Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Abstract

Consider a weakly nonlinear CGL equation on the torus $\mathbb {T}^d$:

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here $u=u(t,x)$, $x\in \mathbb {T}^d$, $0<\epsilon <<1$, $\mu \geqslant 0$, $b,c\in \mathbb {R}$ and $m,p,q\in \mathbb {N}$. Define $I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)$, where $I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2$ and $v_{\mathbf {k}}$, $\mathbf {k}\in \mathbb {Z}^d$, 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 effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term $F(u)$ can be constructed through a kind of resonant averaging of the nonlinearity $b|u|^{2p}+ ic|u|^{2q}u$.