Asymptotic stability of solitons for the Benjamin-Ono equation

Revista Matematica Iberoamericana

Volumn 25, Issue 3, 2009, Pages 909-970

  Kenig, Carlos E ^a   Martel, Yvan ^b  

^a UNIVERSITY OF CHICAGO   (United States)

^b LABORATOIRE DES SCIENCES DU CLIMAT ET DE L ENVIRONNEMENT   (France)

Author keywords

Asymptotic stability; Benjamin Ono equation; Liouville theorem; Soliton

Indexed keywords

EID: 72849111107     PISSN: 02132230     EISSN: 22350616     Source Type: Journal    
DOI: 10.4171/RMI/586     Document Type: Article

References (31)

1. AMICK, C. J. AND TOLAND, J. F.: Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane. Acta Math. 167 (1991), 107-126.
2. BENJAMIN, T. B.: Internal waves of permanent form in fluids of great depth. Journal of Fluid Mechanics 29 (1967), 559-592.
3. BENNETT, D. P., BROWN, R. W., STANSFIELD, S.E., STROUGHAIR, J. D. AND BONA, J. L.: The stability of internal solitary waves. Math. Proc. Cambridge Philos. Soc. 94 (1983), 351-379.
4. BONA, J. L., SOUGANIDIS, P.E. AND STRAUSS, W. A.: Stability and instability of solitary waves of Korteweg-de Vries type. Proc. Roy. Soc. London Ser. A 411 (1987), 395-412.
5. BURQ, N. AND PLANCHON, F.: On well-posedness for the Benjamin-Ono equation. Math. Ann. 340 (2008), no.3, 497-542.
6. CALDERÓN, A.-P.: Commutators of singular integral operators. Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092-1099.
7. COIFMAN, R. R. AND MEYER, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc. 212 (1975), 315-331.
8. HERR, S., IONESCU, A.D., KENIG, CE. AND KOCH, H.: Global solutions to dispersive nonlinear equations. Preprint.
9. GlNIBRE, J. AND VELO, G.: Commutator expansions and smoothing properties of generalized Benjamin-Ono equations. Ann. Inst. H. Poincaré Phys. Théor. 51 (1989), 221-229.
10. 1/2 of the sum of K solitons for the Benjamin-Ono equation. J. Math. Phys. 50 (2009), no.1, 013101, 14 pp.
11. IONESCU, A. D. AND KENIG, C. E.: Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J. Amer. Math. Soc. 20 (2007), 753-798.
12. KATO, T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equation. In Studies in applied mathematics, 93-128. Adv. Math. Suppl. Stud. 8, Academic Press, New York, 1983.
13. KENIG, C. E., PONCE, G. AND VEGA, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46 (1993), 527-620.
14. MARTEL, Y.: Linear problems related to asymptotic stability of solitons of the generalized KdV equations. SIAM J. Math. Anal. 38 (2006), 759-781.
15. MARTEL, Y. AND MERLE, F.: Instability of solitons for the critical generalized Korteweg-de Vries equation. Geom. Funct. Anal. 11 (2001), 74-123. (Pubitemid 33263744)
16. MARTEL, Y. AND MERLE, F.: A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl. (9) 79 (2000), 339-425.
17. MARTEL, Y. AND MERLE, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157 (2001), 219-254. (Pubitemid 33265217)
18. MARTEL, Y. AND MERLE, F.: Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18 (2005), no.1, 55-80.
19. MARTEL, Y. AND MERLE, F.: Asymptotic stability of solitons of the gKd V equations with a general nonlinearity. Math. Ann. 341 (2008), 391-427.
20. MARTEL, Y. AND MERLE, F.: Refined asymptotics around soliton for gKdV equations. Discrete Contin. Dyn. Syst. 20 (2008), 177-218.
21. MARTEL, Y., MERLE, F. AND TSAI, T.-P.: Stability and asymptotic stability in the energy space of the sum of N solitons for the subcritical gKdV equations. Comm. Math. Phys. 231 (2002), 347-373. (Pubitemid 36195010)
22. MATSUNO, Y.: The Lyapunov stability of the IV-soliton solutions in the Lax hierarchy of the Benjamin-Ono equation. J. Math. Phys. 47 (2006), 103505, 13pp.
23. NEVES, A. AND LOPES, O.: Orbital stability of double solitons for the Benjamin-Ono equation. Comm. Math. Phys. 262 (2006), 757-791.
24. PEGO, R. L. AND WEINSTEIN, M. I.: Asymptotic stability of solitary waves. Comm. Math. Phys. 164 (1994), 305-349.
25. PONCE, G.: Smoothing properties of solutions to the Benjamin-Ono equation. In Analysis and partial differential equations, 667-679. Lecture Notes in Pure and Appl. Math. 122. Dekker, New York, 1990.
26. STEIN, E.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series 30. Princeton Univ. Press, Princeton, 1970.
27. 1 (R). J. Hyperbohc Differ. Equ. 1 (2004), 27-49.
28. TOLAND, J. F.: The Peierls-Nabarro and Benjamin-Ono equations. J. Funct. Anal. 145 (1997), 136-150.
29. WEINSTEIN, M. I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985), 472-491.
30. WEINSTEIN, M. I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39 (1986), 51-67.
31. WEINSTEIN, M. I.: Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation. Comm. Partial Differential Equations 12 (1987), 1133-1173.