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The Korteweg-de Vries equation and beyond. (English) Zbl 0842.58045
The author reviews a new method for linearizing the initial-boundary value problem of the Korteweg-de Vries (KdV) equation on the semi-infinite line for decaying initial and boundary data. The author also presents a novel class of physically important integrable equations. These equations, which include generalizations of the KdV, of the modified KdV, of the nonlinear Schrödinger and of the \(N\)-wave interactions, are as generic as their celebrated counterparts and it appears that they describe certain physical situations more accurately.
Reviewer: Y.Kozai (Tokyo)

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI
[1] Gardner, C. S., Green, J. M., Kruskal, M. D., and Miura, R. M.:Phys. Rev. Lett. 19 (1967), 1095. · Zbl 1103.35360 · doi:10.1103/PhysRevLett.19.1095
[2] For recent developments, see A. S. Fokas and V. E. Zakharov (eds),Important Developments in Soliton Theory, Springer-Verlag, New York, 1993. · Zbl 0801.00009
[3] Lax, P.:Comm. Pure Appl. Math. 21 (1968), 467. · Zbl 0162.41103 · doi:10.1002/cpa.3160210503
[4] Bona, J., Pritchard, W. G., and Scott, L. R.:Philos. Trans. Royal Soc. London, A 302 (1981), 457-510. · Zbl 0497.76023 · doi:10.1098/rsta.1981.0178
[5] Fokas, A. S.:Proceedings of the III Potsdam-V Kiev International Workshop 1991, Springer-Verlag, Berlin, 1992;
[6] Fokas, A. S. and Its, A. R.:Phys. Rev. Lett. 68 (1992), 3117. · Zbl 0969.35537 · doi:10.1103/PhysRevLett.68.3117
[7] Fokas, A. S. and Its, A. R.: An initial-boundary value problem for the KdV equation, Preprint, 1993. · Zbl 0826.35119
[8] Taniuti, T. and Wei, C. C.:J. Phys. Soc. Japan 24 (1968), 941; · doi:10.1143/JPSJ.24.941
[9] Taniuti, T.:Progr. Theor. Phys. 55 (1974), 1; · doi:10.1143/PTPS.55.1
[10] Kodama, Y. and Taniuti, T.:J. Phys. Soc. Japan 45 (1978), 298. · Zbl 1334.76103 · doi:10.1143/JPSJ.45.298
[11] Calogero, F.: Why are certain nonlinear PDEs both widely applicable and integrable, in V. E. Zakharov (ed.),What is Integrability, Springer-Verlag, New York, 1992. · Zbl 0808.35001
[12] Fokas, A. S.: Moderately long waves of moderate amplitude and integrable, Preprint, 1994.
[13] Fokas, A. S.: Asymptotic integrability, Preprint, 1994.
[14] Whitham, G. B.:Linear and Nonlinear Waves, Wiley-Intersci., New York, 1974. · Zbl 0373.76001
[15] Bona, J. and Winther, R.:SIAM J. Math. Anal. 14 (1983), 1056-1106. · Zbl 0529.35069 · doi:10.1137/0514085
[16] Bona, J. and Winther, R.:Differ. Integral Equations 2 (1989), 228-250.
[17] Fokas, A. S. and Its, A. R.: The linearization of the initial-boundary value problem of the nonlinear Schrödinger equation, to appear inSIAM J. Math. Anal. (1994). · Zbl 0832.35125
[18] Sung, L.: Solution of the initial-boundary value problem of NLS using PDE techniques, Preprint, Clarkson University, 1994.
[19] Fuchssteiner, B. and Fokas, A. S.:Physica 4D (1981), 47.
[20] Holm, D. and Camassa, R.:Phys. Rev. Lett. 71 (1993), 1671.
[21] Fokas, A. S. and Fuchssteiner, B.:Lett. Nuovo Cimento 28 (1980), 299. · doi:10.1007/BF02798794
[22] Gel’fand, I. M. and Dorfman, I.:Funct. Anal. Appl. 13 (1979), 13;14 (1980), 71.
[23] Rosenau, P.:Phys. Rev. Lett. (1994).
[24] Fokas, A. S. and Santini, P. M.: An inverse acoustic problem and linearization of moderate amplitude dispersive waves, Preprint, 1994.
[25] Kodama, Y.:Phys. Lett A 112 (1985), 193. · doi:10.1016/0375-9601(85)90500-6
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