2014/11/08 by Guan Huang, Huang, Guan, Sergei Kuksin +3 · 1 citation
Mathematics · Physics and Astronomy · #35B20 #35B34 #35B35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math.AP #msc:35B20 #msc:35B34 #msc:35B35
paper · pdf · doi:10.48550/arxiv.1411.2143
openalex publication_date 2014/11/08 · arxiv created 2015/12/11 · arxiv updated 2015/12/14 · openalex created_date 2022/11/08 · openalex updated_date 2026/07/28
Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: ut+i(-Δu+V(x)u)=εμΔu+εP( u), x∈ Rd , (*) under the periodic boundary conditions, where μ\geqslant0 and P is a smooth function. Let \ζ1(x),ζ2(x),…\ be the L2-basis formed by eigenfunctions of the operator -Δ+V(x). For a complex function u(x), write it as u(x)=∑k\geqslant1vkζk(x) and set Ik(u)=(1)/(2)|vk|2. Then for any solution u(t,x) of the linear equation (*)ε=0 we have I(u(t,⋅))=const. In this work it is proved that if equation (*) with a sufficiently smooth real potential V(x) is well posed on time-intervals t\lesssim ε-1, then for any its solution uε(t,x), the limiting behavior of the curve I(uε(t,⋅)) on time intervals of order ε-1, as ε→0, can be uniquely characterized by a solution of a certain well-posed effective equation: ut=εμ\triangle u+εF(u), where F(u) is a resonant averaging of the nonlinearity P(u). We also prove a similar results for the stochastically perturbed equation, when a white in time and smooth in x random force of order √ε is added to the right-hand side of the equation. The approach of this work is rather general. In particular, it applies to equations in bounded domains in Rd under Dirichlet boundary conditions.