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Pseudo-spectral solution of nonlinear Schrödinger equations. (English) Zbl 0691.65090

This paper compares four discretization methods for solving the generalized nonlinear Schrödinger equation \(iu_ t+u_{xx}+q_ c| u|^ 2u+q_ q| u|^ 4u+iq_ m| u|^ 2_ xu+iq_ u| u|^ 2u_ x=0\) where \(q_ c\), \(q_ q\), \(q_ m\) and \(q_ u\) are real parameters. An initial value problem is considered so that \(u(x,0)=u_ 0(x)\) is specified. The solution may be represented in a Fourier series where the coefficients depend on time and the methods differ on their formalism connecting the time variable with the space function discretization at n collocation points. Numerical examples are given.
Reviewer: B.Burrows

MSC:

65Z05 Applications to the sciences
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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[1] Johnson, R. S., (Proc. Roy. Soc. London A, 375 (1977)), 131
[2] Kakutani, T.; Michihiro, K., J. Phys. Soc. Japan, 52, 4129 (1983)
[3] Calogero, F.; Eckhaus, W., Inverse Probl., 3, 229 (1987) · Zbl 0645.35087
[4] Hasimoto, H.; Ono, H., J. Phys. Soc. Japan, 33, 805 (1972)
[5] Strauss, W. A., The Nonlinear Schrödinger Equation, (de la Penha, G. M.; Medeiros, L. A., Contemporary Developments in Continuum Mechanics (1978), North-Holland: North-Holland New York), 452
[6] Lamb, G. L., Elements of Soliton Theory (1980), Wiley: Wiley Toronto · Zbl 0445.35001
[7] Kaup, D.; Newell, A. C., J. Math. Phys., 19, 798 (1978) · Zbl 0383.35015
[8] Tanaka, M., J. Phys. Soc. Japan, 51, 2686 (1982)
[9] Cowan, S.; Enns, R. H.; Rangnekar, S. S.; Sanghera, S. S., Canad. J. Phys., 64, 311 (1986)
[10] Kundu, A., Physica D, 25, 399 (1987) · Zbl 0612.76002
[11] Calogero, F.; Eckhaus, W., Inverse Probl., 3, L27 (1987) · Zbl 0645.35089
[12] Pathria, D.; Morris, J. Ll., Exact solutions for a generalized nonlinear Schrödinger equation, Phys. Scr., 39, 673 (1989)
[13] Sanz-Serna, J. M.; Manoranjan, V. S., J. Comput. Phys., 52, 273 (1983) · Zbl 0514.65085
[14] Delfour, M.; Fortin, M.; Payre, G., J. Comput. Phys., 44, 277 (1981) · Zbl 0477.65086
[15] Sanz-Serna, J. M.; Verwer, J. G., IMA J. Numer. Anal., 6, 25 (1986) · Zbl 0593.65087
[16] Tourigny, Y.; Morris, L. Ll., J. Comput. Phys., 76, 103 (1988) · Zbl 0641.65090
[17] Cooley, J. W.; Tukey, J. W., Math. Comput., 19, 297 (1965) · Zbl 0127.09002
[18] Tadmor, E., SIAM J. Numer. Anal., 23, 1 (1986) · Zbl 0613.65017
[19] Strang, G., SIAM J. Numer. Anal., 5, 506 (1968) · Zbl 0184.38503
[20] Hardin, R. H.; Tappert, F. D., SIAM Rev. Chron., 15, 423 (1973)
[21] Taha, T. R.; Ablowitz, M. J., J. Comput. Phys., 55, 192 (1984) · Zbl 0541.65081
[22] Weideman, J. A.C.; Herbst, B. M., SIAM J. Numer. Anal., 23, 485 (1986) · Zbl 0597.76012
[23] Cloot, A.; Herbst, B. M.; Weideman, J. A.C., (Leon, J. J.-P., Nonlinear Evolutions—Proceedings of the IVth Workshop in Nonlinear Evolution Equations and Dynamical Systems (1987), World Scientific: World Scientific Singapore), 637
[24] Herbst, B. M.; Mitchell, A. R.; Weideman, J. A.C., J. Comput. Phys., 60, 263 (1985) · Zbl 0589.65083
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