Hoyle, Rebecca B. (1993) Long wavelength instabilities of square patterns. Physica D: Nonlinear Phenomena, 67 (1-3), 198-223. (doi:10.1016/0167-2789(93)90206-G).
Abstract
The long wavelength instabilities of square and rectangular planforms are studied analytically and numerically, using amplitude equations which describe the general interaction of two orthogonal coupled roll patterns. The zigzag and two-dimensional Eckhaus instabilities are found, and in addition it is discovered that the three-dimensional equivalent of the Eckhaus instability splits into two variants. The square Eckhaus instability is the direct equivalent of the two-dimensional case, whereas the rectangular Eckhaus instability is truly three-dimensional in character. In the case of square patterns, nonlinear phase diffusion equations are derived close to the onset of the instabilities. A short wavelength cross square mode is also discussed briefly.
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