Damped-driven KdV and effective equations for long-time behaviour of its solutions. (English) Zbl 1231.35205

The author considers a sort of damped-driven KdV equation, and studies the limiting long-time behaviour of the KdV integrals of motion along a solution to the considered damped-driven KdV equation. The obtained limit is represented by a system of so-called effective equations for the damped-driven KdV equation, which are quasilinear stochastic heat equations with non-local nonlinearity.
Reviewer: Ma Wen-Xiu (Tampa)


35Q53 KdV equations (Korteweg-de Vries equations)
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
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[1] Diop M., Iftimie B., Pardoux E., Piatnitski A.: Singular homogenization with stationary in time and periodic in space coefficients. Journal of Functional Analysis 231, 1–46 (2006) · Zbl 1113.35015 · doi:10.1016/j.jfa.2005.02.007
[2] Dubrovin B.A., Novikov S.P.: Hydrodynamics of weakly deformed soliton lattices, differential geometry and Hamiltonian theory. Russ. Math. Surv. 44, 35–124 (1989) · Zbl 0712.58032 · doi:10.1070/RM1989v044n06ABEH002300
[3] Freidlin M.I., Wentzell A.D.: Averaging principle for stochastic perturbations of multifrequency systems. Stochastics and Dynamics 3, 393–408 (2003) · Zbl 1050.60078 · doi:10.1142/S0219493703000747
[4] Ikeda N., Watanabe S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam (1989) · Zbl 0684.60040
[5] T. Kappeler, J. Pöschel, KAM & KdV, Springer, 2003.
[6] Karatzas I., Shreve S.: Brownian Motion and Stochastic Calculus, 2nd ed. Springer-Verlag, Berlin (1991) · Zbl 0734.60060
[7] Khasminski R.: On the averaging principle for Ito stochastic differential equations. Kybernetika 4, 260–279 (1968) (in Russian)
[8] Yu. Kifer, Some recent advances in averaging, in ”Modern Dynamical Systems and Applications” (Ya. Pesin, M. Brin, B. Hasselblatt, eds.), Cambridge University Press, Cambridge (2004), 385–403. · Zbl 1147.37318
[9] Krichever I.M.: The averaging method for two-dimensional ”integrable” equations. Funct. Anal. Appl. 22, 200–213 (1988) · Zbl 0688.35088 · doi:10.1007/BF01077626
[10] N.V. Krylov, Controlled Diffusion Processes, Springer, 1980. · Zbl 0436.93055
[11] N.V. Krylov, Introduction to the Theory of Diffusion Processes, AMS Translations of Mathematical Monographs, 142, Providence, RI, 2003.
[12] Kuksin S.B.: Analysis of Hamiltonian PDEs. Oxford University Press, Oxford (2000) · Zbl 0960.35001
[13] Kuksin S.B.: Eulerian limit for 2D Navier-Stokes equations and damped/driven KdV equation as its model. Proc. Steklov Inst. Math. 259, 128–136 (2007) · Zbl 1161.35461 · doi:10.1134/S0081543807040098
[14] S.B. Kuksin, Dissipative perturbations of KdV, Proceedings of the 16th International Congress on Mathematical Physics (Prague 2009) (P. Exner, ed.), World Scientific (2010), 323–327. · Zbl 1204.35147
[15] Kuksin S.B., Perelman G.: Vey theorem in infinite dimensions and its application to KdV. DCDS-A 27, 1–24 (2010) · Zbl 1193.37076 · doi:10.3934/dcds.2010.27.1
[16] Kuksin S.B., Piatnitski A.L.: Khasminskii–Whitham averaging for randomly perturbed KdV equation. J. Math. Pures Appl. 89, 400–428 (2008) · Zbl 1148.35077
[17] P.D. Lax, C.D. Levermore, S. Venakides, The generation and propagation of oscillations in dispersive IVPs and their limiting behavior, Dispersive IVPs and Their Limiting Behavior (T. Fokas, V.E. Zakharov, eds.), Springer-Verlag, Berlin, 1993, 205–241. · Zbl 0819.35122
[18] McKean H., Trubowitz E.: Hill’s operator and hyperelliptic function theory in the presence of infinitely many branching points. Comm. Pure Appl. Math. 29, 143–226 (1976) · Zbl 0339.34024 · doi:10.1002/cpa.3160290203
[19] Schonbek M.E.: Convergence of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations 7, 959–1000 (1982) · Zbl 0496.35058 · doi:10.1080/03605308208820242
[20] Yor M.: Existence et unicité de diffusion à valeurs dans un espace de Hilbert. Ann. Inst. Henri Poincaré Sec. B 10, 55–88 (1974) · Zbl 0281.60094
[21] Zakharov V.E., Manakov V.E., Novikov S.P., Pitaevskij L.P.: Theory of Solitons. Plenum Press, New York (1984) · Zbl 0598.35002
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