Large Time Asymptotics of Solutions to the Generalized Korteweg-de Vries Equation

Journal of Functional Analysis

Volumn 159, Issue 1, 1998, Pages 110-136

  Hayashi, Nakao ^a   Naumkin, Pavel I ^b  

^a TOKYO UNIVERSITY OF SCIENCE   (Japan)

^b UNIVERSIDAD MICHOACANA DE SAN NICOLÁS DE HIDALGO   (Mexico)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0007323978     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1006/jfan.1998.3291     Document Type: Article

References (30)

1. Ablowitz M J., Segur H. Solitons and the Inverse Scattering Transform. 1981;SIAM, Philadelphia.
2. Bergh J., Löfström J. Interpolation Spaces. 1976;Springer-Verlag, Berlin/Heidelberg/New York.
3. Bona J. L., Saut J.-C. Dispersive blow-up of solutions of generalized Korteweg-de Vries equation. J. Differential Equations. 103:1993;3-57.
4. de Bouard A., Hayashi N., Kato K. Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 12:1995;673-725.
5. Christ F. M., Weinstein M. I. Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation. J. Funct. Anal. 100:1991;87-109.
6. Constantin P., Saut J.-C. Local smoothing properties of dispersive equations. J. Amer. Math. Soc. 1:1988;413-446.
7. Craig W., Kapeller K., Strauss W. A. Gain of regularity for solutions of KdV type. Ann. Inst. H. Poincaré Anal. Non Linéaire. 9:1992;147-186.
8. Deift P., Zhou X. A steepest descent method for oscillatory Riemann-Hilbert problems Asymptotics for the MKdV equation. Ann. Math. 137:1992;295-303.
9. Fedoryuk M. V. Asymptotics: Integrals and Series. 1987;Nauka, Moscow.
10. Friedman A. Partial Differential Equations. 1969;Holt, Rinehart and Winston, New York.
11. Ginibre J., Tsutsumi Y., Velo G. Existence and uniqueness of solutions for the generalized Korteweg-de Vries equation. Math Z. 203:1990;9-36.
12. Hayashi N. Analyticity of solutions of the Korteweg-de Vries equation. SIAM J. Math. Anal. 22:1991;1738-1745.
13. N. Hayashi, P. I. Naumkin, Asymptotics for large time behavior of solutions to the generalized Benjamin-Ono equation, 1996.
14. Kato T. On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Studies in Applied Mathematics. Advances in Mathematics Supplementary Studies. 8:1983;. p. 93-128.
15. Kenig C. E., Ponce G., Vega L. On the (generalized) Korteweg-de Vries equation. Duke Math. J. 59:1989;585-610.
16. Kenig C. E., Ponce G., Vega L. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle. Comm. Pure Appl. Math. 46:1993;527-620.
17. Klainerman S. Long time behavior of solutions to nonlinear evolution equations. Arch. Rat. Mech. Anal. 78:1982;73-89.
18. Klainerman S., Ponce G. Global small amplitude solutions to nonlinear evolution equations. Comm. Pure Appl. Math. 36:1983;133-141.
19. Kruzhkov S. N., Faminskii A. V. Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation. Mat. Sb. 48:1984;391-421.
20. Lions J. L. Quelques méthodes de résolution des problèmes aux limites non linéaires. 1969;Dunod and Gauthier-Villars, Paris.
21. Naumkin P. I., Shishmarev I. A. Asymptotic behaviour ast. Mat. Sb. 187:1996;695-733.
22. Ponce G., Vega L. Nonlinear small data scattering for the generalized Korteweg- de Vries equation. J. Funct. Anal. 90:1990;445-457.
23. Rammaha M. A. On the asymptotic behavior of solutions of generalized Korteweg- de Vries equation. J. Math. Anal. Appl. 140:1989;228-240.
24. Saut J. C. Sur quelques généralisations de l'équation de Korteweg-de Vries. J. Math. Pure Appl. 58:1979;21-61.
25. Sidi A., Sulem C., Sulem P. L. On the long time behavior of a generalized KdV equation. Acta Applicandae Math. 7:1986;35-47.
26. Shatah J. Global existence of small solutions to nonlinear evolution equations. J. Differential Equations. 46:1982;409-425.
27. Stein E. M. Singular Integral and Differentiability Properties of Functions. Princeton Math. Series. 30:1970;Princeton Univ. Press, Princeton.
28. Strauss W. A. Dispersion of low-energy waves for two conservative equations. Arch. Rational Mech. Anal. 55:1974;86-92.
29. Strauss W. A. Nonlinear scattering theory at low energy. J. Funct. Anal. 41:1981;110-133.
30. Tsutsumi M. On global solutions of the generalized Korteweg-de Vries equation. Publ. Res. Inst. Math. Sci. 7:1972;329-344.