Long-time dynamics of resonant weakly nonlinear CGL equations

Abstract

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