Quasiperiodic solutions for the cubic complex Ginzburg–Landau equation

Abstract

The existence of two-dimensional Kolmogorov–Arnold–Moser invariant tori is proved for cubic complex Ginzburg–Landau equation of higher spatial dimension. As a consequence, the equation possesses a Cantorian branch of nontraveling-wave quasiperiodic solutions of two-dimensional frequency vector.

Quasi-periodic solutions for the quasi-periodically forced cubic complex Ginzburg-Landau equation on $\mathbb {T}^{d}$Td

“…(2.22) with a more complex variables transformation (2.23) with respect to Ref. 12. Although the strategy that we obtain the normal form (which is expressed in the action-angle variables in the tangent directions and Cartesian coordinates in normal directions) in this paper is similar to the one in Ref.…”

“…It is inspired by the paper 12 by Cong, Liu, and Yuan, where the autonomous case is studied. It should be worth emphasizing that the main novelties of this paper and of the techniques with respect to the paper 12 not only depends on time in a quasi-periodic way but also on spatial variable x explicitly; (ii) In order to obtain a partial Birkhoff normal form of Eq. (1.2), we have to reduce the linear equation (1.3) into a constant coefficient equation via a quasi-periodic transformation with the same basic frequencies as the initial equation (see Lemma 2.6), and to reduce Eq.…”

“…Although the strategy that we obtain the normal form (which is expressed in the action-angle variables in the tangent directions and Cartesian coordinates in normal directions) in this paper is similar to the one in Ref. 12, the details are quite different (see Subsections II C and II D); (iii) Since the perturbation term (1.4) is related to the time variable t and the spatial variable x, we have to show that the vector field in (2.20) is analytic from l a, s to l a, s when h(θ , x) is analytic in (θ, x) ∈ T m × T d (see Lemma 2.8); (iv) We assume that the forced frequencies are some special forms in Eq. (1.2) and depend on one real parameter ξ ∈ [0, 1] (see (H1)), such that they are non-degenerate, then by using the idea of the measure estimates applied in the paper 13 by Bambusi and Berti, where a degenerate KAM theory for lower dimensional elliptic tori of infinite dimensional Hamiltonian systems depending on one parameter only is studied, we construct a KAM theorem for a dissipative system with a quasi-periodic time-dependent perturbation and one real parameter ξ , and obtain a Cantorian branch of m + 2-dimensional invariant tori for the equation.…”

“…12 that when we choose some special n 1 , n 2 such that R 2 is empty and R = R 1 , see Ref. 12 for details.…”

Response solution to complex Ginzburg–Landau equation with quasi-periodic forcing of Liouvillean frequency

“…When x ∈ T d := (R/2πZ) d , there are some papers concerning the existence of KAM-type tori for (1.1). More concretely, Chung and Yuan [9] and Cong, Liu and Yuan [10] proved the existence of quasiperiodic solutions which are not traveling waves for d = 1 and d ≥ 2 respectively in the case of the group velocity m = 0 by KAM-type theorems. See also [11][12][13].…”