Random Attractors for the Dissipative Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities with Additive Noise

Abstract

In this paper, we study the random dynamical system (RDS) generated by the dissipative Hamiltonian amplitude equation with additive noise defined on the periodic boundaries. We investigate the existence of a compact random attractor for the RDS associated with the equation through introducing two functions and one process in E0=H1×L2. The compactness of the RDS is established by the decomposition of solution semigroup.

Share and Cite:

Yin, J. , Li, Y. and Zhao, H. (2013) Random Attractors for the Dissipative Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities with Additive Noise. Journal of Applied Mathematics and Physics, 1, 37-46. doi: 10.4236/jamp.2013.13007.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] M. Tanaka and N. Yajima, “Soliton Modes in an Unstable Plasma Nonlinear Phenomena in an Electron-Beam Plasma,” Progress of Theoretical Physics Supplement, Vol. 94, 1988, pp. 138-162.
[2] T. Yajima and M. Wadati, “Solitons in an Unstable Medium,” Journal of the Physical Society of Japan, Vol. 56, 1987, pp. 3069-3081.
[3] M. Wadati, H. Segur and M. J. Ablowitz, “A New Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities,” Journal of the Physical Society of Japan, Vol. 61, No. 4, 1992, pp. 1187-1193. doi:10.1143/JPSJ.61.1187
[4] B. L. Guo and Z. D. Dai, “Attractor for the Dissipative Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities,” Discrete and Continuous Dynamical Systems, Vol. 4, No. 4, 1998, pp. 783-793. doi:10.3934/dcds.1998.4.783
[5] Z. D. Dai, “Regularity of the Attractor for the Dissipative Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities,” Acta Mathematicae Applicatae Sinica (English Series), Vol. 18, No. 2, 2002, pp. 263-272. doi:10.1007/s102550200025
[6] Z. D. Dai, L. Yang and J. Huang, “Attractor for the Unperturbed Dissipative Hamiltonian Amplitude Wave,” Acta Mathematicae Applicatae Sinica, Vol. 27, No. 4, 2004, pp. 577-592.
[7] L. Yang and Z. D. Dai, “Finite Dimension of Global Attractors for Dissipative Equations Governing Modulated Wave,” Applied Mathematics: A Journal of Chinese Universities, Vol. 18, No. 4, 2003, pp. 421-430. doi:10.1007/s11766-003-0069-3
[8] Y. R. Li and B. L. Guo, “Random Attractors of Boussinesq Equations with Multiplicative Noise,” Acta Mathematicae Sinica (English Series), Vol. 25, No. 3, 2009, pp. 481-490. doi:10.1007/s10114-008-6226-0
[9] Y. R. Li, and B. L. Guo, “Random Attractors for Quasi-Continuous Random Dynamical Systems and Applications to Stochastic Reaction-Diffusion Equations,” Journal of Differential Equations, Vol. 245, No. 7, 2008, pp. 1775-1800. doi:10.1016/j.jde.2008.06.031
[10] X. M. Fan, “Random Attractor for a Damped Sine-Gordon Equation with White Noise,” Pacific Journal of Mathematics, Vol. 216, No. 1, 2004, pp. 63-76. doi:10.2140/pjm.2004.216.63
[11] H. Crauel and F. Flandoli, “Attracors for Random Dynamical Systems,” Probability Theory and Related Fields, Vol. 100, No. 3, 1994, pp. 365-393. doi:10.1007/BF01193705
[12] H. Crauel and F. Flandoli, “Random Attractors,” Journal of Dynamics and Differential Equations, Vol. 9, No. 2, 1997, pp. 307-341. doi:10.1007/BF02219225
[13] L. Arnold, “Random Dynamical Systems,” Springer-Verlag, New York, 1998.
[14] R. Temam, “Infinite-Dimensional Dynamical System in Mechanics and Physics,” Springer-Verlag, New York, 1988, pp. 90-226. doi:10.1007/978-1-4684-0313-8

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.