Document Zbl 0811.65116

Automatic algorithm for the numerical inverse scattering transform of the Korteweg-de Vries equation. (English) Zbl 0811.65116

Math. Comput. Simul. 37, No. 4-5, 431-450 (1994).

Summary: The inverse scattering transform (IST) for the periodic Korteweg-de Vries (KdV) equation in the \(\mu\)-function representation is considered and numerical formulations are given (1) for determining the direct scattering transform spectrum of an input discrete wave train and (2) for reconstructing the wave train from the spectrum via the inverse scattering problem.
The advantage of the present method over previous approaches is that the numerical computations are automatic, i.e. one is guaranteed that the algorithm will search out and find all the nonlinear modes (to within the input numerical precision) of a given arbitrary, \(N\) degree-of-freedom wave train. The algorithms are most appropriate for the time series analysis of measured and computed data. One is thus numerically able to analyze an input time series with \(M\) discrete points; (1) to construct the IST spectrum, (2) to determine the \(N= M/2\) hyperelliptic function oscillation modes, (3) to reconstruct the input wave train by a linear superposition law and (4) to nonlinearly filter the input wave train.
Numerical details of the algorithm are discussed and an example for which \(N= 128\) is given.

Cited in 4 Documents

MSC:

65Z05	Applications to the sciences
65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53	KdV equations (Korteweg-de Vries equations)

Keywords:

automatic algorithm; periodic Korteweg-de Vries equation; inverse scattering transform; direct scattering transform spectrum; wave train; time series analysis

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References:

[1]	Korteweg, D. J.; de Vries, G., Philos. Mag. Ser. 5, 39, 422 (1895) · JFM 26.0881.02
[2]	Whitham, G. B., Linear and Nonlinear Waves (1974), Wiley-Interscience: Wiley-Interscience New York · Zbl 0373.76001
[3]	Miles, J. W., Ann. Rev. Fluid Mech., 12, 11 (1980) · Zbl 0463.76026
[4]	Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Phys. Rev. Lett., 19, 1095 (1967) · Zbl 1061.35520
[5]	Zakharov, V. E.; Manakov, S. V.; Novikov, S. P.; Pitayevsky, M. P., Theory of Solitons. The Method of the Inverse Scattering Problem (1984), Consultants Bureau: Consultants Bureau New York · Zbl 0598.35002
[6]	Ablowitz, M. J.; Segur, H., Solitons and the Inverse Scattering Transform (1981), SIAM: SIAM Philadelphia · Zbl 0299.35076
[7]	Dodd, R. K.; Eilbeck, J. E.; Gibbon, J. D.; Morris, H. C., Solitons and Nonlinear Wave Equations (1982), Academic Press: Academic Press London · Zbl 0496.35001
[8]	Newell, A. C., Solitons in Mathematics and Physics (1985), SIAM: SIAM Philadelphia · Zbl 0565.35003
[9]	Degasperis, A., (Osborne, A. R., Nonlinear Topics in Ocean Physics (1991), Elsevier: Elsevier Amsterdam)
[10]	Dubrovin, B. A.; Novikov, S. P., (Sov. Phys. JETP, 40 (1974)), 1058
[11]	Dubrovin, B. A.; Matveev, V. B.; Novikov, S. P., Russian Math. Surv., 31, 59 (1976) · Zbl 0346.35025
[12]	Flaschka, H.; McLaughlin, D. W., Prog. Theoret. Phys., 55, 438 (1976) · Zbl 1109.35374
[13]	McKean, H. P.; Trubowitz, E., Comm. Pure Appl. Math., 29, 143 (1976) · Zbl 0339.34024
[14]	Taha, T. R.; Ablowitz, M. J., J. Comp. Phys., 55, 192 (1984) · Zbl 0541.65081
[15]	Taha, T. R.; Ablowitz, M. J., J. Comp. Phys., 55, 231 (1984) · Zbl 0541.65083
[16]	Bishop, A. R.; Forest, M. G.; McLaughlin, D. W.; Overman, E. A., Physica D, 18, 293 (1986)
[17]	Bishop, A. R.; Lomdahl, P. S., Physica D, 18, 26 (1986)
[18]	Flesch, R.; Forest, M. G.; Sinha, A., Physica D, 48, 169 (1991) · Zbl 0728.65098
[19]	McLaughlin, D. W.; Schober, C. M., Physica D, 57, 447 (1992) · Zbl 0760.35045
[20]	Terrones, G.; McLaughlin, D. W.; Overman, E. A.; Pearlstein, A., SIAM J. Appl. Math., 50, 791 (1990) · Zbl 0702.58067
[21]	Osborne, A. R., (Osborne, A. R., Nonlinear Topics in Ocean Physics (1991), North-Holland: North-Holland Amsterdam), 669
[22]	Osborne, A. R., (J. Comp. Phys., 94 (1991)), 284 · Zbl 0729.65097
[23]	Provenzale, A.; Osborne, A. R., (J. Comp. Phys., 94 (1991)), 314 · Zbl 0729.65098
[24]	Osborne, A. R.; Bergamasco, L., Nuovo Cimento B, 85, 229 (1985)
[25]	Osborne, A. R.; Bergamasco, L., Physica D, 18, 26 (1986) · Zbl 0601.35101
[26]	Osborne, A. R.; Segre, E., Physica D, 44, 575 (1990) · Zbl 0707.65089
[27]	Osborne, A. R., Phys. Rev. Lett., 3115 (1993)
[28]	Zabusky, N. J.; Kruskal, M. D., Phys. Rev. Lett., 15, 240 (1965) · Zbl 1201.35174
[29]	Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T., Numerical Recipes (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0727.65001
[30]	Osborne, A. R.; Segre, E.; Boffeta, G.; Cavaleri, L., Phys. Rev. Lett., 64, 1733 (1991)
[31]	Osborne, A. R., J. Comp. Phys. (1993)
[32]	Ablowitz, M. J.; Ladik, J., (J. Math. Phys., 16 (1975)), 598 · Zbl 0296.34062
[33]	Ablowitz, M. J.; Ladik, J., (J. Math. Phys., 17 (1976)), 1011 · Zbl 0322.42014
[34]	Osborne, A. R.; Petti, M., Phys. Rev. E, 47, 1035 (1993)
[35]	Osborne, A. R., Phys. Rev. E, 48, 296 (1993)
[36]	Osborne, A. R., Phys. Lett. A, 176, 75 (1993)
[37]	Osborne, A. R., (Müller, P.; Henderson, D., Proceedings of the Aha Huliko’a Hawaiian Winter Workshop (1993), University of Hawaii Press)

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