Abstract
We study a model of small amplitude traveling waves arising in a supercritical Hopf bifurcation, that are coupled to a slowly varying, real field. The field is advected by the waves and, in turn, affects their stability via a coupling to the growth rate. In the absence of dispersion, we identify two distinct short-wave instabilities. One instability induces a phase slip of the waves and a corresponding reduction of the winding number, while the other leads to a modulated wave structure. The bifurcation to modulated waves can be either forward or backward, in the latter case permitting the existence of localized, traveling pulses which are bistable with the basic, conductive state.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 19-38 |
| Number of pages | 20 |
| Journal | Physica D: Nonlinear Phenomena |
| Volume | 156 |
| Issue number | 1-2 |
| DOIs | |
| State | Published - Aug 1 2001 |
Funding
We thank Blas Echebarria for many interesting and helpful discussions. This work was supported by the Engineering Research Program of the Office of Basic Energy Sciences at the Department of Energy (DE-FG02-92ER14303), NSF grant DMS 9804673, and the NSF IGERT grant DGE-9987577.
Keywords
- Ginzburg-Landau equation
- Long-wave mode
- Short-wave instability
- Traveling waves
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics