Attractor for nonlinear Schrödinger equation coupling with stochastic weakly damped, forced KdV equation

Abstract

The nonlinear Schrödinger equation coupling with stochastic weakly damped, forced KdV equation with additive noise can be solved pathwise, and the unique solution generates a random dynamical system. Then we prove that the system possesses a global weak random attractor.

Keywords

Korteweg-de Vries (KdV) equation / nonlinear Schrödinger equation / additive noise

References

[1] Appert K., Vaclavik J. Dynamics of coupled solitons. Physics Fluids, 1977, 20: 1845-1949.

[2] Arnold L. Random Dynamical System, 1998, Berlin: Springer.

[3] Bona J. L., Smith R. The initial-value problem for the Korteweg-de Vries equation, 1975, 278: 555-604.

[4] Chang H. Y., Lien C., Sukarto S. Propagation of ion-acoustic solitons in a non-quiescent plasma. Plasma Phys Control Fusion, 1986, 28: 675-681.

[5] Crauel H., Debussche A., Franco F. Random attractors. J Dyn Diff Eq, 1997, 9(2): 307-341.

[6] Crauel H., Flaudoli F. Attractors for random dynamical systems. Probab Th Rel Fields, 1994, 100: 365-393.

[7] De Bouard A., Debussche A. On the stochastic Korteweg-de Vries equation. J Funct Anal, 1998, 154: 215-251.

[8] De Bouard A., Debussche A., Tsutsumi Y. White noise driven Korteweg-de Vries equation. J Funct Anal, 1999, 169: 532-558.

[9] Ghidaglia J. M. Finite dimensional behavior for weakly damped driven Schrödinger equations. Ann Inst Henri Poincaré, Analyse Non Linéaire, 1988, 5(4): 365-405.

[10] Gibbous J. On the theory of Langmuir solitons. J Plasma Phys, 1977, 17(2): 153-170.

[11] Grimshaw R., Pelinovsky E., Tian X. Interaction of a solitary wave with an external force. Physica D, 1994, 77: 405-433.

[12] Guo B., Chen F. Finite dimensional behavior of global attractors for weakly damped and forced KdV equations coupling with nonlinear Schrödinger equations. Nonlinear Analysis TMA, 1997, 29(5): 569-584.

[13] Guo B, Shen L. The periodic initial value problem and the initial value problem for the system of KdV equation coupling with nonlinear Schrödinger equations. In: Proceedings of DD-3 Symposium, Chang Chun. 1982, 417–435

[14] Herman R. The stochastic, damped Korteweg-de Vries equation. J Phys A, 1990, 23: 1063-1084.

[15] Makhankov V. G. Dynamics of classical solitons. Physics Reports (Section C of Physics Letters), 1978, 35(1): 1-128.

[16] Rosa R. The global attractor of a weakly damped, forced Korteweg-de Vries equation in H¹(ℝ. Mat Contemp, 2000, 19: 129-152.

[17] Scalerandi M., Romano A., Condat C. A. Korteweg-de Vries solitons under additive stochastic perturbations. Phys Rev E, 1998, 58: 4166-4173.

[18] Temam R. Infinite-Dimensional Systems in Mechanics and Physics. Applied Math Sciences, 1988, New York: Springer.

[19] Yang D. The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise. J Math Phys, 2004, 45(11): 4064-4076.