zbMATH — the first resource for mathematics

Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation. (English) Zbl 0541.65082
Summary: [For part I see the article reviewed above.]
Various numerical methods are employed in order to approximate the nonlinear Schrödinger equation, namely: (i) The classical explicit method, (ii) hopscotch method, (iii) implicit-explicit method, (iv) Crank-Nicolson implicit scheme, (v) the Ablowitz-Ladik scheme, (vi) the split step Fourier method, and (vii) pseudospectral (Fourier) method. Comparisons between the Ablowitz-Ladik scheme, which was developed using notions of the inverse scattering transform, and the other utilized schemes are obtained.

65Z05 Applications to the sciences
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
Full Text: DOI
[1] Yajima, N.; Oikawa, M.; Satsuma, J.; Namba, C., Modulated langmiur waves and nonlinear Landau damping, Rep. res. inst. appl. mech., XXII, No. 70, (1975)
[2] Karpman, V.I.; Krushkal, E.M., Modulated waves in nonlinear dispersive media, Soviot phys. JETP, 28, 277, (1969)
[3] Yajima, N.; Outi, A., A new example of stable solitary waves, Prog. theor. phys., 45, 1997, (1971)
[4] Satsuma, J.; Yajima, N., Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media, Prog. theor. phys. supp., 55, (1974)
[5] \scF. Tappert, private communication, 1981.
[6] Hardin, R.H.; Tappert, F.D.; Hardin, R.H.; Tappert, F.D., Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, (), SIAM rev. chronicle, 15, 423, (1973)
[7] Ablowitz, M.; Ladik, J., Stud. appl. math., 55, 213, (1976)
[8] Ablowitz, M.; Ladik, J., Stud. appl. math., 57, 1-12, (1977)
[9] Eilbeck, J.C., Numerical studies of solitons, (), 28-43 · Zbl 0325.65054
[10] Greig, I.S.; Morris, J.Ll, J. comp. phys., 20, 60-84, (1976)
[11] Fornberg, B.; Whitham, G.B., Phil. trans. roy. soc., 289, 373, (1978)
[12] \scH. Segur, private communication, 1981.
[13] Hirota, R., Exact envelope-soliton solutions of a nonlinear wave equation, J. math. phys., 14, 805, (1973) · Zbl 0257.35052
[14] Richtmyer, R.D.; Morton, K.W., Difference methods for initial value problems, (1967), Wiley-Interscience New York · Zbl 0155.47502
[15] \scM. D. Kruskal, private communication, 1981.
[16] Smith, G.D., Numerical solution of partial differential equations, (1965), Oxford Univ. Press New York · Zbl 0123.11806
[17] Cooley, J.W.; Lewis, P.A.W.; Welch, P.D., IEEE trans. educ. E-12, 1, 27-34, (1969)
[18] McCraken, D.D.; William, S.D., Numerical methods and FORTRAN programming with applications in engineering and science, (1964), Wiley New York
[19] Delfour, M.; Fortin, M.; Payre, G., Finite-difference solutions of a nonlinear Schrödinger equation, J. comp. phys., 44, 277-288, (1981) · Zbl 0477.65086
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.