. Cox and P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002), pp. 430–455. doi: 10.1006/jcph.2002.6995
  • I. Dăg, A quadratic B-spline finite element method for solving nonlinear Schrödinger equation, Comput. Methods Appl. Mech. Eng. 174 (1999), pp. 247–258. doi: 10.1016/S0045-7825(98)00257-6
  • Y. Dereli, The meshless kernel-based method of lines for the numerical solution of the nonlinear Schrödinger equation, Eng. Anal. Bound. Elem. 36 (2012), pp. 1416–1423. doi: 10.1016/j.enganabound.2012.02.018
  • N.M. Duy, D. Pan, N. Phan-Thien, and B.C. Khoo, Dissipative particle dynamics modeling of low Reynolds number incompressible flows, J. Rheol. 57(2) (2013), pp. 585–604. doi: 10.1122/1.4789444
  • K.B. Dyshe, Note on a modification to the nonlinear Schrodinger equation for application to deep water waves, P Roy. Soc. Lond. A Mat. 369(1736) (1979), pp. 105–114.
  • A.E.K. Elcoot, Nonlinear stability of an axial electric field: Effect of interfacial charge relaxation, Appl. Math. Modell. 34 (2010), pp. 1965–1983. doi: 10.1016/j.apm.2009.10.035
  • G. Fairweather and M. Khebchareon, Numerical methods for Schrödinger-type problems, in Trends in Industrial and Applied Mathematics, A.H. Siddiqi and M. Kocvara, eds., Kluwer Academic Publishers, Boston, 2002, pp. 219–250.
  • L.R.T. Gardner, G.A. Gardner, S.I. Zaki, and Z. El Sahrawi, B-spline finite element studies of the non-linear Schrödinger equation, Comput. Methods Appl. Mech. Eng. 108 (1993), pp. 303–318. doi: 10.1016/0045-7825(93)90007-K
  • A. Giusti-Suzor, F.H. Mies, L.F. DiMauro, E. Charron, and B. Yang, Dynamics of H2+ in intense laser fields, J. Phys. B: At. Mol. Opt. Phys. 28 (1995), pp. 309–339. doi: 10.1088/0953-4075/28/3/006
  • D.F. Griffiths, A.R. Mitchell, and J.LI. Morris, A numerical study of the non-linear Schrödinger equation, Comput. Meths. Appl. Mech. Engrg. 45 (1984), pp. 177–215. doi: 10.1016/0045-7825(84)90156-7
  • M.S. Ismail and T.R. Taha, A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation, Math. Comput. Simul. 74 (2007), pp. 302–311. doi: 10.1016/j.matcom.2006.10.020
  • F. Ivanauskas and M. Radz˘iūnas, On convergence and stability of the explicit difference method for solution of nonlinear Schrödinger equations, SIAM J. Numer. Anal. 36 (1999), pp. 1466–1481. doi: 10.1137/S0036142997319728
  • A.-K. Kassam and L.N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput. 26 (2005), pp. 1214–1233. doi: 10.1137/S1064827502410633
  • A. Korkmaz and I. Dağ, A differential quadrature algorithm for simulations of nonlinear Schrödinger equation, Comput. Math. Appl. 56(9) (2008), pp. 2222–2234. doi: 10.1016/j.camwa.2008.03.047
  • H. Lange, On Dysthe's nonlinear Schrodinger equation for deep water waves, Transp. Theory Statist. Phys. 29(3–5) (2000), pp. 509–524. doi: 10.1080/00411450008205888
  • X. Liang, A.Q.M. Khaliq, and Q. Sheng, Exponential time differencing Crank–Nicolson method with a quartic spline approximation for nonlinear Schrödinger equations, Appl. Math. Comput. 235 (2014), pp. 235–252. doi: 10.1016/j.amc.2014.02.063
  • Y.-X. Liu and H.-D. Zhang, Exponential time differencing methods with Chebyshev collocation for polymers confined by interacting surfaces, J. Chem. Phys. 140 (2014), p. 224101.
  • B.A. Malomed, D. Mihalache, F. Wise, and L. Torner, Spatiotemporal optical solitons, J. Opt. B: Quantum Semiclass. Opt. 7 (2005), pp. R53–R72. doi: 10.1088/1464-4266/7/5/R02
  • D.R. Mott, E.S. Oran, and B. van Leer, A quasi-steady-state solver for the stiff ordinary differential equations of reaction kinetics, J. Comput. Phys. 164 (2000), pp. 407–428. doi: 10.1006/jcph.2000.6605
  • P.L. Nash and L.Y. Chen, Efficient finite difference solutions to the time-dependent Schrödinger equation, J. Comput. Phys. 130 (1997), pp. 266–268. doi: 10.1006/jcph.1996.5589
  • D. Pathria and J.L.I. Morris, Pseudo–spectral solution of nonlinear Schrödinger equations, J. Comput. Phys. 87 (1990), pp. 108–125. doi: 10.1016/0021-9991(90)90228-S
  • Y.M. Samrout, New second- and fourth-order accurate numerical schemes for the nonlinear cubic Schrödinger equation, Int. J. Comput. Math. 84(11) (2007), pp. 1625–1651. doi: 10.1080/00207160701546668
  • A. Taflove, Computational Electrodynamics: The Finite Difference Time-Domain Method, Artech House, Boston, 1995.
  • E.H. Twizell, A.G. Bratsos, and J.C. Newby, A finite-difference method for solving the cubic Schrödinger equation, Math. Comput. Simul. 43 (1997), pp. 67–75. doi: 10.1016/S0378-4754(96)00056-0
  • M.C. Villet and G.H. Fredrickson, Efficient field-theoretic simulation of polymer solutions, J. Chem. Phys. 141 (2014), p. 224115. doi: 10.1063/1.4902886
  • A.M. Wazwaz, A reliable technique for solving linear and nonlinear Schrödinger equations by Adomian decomposition method, Bull. Inst. Math. Acad. Sinica 29(2) (2001), pp. 125–134.
  • A.-M. Wazwaz, A study on linear and nonlinear Schrödinger equations by the variational iteration method, Chaos Soliton Fract. 37(4) (2008), pp. 1136–1142. doi: 10.1016/j.chaos.2006.10.009
  • Z. Yan, Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres, Chaos Soliton Fract. 16 (2003), pp. 759–766. doi: 10.1016/S0960-0779(02)00435-6
  • J. Yang, Multisoliton pertubation theory for the Manakov equations and its application to nonlinear optics, Phys. Rev. E 59 (1999), pp. 2393–2405. doi: 10.1103/PhysRevE.59.2393
  • H.W. Yang, Z.K. Yang, C.-K. Zhu, Ai.P. Li, and X. You, Analysis of the photonic band gap of plasma photonic crystals with filmy structure, Optik 125 (2014), pp. 532–535. doi: 10.1016/j.ijleo.2013.07.032
  • V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media, Zh. Eksper. Teoret. Fiz. 61 (1971), pp. 118–134; Soviet Physics JETP, 34, 62–69 (1972).
  • X. Zhuansun, X. Ma, and Q. Liu, An exponential time differencing algorithm for the FDTD-PML analysis of nonlinear photonic bandgap structures, IEEE Trans. Magn. 50(2) (2014), p. 7004604. doi: 10.1109/TMAG.2013.2283227

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