Hostname: page-component-cb9f654ff-kl2l2 Total loading time: 0 Render date: 2025-08-03T07:54:23.018Z Has data issue: false hasContentIssue false
  • English
  • Français

The Global Attractor of a Damped, Forced Hirota Equation in H 1

Published online by Cambridge University Press:  20 November 2018

Boling Guo  and
Zhaohui Huo
Boling Guo
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing, 100088, P.R. China e-mail: gbl@iapcm.ac.cn
Zhaohui Huo
Affiliation:
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, P.R. China, and Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, P.R. China e-mail: huozhaohui@yahoo.com.cn Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, P.R. China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The existence of the global attractor of a damped forced Hirota equation in the phase space ${{H}^{1}}\left( \mathbb{R} \right)$ is proved. The main idea is to establish the so-called asymptotic compactness property of the solution operator by energy equation approach.

Keywords

35Q5335B4035B4137L30global attractorFourier restriction normdamping systemasymptotic compactness

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 2 , 01 June 2010 , pp. 295 - 310
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Abergel, F., Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains. J. Differential Equations 83(1990), no. 1, 85–108. doi:10.1016/0022-0396(90)90070-6Google Scholar
[2] Bourgain, J., Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equation. Geom. Funct. Anal. 3(1993), no. 2, 107–156. doi:10.1007/BF01896020Google Scholar
[3] Bourgain, J., Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equation. II. The KdV equation. Geom. Funct. Anal. 3(1993), no. 3, 209–262. doi:10.1007/BF01895688Google Scholar
[4] Ghidaglia, J.-M., Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time. J. Differential Equations 74(1988), no. 2, 369–390. doi:10.1016/0022-0396(88)90010-1Google Scholar
[5] Ghidaglia, J.-M., A note on the strong convergence towards attractors of damped forced KdV equations. J. Differential Equations 110(1994), no. 2, 356–359. doi:10.1006/jdeq.1994.1071Google Scholar
[6] Goubet, O., Regularity of the global attractor for a weakly damped nonlinear Schrödinger equation in ℝ2 . Adv. Differential Equations 3(1998), no. 3, 337–360.Google Scholar
[7] Goubet, O., Asymptotic smoothing effect for weakly damped Korteweg-de Vries equations. Discrete Contin. Dynam. Systems, 6(2000), no. 3, 625–644.Google Scholar
[8] Goubet, O. and Rosa, R., Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line. J. Differential Equations 185(2002), no. 1, 25–53. doi:10.1006/jdeq.2001.4163Google Scholar
[9] Guo, B. and Tan, S., Global smooth solution for nonlinear evolution equation of Hirota type. Sci. China, Ser. A 35(1992), no. 12, 1425–1433.Google Scholar
[10] Hasegawa, A., Optical Solitons in Fibers. Second edition. Springer, Berlin, 1990.Google Scholar
[11] Hasegawa, A. and Kodama, Y., Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Electronics. 23(1987), 510–524.Google Scholar
[12] Huo, Z. and Guo, B., Well-posedness of the Cauchy problem for the Hirota equation in Sobolev spaces Hs . Nonlinear Anal. 60(2005), no. 6, 1093–1110. doi:10.1016/j.na.2004.10.011Google Scholar
[13] Huo, Z. and Jia, Y., Well-posedness for the Cauchy problem to the Hirota equation in Sobolev spaces of negative indices. Chinese Ann. Math. Ser. B 26(2005), no. 1, 75–88. doi:10.1142/S0252959905000075Google Scholar
[14] Karachalios, N. and Stavrakakis, N., Global attractor for the weakly damped driven Schrödinger equation in H2R . NoDEA Nonlinear Differential Equations Appl. 9(2002), no. 3, 347–360. doi:10.1007/s00030-002-8132-yGoogle Scholar
[15] Kenig, C. E., Ponce, G., and Vega, L., The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 71(1993), no. 1, 1–21. doi:10.1215/S0012-7094-93-07101-3Google Scholar
[16] Kenig, C. E., Ponce, G., and Vega, L., A bilinear estimate with applications to the Kdv equation. J. Amer. Math. Soc. 9(1996), no. 2, 573–603. doi:10.1090/S0894-0347-96-00200-7Google Scholar
[17] Kodama, Y., Optical solitons in a monomode fiber. J. Statist. Phys. 39(1985), no. 5–6, 597–614. doi:10.1007/BF01008354Google Scholar
[18] Ladyzhenskaya, O., Attractor for Semigroups and Evolution Equations. Lezioni Lincei, Cambridge University Press, 1991.Google Scholar
[19] Laurey, C., The Cauchy problem for a third order nonlinear Schrödinger equation. Nonlinear Anal. 29(1997), no. 2, 121–158. doi:10.1016/S0362-546X(96)00081-8Google Scholar
[20] Moise, I. and Rosa, R., On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation. Adv. Differential Equations 2(1997), no. 2, 275–296.Google Scholar
[21] Moise, I., Rosa, R., and Wang, X., Attractors for non-compact semigroups via energy equations. Nonlinearity 11(1998), no. 5, 1369–1393. doi:10.1088/0951-7715/11/5/012Google Scholar
[22] Rosa, R., The global attractor of a weakly damped, forced Korteweg-de Vries equation in H 1(ℝ) . Mat. Contemp. 19(2000), 129–152.Google Scholar
[23] Tan, S., Han, Y., Long time behavior of solutions to general nonlinear evolution equations. Chinese Ann. Math. Ser. A 16(1995), no. 2, 127–141.Google Scholar
[24] Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics. Second edition. Applied Mathematical Sciences 68, Springer-Verlag, New York, 1997.Google Scholar