Phase Slips and the Eckhaus Instability

Details Phase Slips and the Eckhaus Instability Phase Slips and the Eckhaus Instability

1995-03-08

arxiv.org/abs/patt-sol/9503001v1

Nonlinear Sciences

Pattern Formation and Solitons

22 pages, Postscript, A4

Scientific paper

10.1088/0951-7715/8/6/004

We consider the Ginzburg-Landau equation, $ \partial_t u= \partial_x^2 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.

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