Periodic and quasi-periodic solutions for the complex Ginzburg–Landau equation

Abstract

In this paper, we prove that there are a continuous branch of periodic solutions and a Cantorian branch of quasi-periodic solutions for the complex Ginzburg-Landau equation for some coefficients of the linear driving term and the dissipation term and these solutions are normally hyperbolic.

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].…”

“…Moreover, we assume the basic frequency ω is Liouvillean. Thus, the method in [9,10,14] cannot be directly applied since in these papers the frequency is Diophantine and the linear system is pure hyperbolic i.e., the real parts of all frequencies are not zero. In a Hamiltonian case like [17], one constructed the symplectic transformation by using the time-1-map of an auxiliary Hamiltonian flow to preserve the Hamiltonian structure in each KAM step.…”

Explicit traveling wave solutions of five kinds of nonlinear evolution equations

“…This section is motivated by Bikbaev [2], Bona and Schonbek [3], Chung and Yuan [7], David, Fernando and Feng [8], Feng [9,[11][12][13][14][15][16], Feng and Meng [19], Geng and Cui [20], Huang and Dai [22], Jacobs, McKinney and Shearer [24], Jeffrey and Xu [26], Ma [31], Parkes [37], Sahu and Roychoudhury [39], Shen [40], Zayko and Nefedov [48], Zhang, Wang and Gao [49], Zhang [50,51], Zhong, Yang and Yang [53]. …”

Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$

“…This equation has a long history in physics as a generic amplitude equation near the onset of instabilities that lead to chaotic dynamics in fluid mechanical systems, as well as in the theory of phase transitions and superconductivity. The existence and stability of periodic solutions of traveling-wave type for (1) have been extensively investigated in many papers. See [3,5,6,7], for example, for more physical and mathematical backgrounds.…”

“…However, there are fewer papers concerning the existence of KAM tori and quasi-periodic solutions for (1), although numerical evidence has shown the existence of 2-and 3-tori. In 1992, a traveling-wave quasi-periodic solution was found in [9]; recently, non-traveling-wave quasi-periodic solutions and KAM 2-tori of (1) have been constructed by means of KAM theory in [1], provided that p = 1.…”