Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus T^d:\begin{aligned} (Formula Presented.) N. Define I(u) = (I_k, k∈ Z^d) , where I_k= v_kv¯ _k/ 2 and v_k, k∈ Z^d, are the Fourier coefficients of the function u we give. Assume that the equation (∗) is well posed on time intervals of order ϵ^{- 1} and its solutions have there a-priori bounds, independent of the small parameter. Let u(t, x) solve the equation Faou, E., Germain, P., Hani, Z.: The weakly nonlinear large box limit of the 2d cubic nonlinear Schrödinger equation. preprint, 2013.∗). If ϵ is small enough, then for t≲ ϵ^{- 1}, the quantity I(u(t, x)) can be well described by solutions of an effective equation: (Formula Presented.),where the term F(u) can be constructed through a kind of resonant averaging of the nonlinearity b|u| ^2p+ ic| u| ^2qu.