Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations

SIAM Journal on Mathematical Analysis

Volumn 32, Issue 5, 2001, Pages 1050-1080

  Mizumachi, Tetsu ^a  

^a YOKOHAMA CITY UNIVERSITY   (Japan)

Author keywords

Generalized Korteweg de Vries equation; Nonlinear scattering; Solitary waves

Indexed keywords

EID: 0035538185     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141098346827     Document Type: Article

References (37)

1. M. J. ABLOWITZ AND H. SEGUR, Solitons and the Inverse Scattering Transform, Stud. Appl. Math. 4, SIAM, Philadelphia, PA, 1981.
2. J. BERGH AND J. LÖFSTRÖM, Interpolation Spaces, Springer-Verlag, Berlin, New York, 1976.
3. T. B. BENJAMIN, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 328 (1972), pp. 153-183.
4. J. L. BONA, On the stability of solitary waves, Proc. Roy. Soc. London Ser. A, 344 (1975), pp. 363-374.
5. J. L. BONA, On solitary waves and their role in the evolution of long waves, in Applications of Nonlinear Analysis in the Physical Sciences, H. Amann, N. Bazley, and K. Kirchgässner, eds., Pitman, London, 1981, pp. 183-205.
6. J. L. BONA, V. A. DOUGALIS, O. A. KARAKASHIAN, AND W. R. MCKINNEY, The effect of dissipation on solutions of the generalized Korteweg-de Vries equation, J. Comput. Appl. Math., 74 (1996), pp. 127-154.
7. J. L. BONA, V. A. DOUGALIS, O. A. KARAKASHIAN, AND W. R. MCKINNEY, Numerical simulation of singular solutions of the generalized Korteweg-de Vries equation, in Mathematical Problems in the Theory of Water Waves Korteweg-de Vries Equation, Luminy, 1995, Contemp. Math. 200, AMS, Providence, RI, 1996, pp. 17-29.
8. J. L. BONA, P. E. SOUGANIDIS, AND W. A. STRAUSS, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London Ser. A, 411 (1987), pp. 395-412.
9. J. L. BONA AND A. SOYEUR, On the stability of solitary-wave solutions of model equations for long waves, J. Nonlinear Sci., 4 (1994), pp. 449-470.
10. V. S. BUSLAEV AND G. S. PERELMAN, Scattering for the nonlinear Schrödinger equation: States close to a soliton, St. Petersburg Math. J., 4 (1993), pp. 1111-1142.
11. F. M. CHRIST AND M. I. WEINSTEIN, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), pp. 87-109.
12. P. DEIFT AND X. ZHOU, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math., 137 (1993), pp. 295-368.
13. C. S. GARDNER, J. M. GREENE, M. D. KRUSKAL, AND R. M. MIURA, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), pp. 1095-1097.
14. C. S. GARDNER, J. M. GREENE, M. D. KRUSKAL, AND R. M. MIURA, Korteweg-de Vries equation and generalizations. VI. Methods for exact solution, Comm. Pure Appl. Math., 27 (1974), pp. 97-133.
15. J. GINIBRE, Y. TSUTSUMI, AND G. VELO, Existence and uniqueness of solutions for the generalized Korteweg-de Vries equation, Math. Z., 203 (1990), pp. 9-36.
16. N. HAYASHI AND P. NAUMKIN, Large time asymptotics of solutions to the generalized Korteweg-de Vries equation, J. Funct. Anal., 159 (1998), pp. 110-136.
17. L. HÖRMONDER, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, Heidelberg, 1983.
18. A. JENSEN AND T. KATO, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), pp. 583-611.
19. T. KATO, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Appl. Math., 8 (1983), pp. 93-128.
20. C. KENIG, G. PONCE, AND L. VEGA, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), pp. 527-620.
21. S. KLAINERMAN, Long time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal., 78 (1982), pp. 73-89.
22. S. KLAINERMAN AND G. PONCE, Global small amplitude solutions to nonlinear evolution equations, Comm. Pure. Appl. Math., 36 (1983), pp. 133-141.
23. D. J. KORTEWEG 41ND G. DE VRIES, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos. Mag. Ser. 5, 39 (1895), pp. 422-443.
24. J. R. MILLER, Spectral properties and time decay for an Airy operator with potential, J. Differential Equations, 141 (1997), pp. 102-121.
25. M. MURATA, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), pp. 10-56.
26. A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, Berlin, Heidelberg, 1983.
27. R. L. PEGO AND M. I. WEINSTEIN, Eigenvalues and instabilities of solitary waves, Philos. Trans. Roy. Soc. London A, 340 (1992), pp. 47-94.
28. R. L. PEGO AND M. I. WEINSTEIN, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), pp. 305-349.
29. G. PONCE AND L. VEGA, Nonlinear small data scattering for the generalized Korteweg-de Vries equation, J. Funct. Anal., 90 (1990), pp. 445-457.
30. M. A. RAMMAHA, On the asymptotic behavior of solutions of generalized Korteweg-de Vries equations, J. Math. Anal. Appl., 140 (1989), pp. 228-240.
31. P. C. SCHUUR, Asymptotic Analysis of Soliton Problems, Lecture Notes in Math. 1232, Springer-Verlag, Berlin, Heidelberg, New York, 1986.
32. J. SHATAH, Global existence of small solutions to nonlinear evolution equations, J. Differential Equations, 46 (1982), pp. 409-425.
33. A. SOFFER AND M. I. WEINSTEIN, Multichannel nonlinear scattering theory for nonintegrable equations, Comm. Math. Phys., 133 (1990), pp. 119-146.
34. A. SOFFER AND M. I. WEINSTEIN, Multichannel nonlinear scattering theory for nonintegrable equations II: The case of anisotropic potentials and data, 3. Differential Equations, 98 (1992), pp. 376-390.
35. W. A. STRAUSS, Dispersion of low-energy waves for two conservative equations, Arch. Rational Mech. Anal., 55 (1974), pp. 86-92.
36. W. A. STRAUSS, Nonlinear scattering theory, in Scattering Theory in Mathematical Physics, J. A. Lavita and J. P. Marchand, eds., Reidel Publishing, Dordrecht, Holland, 1974, pp. 53-78.
37. M. I. WEINSTEIN, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), pp. 51-68.