Large time asymptotics of solutions to the generalized Benjamin-Ono equation
HTML articles powered by AMS MathViewer
- by Nakao Hayashi and Pavel I. Naumkin PDF
- Trans. Amer. Math. Soc. 351 (1999), 109-130 Request permission
Abstract:
We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: $u_{t} + (|u|^{\rho -1}u)_{x} + \mathcal {H} u_{xx} = 0$, where $\mathcal {H}$ is the Hilbert transform, $x, t \in {\mathbf {R}}$, when the initial data are small enough. If the power $\rho$ of the nonlinearity is greater than $3$, then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. In the case $\rho =3$ critical for the asymptotic behavior i.e. for the modified Benjamin-Ono equation, we prove that the solutions have the same $L^{\infty }$ time decay as in the corresponding linear BO equation. Also we find the asymptotics for large time of the solutions of the Cauchy problem for the BO equation in the critical and noncritical cases. For the critical case, we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted Sobolev spaces. These modified scattering states differ from the free scattering states by a rapidly oscillating factor.References
- Laziz Abdelouhab, Nonlocal dispersive equations in weighted Sobolev spaces, Differential Integral Equations 5 (1992), no. 2, 307–338. MR 1148220
- L. Abdelouhab, J. L. Bona, M. Felland, and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D 40 (1989), no. 3, 360–392. MR 1044731, DOI 10.1016/0167-2789(89)90050-X
- A. S. Fokas and M. J. Ablowitz, The inverse scattering transform for the Benjamin-Ono equation—a pivot to multidimensional problems, Stud. Appl. Math. 68 (1983), no. 1, 1–10. MR 686248, DOI 10.1002/sapm19836811
- C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation—a nonlinear Neumann problem in the plane, Acta Math. 167 (1991), no. 1-2, 107–126. MR 1111746, DOI 10.1007/BF02392447
- T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-592.
- T. L. Bock and M. D. Kruskal, A two-parameter Miura transformation of the Benjamin-Ono equation, Phys. Lett. A 74 (1979), no. 3-4, 173–176. MR 591320, DOI 10.1016/0375-9601(79)90762-X
- K. M. Case, Benjamin-Ono-related equations and their solutions, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 1, 1–3. MR 516140, DOI 10.1073/pnas.76.1.1
- Ronald R. Coifman and Mladen Victor Wickerhauser, The scattering transform for the Benjamin-Ono equation, Inverse Problems 6 (1990), no. 5, 825–861. MR 1073870, DOI 10.1088/0266-5611/6/5/011
- P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), no. 2, 413–439. MR 928265, DOI 10.1090/S0894-0347-1988-0928265-0
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n\geq 2$, Comm. Math. Phys. 151 (1993), no. 3, 619–645. MR 1207269, DOI 10.1007/BF02097031
- J. Ginibre and G. Velo, Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation, J. Differential Equations 93 (1991), no. 1, 150–212. MR 1122309, DOI 10.1016/0022-0396(91)90025-5
- Jean Ginibre and Giorgio Velo, Propriétés de lissage et existence de solutions pour l’équation de Benjamin-Ono généralisée, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 11, 309–314 (French, with English summary). MR 989894
- J. Ginibre and G. Velo, Commutator expansions and smoothing properties of generalized Benjamin-Ono equations, Ann. Inst. H. Poincaré Phys. Théor. 51 (1989), no. 2, 221–229 (English, with French summary). MR 1033618
- Nako Hayashi, Keiichi Kato, and Tohru Ozawa, Dilation method and smoothing effects of solutions to the Benjamin-Ono equation, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 2, 273–285. MR 1386863, DOI 10.1017/S0308210500022733
- N.Hayashi and P.I.Naumkin, Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation, Ann. Inst. H. Poincaré (Physique Theorique) (to appear).
- Nakao Hayashi and Tohru Ozawa, Modified wave operators for the derivative nonlinear Schrödinger equation, Math. Ann. 298 (1994), no. 3, 557–576. MR 1262776, DOI 10.1007/BF01459751
- Rafael José Iório Jr., On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential Equations 11 (1986), no. 10, 1031–1081. MR 847994, DOI 10.1080/03605308608820456
- Rafael José Iório Jr., The Benjamin-Ono equation in weighted Sobolev spaces, J. Math. Anal. Appl. 157 (1991), no. 2, 577–590. MR 1112336, DOI 10.1016/0022-247X(91)90108-C
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33–69. MR 1101221, DOI 10.1512/iumj.1991.40.40003
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc. 342 (1994), no. 1, 155–172. MR 1153015, DOI 10.1090/S0002-9947-1994-1153015-4
- P.I.Naumkin, Asymptotics for large time for nonlinear Schrödinger equation, The Proceedings of the 4th MSJ International Reserch Institute on "Nonlinear Waves", GAKUTO International Series, Mathematical Sciences and Applications, Gakkotosho, 1996 (to appear).
- P.I.Naumkin, Asymptotics for large time for nonlinear Schrödinger equation (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 4, 81–118; English transl., to appear in Russian Acad. Sci. Izv. Math.
- V.L.Nunes Wagner, On the well-posedness and scattering for the transitional Benjamin-Ono equation, Second Workshop on PDE (Rio de Janeiro, 1991), Mat. Contemp. 3 (1992), 127-148.
- Hiroaki Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), no. 4, 1082–1091. MR 398275, DOI 10.1143/JPSJ.39.1082
- Tohru Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys. 139 (1991), no. 3, 479–493. MR 1121130, DOI 10.1007/BF02101876
- Akira Nakamura, Bäcklund transform and conservation laws of the Benjamin-Ono equation, J. Phys. Soc. Japan 47 (1979), no. 4, 1335–1340. MR 550203, DOI 10.1143/JPSJ.47.1335
- Gustavo Ponce, Regularity of solutions to nonlinear dispersive equations, J. Differential Equations 78 (1989), no. 1, 122–135. MR 986156, DOI 10.1016/0022-0396(89)90078-8
- Gustavo Ponce, Smoothing properties of solutions to the Benjamin-Ono equation, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 667–679. MR 1044813
- Gustavo Ponce, On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations 4 (1991), no. 3, 527–542. MR 1097916
- J.-C. Saut, Sur quelques généralisations de l’équation de Korteweg-de Vries, J. Math. Pures Appl. (9) 58 (1979), no. 1, 21–61 (French). MR 533234
- P. M. Santini, M. J. Ablowitz, and A. S. Fokas, On the limit from the intermediate long wave equation to the Benjamin-Ono equation, J. Math. Phys. 25 (1984), no. 4, 892–899. MR 739238, DOI 10.1063/1.526243
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Mitsuhiro Tanaka, Nonlinear self-modulation problem of the Benjamin-Ono equation, J. Phys. Soc. Japan 51 (1982), no. 8, 2686–2692. MR 678043, DOI 10.1143/JPSJ.51.2686
- Michael Mudi Tom, Smoothing properties of some weak solutions of the Benjamin-Ono equation, Differential Integral Equations 3 (1990), no. 4, 683–694. MR 1044213
Additional Information
- Nakao Hayashi
- Affiliation: Department of Applied Mathematics, Science University of Tokyo, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162, Japan
- Email: nhayashi@rs.kagu.sut.ac.jp
- Pavel I. Naumkin
- Affiliation: Instituto de Fisica y Matematica, Universidad Michoacana, AP 2-82, CP 58040, Morelia, Michoacana, Mexico
- Email: naumkin@ifm1.ifm.umich.mx
- Received by editor(s): August 9, 1996
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 109-130
- MSC (1991): Primary 35Q55
- DOI: https://doi.org/10.1090/S0002-9947-99-02285-0
- MathSciNet review: 1491867