Long-time dynamics of resonant weakly nonlinear CGL equations

TL;DR

This paper studies the long-time behavior of solutions to a weakly nonlinear complex Ginzburg-Landau equation on a torus, showing that for small epsilon, the dynamics can be approximated by an effective equation derived through resonant averaging.

Contribution

It introduces an effective equation capturing the long-time dynamics of the weakly nonlinear CGL equation via resonant averaging, extending understanding of its asymptotic behavior.

Findings

Solutions are well approximated by the effective equation for times up to epsilon^{-1}.

The effective equation is derived through resonant averaging of the nonlinear terms.

The approach provides a framework for analyzing long-time dynamics of similar PDEs.

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 $b|u|^{2p}+ ic|u|^{2q}u$.