On long time dynamics of perturbed KdV equations

2013/10/21 by Guan Huang, Huang, Guan
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.1310.5462

arxiv created 2013/12/06 · arxiv updated 2013/12/09

Abstract

Consider perturbed KdV equations: u_t+u_xxx-6uu_x=εf(u(⋅)), x∈\mathbbT=ℝ/ℤ, ∫_\mathbbTu(x,t)dx=0, where the nonlinearity defines analytic operators u(⋅)↦ f(u(⋅)) in sufficiently smooth Sobolev spaces. Assume that the equation has an ε-quasi-invariant measure μ and satisfies some additional mild assumptions. Let u^ε(t) be a solution. Then on time intervals of order ε^-1, as ε→0, its actions I(u^ε(t,⋅)) can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is μ-typical.

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