Long-time dynamics of resonant weakly nonlinear CGL equations

2014/07/04 by Guan Huang, Huang, Guan
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.1407.1156

arxiv created 2014/07/04 · arxiv updated 2014/07/07

Abstract

Consider a weakly nonlinear CGL equation on the torus~\mathbbT^d: u_t+iΔu=ε[μ(-1)^m-1Δ^m u+b|u|^2pu+ ic|u|^2qu].\eqno(*) Here u=u(t,x), x∈\mathbbT^d, 0<ε<<1, μ\geqslant0, b,c∈ℝ and m,p,q∈ℕ. Define \mboxI(u)=(I_\dk,\dk∈ℤ^d), where I_\dk=v_\dkv_\dk/2 and v_\dk, \dk∈ℤ^d, are the Fourier coefficients of the function~u we give. Assume that the equation (*) is well posed on time intervals of order ε^-1 and its solutions have there a-priori bounds, independent of the small parameter. Let u(t,x) solve the equation (*). If ε is small enough, then for t\lesssimε^-1, the quantity I(u(t,x)) can be well described by solutions of an \it effective equation: u_t=ε[μ(-1)^m-1Δ^m u+ F(u)], where the term F(u) can be constructed through a kind of resonant averaging of the nonlinearity b|u|^2p+ ic|u|^2qu.

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