We present results on stripe formation in the Swift-Hohenberg equation with a directional quenching term. Stripes are "grown" in the wake of a moving parameter step line, and we analyze how the orientation of stripes changes depending on the speed of the quenching line and on a lateral aspect ratio. We observe stripes perpendicular to the quenching line, but also stripes created at oblique angles, as well as periodic wrinkles created in an otherwise oblique stripe pattern. Technically, we study stripe formation as traveling-wave solutions in the Swift-Hohenberg equation and in reduced Cahn-Hilliard and Newell-Whitehead-Segel models, analytically, through numerical continuation, and in direct simulations.
Bibliographical noteFunding Information:
\ast Received by the editors October 22, 2018; accepted for publication (in revised form) April 5, 2019; published electronically June 4, 2019. http://www.siam.org/journals/siads/18-2/M122198.html Funding: The work of the second author was partially supported by NSF grant DMS-1603416. The work of the fifth author was partially supported by NSF grant DMS-1612441. \dagger School of Mathematics, University of Minnesota, Minneapolis, MN 55455 (firstname.lastname@example.org; email@example.com, http://www.umn.edu/\sim scheel). \ddagger Department of Mathematics and Statistics, Boston University, Boston, MA 02215 (firstname.lastname@example.org). \S School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804 (ogoodloe@ asu.edu). \P School of Mathematics, University of Bristol, Bristol, BS8 1QU, UK (email@example.com).
- Growing domains
- Stripe selection
- Zigzag instabilities