Data di Pubblicazione:
2015
Citazione:
Time-averaging forweakly nonlinear CGL equations with arbitrary potentials / G. Huang, S. Kuksin, A. Maiocchi (FIELDS INSTITUTE COMMUNICATIONS). - In: Hamiltonian Partial Differential Equations and Applications / [a cura di] P. Guyenne, D. Nicholls, C. Sulem. - [s.l] : Springer, 2015. - ISBN 978-1-4939-2949-8. - pp. 323-349 [10.1007/978-1-4939-2950-4_11]
Abstract:
Consider weakly nonlinear complex Ginzburg–Landau (CGL) equation of the form: ut + i(-Δu + V(x)u) = ε μΔu + ε(∇u, u), x ∈ Rd ; (*) under the periodic boundary conditions, where μ ≥0 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 ≥1 vk ζk (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 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 =ε μ Δ Dμ + ε F(u); where F(u), is a resonant averaging of the nonlinearity P(∇u; u). We also prove 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.
Tipologia IRIS:
03 - Contributo in volume
Keywords:
Mathematics (all)
Elenco autori:
G. Huang, S. Kuksin, A. Maiocchi
Link alla scheda completa:
Titolo del libro:
Hamiltonian Partial Differential Equations and Applications