Tbilisi Mathematical Journal

Long-time behaviour for Hirota equation

Xianyun Guo and Chaosheng Zhu

Full-text: Open access

Abstract

In this paper, we investigate the long-time behavior of the solutions for the Hirota equation with the periodic boundary condition. At first, by time uniform priori estimates of solutions, we obtain the existence of global solutions. Furthermore, we prove the existence of a global attractor. Finally, by squeezing property and Lipschitz continuity, we prove the existence of an exponential attractor of finite fractal dimension which contains the global attractor.

Article information

Source
Tbilisi Math. J., Volume 5, Issue 1 (2012), 51-64.

Dates
Received: 21 December 2011
Revised: 10 July 2012
Accepted: 13 September 2012
First available in Project Euclid: 12 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1528768889

Mathematical Reviews number (MathSciNet)
MR3006758

Zentralblatt MATH identifier
1282.35065

Subjects
Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35B41: Attractors 35B45: A priori estimates

Keywords
Hirota equation global attractor exponential attractor

Citation

Guo, Xianyun; Zhu, Chaosheng. Long-time behaviour for Hirota equation. Tbilisi Math. J. 5 (2012), no. 1, 51--64. https://projecteuclid.org/euclid.tbilisi/1528768889


Export citation

References

  • N. Akroune, Regularity of the attractor for a weakly damped Schrödinger equation on $\mathbb{R}$, Applied Math. Letters, 12, (1999), 45-48
  • A. Eden, C. Foias, B. Nicolaenko and R. Temam, Inertial sets for dissipative evolution equations, Part I: Construction and applications, IMA Preprint Series 812, University Minnesota, (1991).
  • X. M. Fan, H. Yang, Exponential attractor and its fractal dimension for a second order lattice dynamical system, J. Math. Anal. Appl., 367 (2010), 350-359.
  • H. J. Gao, Exponential attractors for a generalized Ginzburg-Laudau equation, Applied Mathematics and Mechanics, English Edition, 16(9) (1995), 877-882.
  • O. Goubet, L. Molinet, Global attractor for weakly damped nonlinear Schrödinger equations in $L^{2}(\mathbb{R})$, Nonlinear Analysis, 71 (2009), 317-320.
  • B. L. Guo, H. Y. Huang, M. Y. Jiang, Ginzburg-Laudau equation, Beijing: Science Press, (2002), 27-34.
  • B. L. Guo, B. X. Wang, Exponential attractors for the generalized Ginzburg-Landau equation, Acta Mathematica Sinica, English Series, (2000), 515-526.
  • C. E. Kenig, G. Ponce, and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), no.2, 323-347.
  • C. E. Kenig, G. Ponce, and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.
  • C. E. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.
  • C. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no. 3, 617-633.
  • Y. S. Li, Q. Y. Chen, Finite dimensional global attractor for dissipative Schrödinger Boussinesq equations, J. Math. Anal. Appl., 205 (1997), 107-132.
  • S. J. Lü, Q. S. Lu, Exponential attractor for the 3$D$ Ginzburg-Laudau type equation, Nonlinear Analysis, 67 (2007), 3116-3135.
  • A. Miranville, Exponential attractors for non-autonomous evolution equations, Appl. Math. Lett., 11(2), (1988), 19-22.
  • M. Poulou, N. M. Stavrakakis, Global attractor for a system of Klein-Gordon-Schrödinger type in all $\mathbb{R}$, Nonlinear Analysis, 74 (2011), 2548-2562.
  • R. Temam, Infinite dimensional dynamical dystems in dechanics and dhysics, New York: Springer-Verlag, (2000).
  • C. S. Zhu, C. L. Mu, Z. L. Pu, Attractor for the nonlinear Schrödinger equation with a non-local nonlinear term, J. Dyna. Contr. Syst., 16 (4) (2010), 585-603.