Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions |
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| Authors: | Sergei B. Kuksin |
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| Affiliation: | (1) Institute of Mathematics of Ukrainian National Academy of Sciences, 01601 Kyiv, Ukraine;(2) Department of Mathematical Analysis Faculty of Mechanics and Mathematics, National Taras Shevchenko University of Kyiv, 01033 Kyiv, Ukraine;(3) Department of Probability Theory and Mathematical Statistics Faculty of Mechanics and Mathematics, National Taras Shevchencko University of Kyiv, 01033 Kyiv, Ukraine |
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| Abstract: | For the damped-driven KdV equation $ dot{u}-{nu}u_{xx} + u_{xxx} - 6uu_{x} = sqrt{nu},eta(t, x), x in S^1, int udx equiv int eta dx equiv 0, $ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I 1, I 2, . . . ), evaluated along a solution u ν (t, x), as ν → 0. We prove that for ${0 leq tau := {nu}t lesssim 1}For the damped-driven KdV equation | [(u)dot]-nuxx + uxxx - 6uux = ?{n} h(t, x), x ? S1, òudx o òhdx o 0, dot{u}-{nu}u_{xx} + u_{xxx} - 6uu_{x} = sqrt{nu},eta(t, x), x in S^1, int udx equiv int eta dx equiv 0, |
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