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A numerical study of the nonlinear Schrödinger equation involving quintic terms. (English) Zbl 0685.65110

Solutions of the cubic-quintic Schrödinger equation of the form \(u_ t=i{\mathcal L}u+i{\mathcal N}(u)u\) where \({\mathcal L}u:=u_{xx}\) and \({\mathcal N}(u):=q_ c| u|^ 2+q_ q| u|^ 4\) are studied. Sufficient conditions for bounded solutions are derived by using conservation properties (conservation of energy and of the second quantity). A pseudospectral scheme is proposed which adaptively adjusts the number of degrees of freedom. The number of active Fourier modes may vary considerably during the computation. Boundedness of the solution in the cases of \(q_ c>0\) and \(q_ q>0,\) \(q_ c<0\) and \(q_ q<0,\) \(q_ c>0\) and \(q_ q<0\) and \(q_ c<0\) and \(q_ q>0\) is discussed. Conditions under which instability remains bounded are analyzed. Numerical examples are computed and shown graphically.
Reviewer: V.Burjan

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] Glassey, R. T., J. Math. Phys., 18, 1794 (1977)
[2] Herbst, B. M.; Morris, J. L.L.; Mitchell, A. R., J. Comput. Phys., 60, 282 (1985)
[3] Johnson, R. S., (Proc. Roy. Soc. London Ser. A, 357 (1977)), 131
[4] Kakutani, T.; Michihiro, K., J. Phys. Soc. Japan, 52, 4129 (1983)
[5] Newell, A. C., (Solitons in Mathematics and Physics Regional Conference Series in Applied Mathematics No. 48 (1985), SIAM, CBMS-NSF: SIAM, CBMS-NSF Philadelphia)
[6] Patterson, G. S.; Orszag, S. A., Phys. Fluids, 14, 2538 (1971)
[7] Salu, Y.; Knorr, G., J. Comput. Phys., 17, 68 (1975)
[8] Sanz-Serna, J. M.; Verwer, J. G., IMA J. Numer. Anal., 6, 25 (1986)
[9] Taha, T. R.; Ablowitz, M. J., J. Comput. Phys., 55, 203 (1984)
[10] Taha, T. R.; Ablowitz, M. J., J. Comput. Phys., 55, 203 (1984)
[11] Tappert, F. D., (Lecture in Appl. Math., Vol. 15 (1974), Amer. Math. Soc.,: Amer. Math. Soc., Cambridge, MA), 215
[12] Thyagaraja, A., Recurrence Phenomena and the Number of Effective Degrees of Freedom in Nonlinear Wave motions, (Debnath, L., Nonlinear Waves (1983), Cambridge Univ. Press: Cambridge Univ. Press London), Chap. 17 · Zbl 0542.76027
[13] Weideman, J. A.C., (Ph.D. thesis (1986), Dept. Applied Mathematics, University of the Orange Free State: Dept. Applied Mathematics, University of the Orange Free State Bloemfontein), (unpublished)
[14] Weinstein, (Ph.D. thesis (1982), New York University)
[15] Yuen, H. C.; Ferguson, W. E., Phys. Fluids, 21, 1275 (1978)
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