Back
 JAMP  Vol.1 No.3 , August 2013
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.
Cite this paper: 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.
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

 
 
Top