Abstract
We review a new method for linearizing the initial-boundary value problem of the KdV on the semi-infinite line for decaying initial and boundary data. We also present 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 theN-wave interactions, are as generic as their celebrated counterparts and, furthermore it appears that they describe certain physical situations more accurately.
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Gardner, C. S., Green, J. M., Kruskal, M. D., and Miura, R. M.:Phys. Rev. Lett. 19 (1967), 1095.
For recent developments, see A. S. Fokas and V. E. Zakharov (eds),Important Developments in Soliton Theory, Springer-Verlag, New York, 1993.
Lax, P.:Comm. Pure Appl. Math. 21 (1968), 467.
Bona, J., Pritchard, W. G., and Scott, L. R.:Philos. Trans. Royal Soc. London, A 302 (1981), 457–510.
Fokas, A. S.:Proceedings of the III Potsdam-V Kiev International Workshop 1991, Springer-Verlag, Berlin, 1992;
Fokas, A. S. and Its, A. R.:Phys. Rev. Lett. 68 (1992), 3117.
Fokas, A. S. and Its, A. R.: An initial-boundary value problem for the KdV equation, Preprint, 1993.
Taniuti, T. and Wei, C. C.:J. Phys. Soc. Japan 24 (1968), 941;
Taniuti, T.:Progr. Theor. Phys. 55 (1974), 1;
Kodama, Y. and Taniuti, T.:J. Phys. Soc. Japan 45 (1978), 298.
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.
Fokas, A. S.: Moderately long waves of moderate amplitude and integrable, Preprint, 1994.
Fokas, A. S.: Asymptotic integrability, Preprint, 1994.
Whitham, G. B.:Linear and Nonlinear Waves, Wiley-Intersci., New York, 1974.
Bona, J. and Winther, R.:SIAM J. Math. Anal. 14 (1983), 1056–1106.
Bona, J. and Winther, R.:Differ. Integral Equations 2 (1989), 228–250.
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).
Sung, L.: Solution of the initial-boundary value problem of NLS using PDE techniques, Preprint, Clarkson University, 1994.
Fuchssteiner, B. and Fokas, A. S.:Physica 4D (1981), 47.
Holm, D. and Camassa, R.:Phys. Rev. Lett. 71 (1993), 1671.
Fokas, A. S. and Fuchssteiner, B.:Lett. Nuovo Cimento 28 (1980), 299.
Gel'fand, I. M. and Dorfman, I.:Funct. Anal. Appl. 13 (1979), 13;14 (1980), 71.
Rosenau, P.:Phys. Rev. Lett. (1994).
Fokas, A. S. and Santini, P. M.: An inverse acoustic problem and linearization of moderate amplitude dispersive waves, Preprint, 1994.
Kodama, Y.:Phys. Lett A 112 (1985), 193.
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Fokas, A.S. The Korteweg-de Vries equation and beyond. Acta Appl Math 39, 295–305 (1995). https://doi.org/10.1007/BF00994638
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DOI: https://doi.org/10.1007/BF00994638