Khasminskii-Whitham averaging for randomly perturbed KdV equation

Sergei B. Kuksin, Andrey L. Piatnitski

Research output: Contribution to journalArticlepeer-review

35 Citations (Scopus)


We consider the damped-driven KdV equation:over(u, ?) - ? ux x + ux x x - 6 u ux = sqrt(?) ? (t, x), x ? S1, ? u d x = ? ? d x = 0, where 0 < ? = 1 and the random process ? is smooth in x and white in t. For any periodic function u (x) let I = (I1, I2, ...) be the vector, formed by the KdV integrals of motion, calculated for the potential u (x). We prove that if u (t, x) is a solution of the equation above, then for 0 = t ? ?-1 and ? ? 0 the vector I (t) = (I1 (u (t, ?)), I2 (u (t, ?)), ...) satisfies the (Whitham) averaged equation. © 2007 Elsevier Masson SAS. All rights reserved.

Original languageEnglish
Pages (from-to)400-428
Number of pages29
JournalJournal de Mathématiques Pures et Appliquées
Issue number4
Publication statusPublished - Apr 2008


  • KdV equation
  • Random process
  • Whitham averaged equation


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