Khasminskii-Whitham averaging for randomly perturbed KdV equation

Sergei B. Kuksin; Andrey L. Piatnitski

doi:10.1016/j.matpur.2007.12.003

Khasminskii-Whitham averaging for randomly perturbed KdV equation

Sergei B. Kuksin, Andrey L. Piatnitski

Research output: Contribution to journal › Article › peer-review

37 Citations (Scopus)

Abstract

We consider the damped-driven KdV equation:over(u, ?) - ? u_{x x} + u_{x x x} - 6 u u_x = 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 language	English
Pages (from-to)	400-428
Number of pages	29
Journal	Journal de Mathématiques Pures et Appliquées
Volume	89
Issue number	4
DOIs	https://doi.org/10.1016/j.matpur.2007.12.003
Publication status	Published - Apr 2008

Keywords

KdV equation
Random process
Whitham averaged equation

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10.1016/j.matpur.2007.12.003

Cite this

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title = "Khasminskii-Whitham averaging for randomly perturbed KdV equation",

abstract = "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. {\textcopyright} 2007 Elsevier Masson SAS. All rights reserved.",

keywords = "KdV equation, Random process, Whitham averaged equation",

author = "Kuksin, {Sergei B.} and Piatnitski, {Andrey L.}",

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T1 - Khasminskii-Whitham averaging for randomly perturbed KdV equation

AU - Kuksin, Sergei B.

AU - Piatnitski, Andrey L.

PY - 2008/4

Y1 - 2008/4

N2 - 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.

AB - 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.

KW - KdV equation

KW - Random process

KW - Whitham averaged equation

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U2 - 10.1016/j.matpur.2007.12.003

DO - 10.1016/j.matpur.2007.12.003

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VL - 89

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JO - Journal de Mathématiques Pures et Appliquées

JF - Journal de Mathématiques Pures et Appliquées

IS - 4

ER -

Khasminskii-Whitham averaging for randomly perturbed KdV equation

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Keywords

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