Long-time dynamics of resonant weakly nonlinear CGL equations

Abstract

Consider a weakly nonlinear CGL equation on the torus~$\mathbb{T}^d$: \[u_t+i\Delta u=\epsilon [\mu(-1)^{m-1}\Delta^{m} u+b|u|^{2p}u+ ic|u|^{2q}u].\eqno{(*)}\] Here $u=u(t,x)$, $x\in\mathbb{T}^d$, $0<\epsilon<<1$, $\mu\geqslant0$, $b,c\in\mathbb{R}$ 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 $(*)$ 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 $(*)$. 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}: \[u_t=\epsilon[\mu(-1)^{m-1}\Delta^m u+ F(u)],\] where the term $F(u)$ can be constructed through a kind of resonant averaging of the nonlinearity $b|u|^{2p}+ ic|u|^{2q}u$.