## Revista Matemática Iberoamericana

### Asymptotic stability of solitons for the Benjamin-Ono equation

#### Abstract

In this paper, we prove the asymptotic stability of the family of solitons of the Benjamin-Ono equation in the energy space. The proof is based on a Liouville property for solutions close to the solitons for this equation, in the spirit of [Martel, Y. and Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157 (2001), 219-254], [Martel, Y. and Merle, F.: Asymptotic stability of solitons of the gKdV equations with a general nonlinearity. Math. Ann. 341 (2008), 391-427]. As a corollary of the proofs, we obtain the asymptotic stability of exact multi-solitons.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 25, Number 3 (2009), 909-970.

Dates
First available in Project Euclid: 3 November 2009

https://projecteuclid.org/euclid.rmi/1257258098

Mathematical Reviews number (MathSciNet)
MR2590690

Zentralblatt MATH identifier
1247.35133

#### Citation

Kenig, Carlos E.; Martel, Yvan. Asymptotic stability of solitons for the Benjamin-Ono equation. Rev. Mat. Iberoamericana 25 (2009), no. 3, 909--970. https://projecteuclid.org/euclid.rmi/1257258098

#### References

• 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.
• Benjamin, T. B.: Internal waves of permanent form in fluids of great depth. Journal of Fluid Mechanics 29 (1967), 559-592.
• 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.
• 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.
• Burq, N. and Planchon, F.: On well-posedness for the Benjamin-Ono equation. Math. Ann. 340 (2008), no. 3, 497-542.
• Calderón, A.-P.: Commutators of singular integral operators. Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092-1099.
• Coifman, R. R. and Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc. 212 (1975), 315-331.
• Herr, S., Ionescu, A. D., Kenig, C. E. and Koch, H.: Global solutions to dispersive nonlinear equations. Preprint.
• Ginibre, J. and Velo, G.: Commutator expansions and smoothing properties of generalized Benjamin-Ono equations. Ann. Inst. H. Poincaré Phys. Théor. 51 (1989), 221-229.
• Gustafson, S., Takaoka, H. and Tsai, T.-P.: Stability in $H^\frac 12$ of the sum of $K$ solitons for the Benjamin-Ono equation. J. Math. Phys. 50 (2009), no. 1, 013101, 14 pp.
• 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.
• 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.
• 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.
• Martel, Y.: Linear problems related to asymptotic stability of solitons of the generalized KdV equations. SIAM J. Math. Anal. 38 (2006), 759-781.
• Martel, Y. and Merle, F.: Instability of solitons for the critical generalized Korteweg-de Vries equation. Geom. Funct. Anal. 11 (2001), 74-123.
• 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.
• Martel, Y. and Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157 (2001), 219-254.
• Martel, Y. and Merle, F.: Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18 (2005), no. 1, 55-80.
• Martel, Y. and Merle, F.: Asymptotic stability of solitons of the gKdV equations with a general nonlinearity. Math. Ann. 341 (2008), 391-427.
• Martel, Y. and Merle, F.: Refined asymptotics around soliton for gKdV equations. Discrete Contin. Dyn. Syst. 20 (2008), 177-218.
• 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.
• Matsuno, Y.: The Lyapunov stability of the $N$-soliton solutions in the Lax hierarchy of the Benjamin-Ono equation. J. Math. Phys. 47 (2006), 103505, 13pp.
• Neves, A. and Lopes, O.: Orbital stability of double solitons for the Benjamin-Ono equation. Comm. Math. Phys. 262 (2006), 757-791.
• Pego, R. L. and Weinstein, M. I.: Asymptotic stability of solitary waves. Comm. Math. Phys. 164 (1994), 305-349.
• 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.
• Stein, E.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series 30. Princeton Univ. Press, Princeton, 1970.
• Tao, T.: Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbbR)$. J. Hyperbolic Differ. Equ. 1 (2004), 27-49.
• Toland, J. F.: The Peierls-Nabarro and Benjamin-Ono equations. J. Funct. Anal. 145 (1997), 136-150.
• Weinstein, M. I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985), 472-491.
• Weinstein, M. I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39 (1986), 51-67.
• 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.