1995/03/08 by J. -P. Eckmann, J -P Eckmann, Th. Gallay +3 · 1 citation
Computer Science · Physics and Astronomy · #Amplitude #Chaos control and synchronization #First order #Instability #Limiting #Nonlinear Dynamics and Pattern Formation #Order (exchange) #Phase (matter) #Quantum chaos and dynamical systems #nlin.PS #patt-sol
paper · pdf · doi:10.1088/0951-7715/8/6/004
22 pages, Postscript, A4
arxiv created 1995/03/08 · openalex publication_date 1995/11/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider the Ginzburg-Landau equation, ∂t u= ∂x2 u + u - u|u|2 , with complex amplitude u(x,t). We first analyze the phenomenon of phase slips as a consequence of the \it local shape of u. We next prove a \it global theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped with these results, we proceed to prove the corresponding phenomena for the fourth order Swift-Hohenberg equation, of which the Ginzburg-Landau equation is the amplitude approximation. This sheds light on how one should deal with local and global aspects of phase slips for this and many other similar systems.