Abstract
In this paper, the existence of a response solution with the Liouvillean frequency vector to the quasi-periodically forced complex Ginzburg–Landau equation, whose linearized system is elliptic–hyperbolic, is obtained. The proof is based on constructing a modified KAM theorem for an infinite-dimensional dissipative system with Liouvillean forcing frequency.
Highlights
Introduction and main result The complexGinzburg–Landau equation ut = ru + (b + iν)∂xxu + m∂xu – (1 + iμ)|u|2u (1.1)is extensively studied in the physics community
If we find the solutions Hj (j = 1, 2, 3) and Fj (j = 1, 2) of the homological equations (4.11)– (4.15), we will obtain a new system with another perturbation, which will be smaller on a small domain
For the homological equation (4.13), we find an approximate solution with suitable small error term using idea in [16, 17]
Summary
2.1 Functional setting Let T2c = C2/(2π Z2) be the two-dimensional complex torus. Given a function f : D(δ) × O → C, which is analytic in θ ∈ D(δ) and CW1 in ξ ∈ O with Fourier expansion f (θ ; ξ ) = k∈Z2 f (k; ξ )ei k,θ , we define its norm as f δ,O :=. For the vector field X : D(δ, s) × O → Pa,p, which is analytic in (θ , ρ, z) ∈ D(δ, s) and depends CW1 smoothly on parameter ξ ∈ O, the weighted norm of X is defined as. Our goal is to show that if the perturbations p, g are small enough, the system (3.1) still admits invariant torus with Liouvillean frequency ω = (1, α) for most of parameter ξ ∈ O (in Lebesgue measure sense) provided that Ω, Λ satisfy some non-degeneracy conditions.
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