Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations

Communications in Mathematical Physics

Volumn 231, Issue 2, 2002, Pages 347-373

  Martel, Yvan ^a   Merle, Frank ^a,b   Tsai, Tai Peng ^c  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

^b INSTITUT UNIVERSITAIRE DE FRANCE   (France)

^c NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036882736     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00220-002-0723-2     Document Type: Article

References (25)

1. Benjamin, T.B.: The stability of solitary waves. Proc. Roy. Soc. London A 328, 153-183 (1972)
2. Bona, J.L., Souganidis, P.E., Strauss, W.A.: Stability and instability of solitary waves of Korteweg-de Vries type. Proc. R. Soc. Lond. 411, 395-412 (1987)
3. Buslaev, V.S., Perelman, G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. (English) In: Nonlinear Evolution Equations, Uraltseva, N.N. ed., Transl., Ser. 2, Am. Math. Soc. 164, Providence, RI: Am. Math. Soc., 1995, pp. 75-98
4. Dix, D.B., McKinney, W.R.: Numerical computations of self-similar blow up solutions of the generalized Korteweg-de Vries equation. Diff. Int. Eq. 11, 679-723 (1998)
5. Ei, S.-I., Ohta, T.: Equation of motion for interacting pulses. Phys. Rev. E 50, 4672-4678 (1994)
6. Ginibre, J., Tsutsumi, Y.: Uniqueness of solutions for the generalized Korteweg-de Vries equation. SIAM J. Math. Anal. 20, 1388-1425 (1989)
7. Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. J. Funct Anal. 74, 160-197 (1987)
8. Kato, T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Adv. in Math. Supplementary Studies, Studies in Applied Math. 8, 93-128 (1983)
9. Kenig, C.E., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46, 527-620 (1993)
10. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 539, 422-443 (1895)
11. Lamb Jr., G.L.: Element of Soliton Theory. New York: John Wiley & Sons, 1980
12. Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467-490 (1968)
13. Maddocks, J.H., Sachs, R.L.: On the stability of KdV multi-solitons. Comm. Pure Appl. Math. 46, 867-901 (1993)
14. Martel, Y., Merle, F.: A Liouville Theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl. 79, 339-425 (2000)
15. Martel, Y., Merle, F.: Instability of solitons for the critical generalized Korteweg-de Vries equation. Geom. Funct. Anal. 11, 74-123 (2001)
16. Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Rational Mech. Anal. 157, 219-254 (2001)
17. Martel, Y., Merle, F.: Stability of the blow up profile and lower bounds on the blow up rate for the critical generalized KdV equation. Ann. of Math. 155, 235-280 (2002)
18. 2-critical generalized KdV equation. J. Am. Math. Soc. 15, 617-664 (2002)
19. Merle, F.: Construction of solutions with exactly k blow up points for the Schrödinger equation with critical nonlinearity. Commun. Math. Phys. 129, 223-240 (1990)
20. Merle, F.: Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Am. Math. Soc. 14, 555-578 (2001)
21. Miura, R.M.: The Korteweg-de Vries equation: a survey of results. SIAM Rev. 18, 412-459 (1976)
22. Pego, R.L., Weinstein, M.I.: Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305-349 (1994)
23. Perelman, G.S.: Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations. (English) In: Spectral Theory, Microlocal Analysis, Singular Manifolds. Demuth, Michael et al., eds., Math. Top. 14, Berlin: Akademie Verlag, 1997, pp. 78-137
24. Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16, 472-491 (1985)
25. Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure. Appl. Math. 39, 51-68 (1986)