Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

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

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

(*)

Here (u=u(t,x)), (xin mathbb {T}^d), (0, (mu geqslant 0), (b,cin mathbb {R}) and (m,p,qin 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 (tlesssim {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).