Phase Turbulence in the Complex Ginzburg-Landau Equation via Kuramoto–Sivashinsky Phase Dynamics
- 105 Downloads
We study the Complex Ginzburg-Landau initial value problem Open image in new window for a complex field u ∈ C, with α,β∈R. We consider the Benjamin–Feir linear instability region Open image in new window We show that for all Open image in new window and for all initial data u 0 sufficiently close to 1 (up to a global phase factor e iφ 0 ,φ0∈R) in the appropriate space, there exists a unique (spatially) periodic solution of space period L 0 . These solutions are small even perturbations of the traveling wave solution, Open image in new window and s,η have bounded norms in various L p and Sobolev spaces. We prove that Open image in new window apart from Open image in new window corrections whenever the initial data satisfy this condition, and that in the linear instability range Open image in new window the dynamics is essentially determined by the motion of the phase alone, and so exhibits ‘phase turbulence’. Indeed, we prove that the phase η satisfies the Kuramoto–Sivashinsky equation Open image in new window for times Open image in new window while the amplitude 1+α2 s is essentially constant.
KeywordsInitial Data Periodic Solution Sobolev Space Wave Solution Travel Wave Solution
Unable to display preview. Download preview PDF.
- 5.Kuramoto, Y., Tsuzuki, T.: Persistent Propagation of Concentration Waves in Dissipative Media Far from Thermal Equilibrium. Prog. Theor. Phys. 55, 356–369 (1976)Google Scholar
- 7.Manneville, P.: Dissipative structures and weak turbulence. Boston, MA: Academic Press, 1990Google Scholar
- 8.Montagne, R., Hernández-García, E., Amengual, A., San Miguel, M.: Wound-up phase turbulence in the complex Ginzburg-Landau equation. Phys. Rev. E (3) 56(1), part A, 151–167 (1997)Google Scholar
- 9.Newell, A.: Lect. Appl. Math. 15, 157 (1974)Google Scholar
- 11.Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Second edition, New York: Springer, 1997Google Scholar