Stability on Time-Dependent Domains

We explore the key differences in the stability picture between extended systems on time-fixed and time-dependent spatial domains. As a paradigm, we take the complex Swift–Hohenberg equation, which is the simplest nonlinear model with a finite critical wavenumber, and use it to study dynamic pattern formation and evolution on time-dependent spatial domains in translationally invariant systems, i.e., when dilution effects are absent. In particular, we discuss the effects of a time-dependent domain on the stability of spatially homogeneous and spatially periodic base states, and explore its effects on the Eckhaus instability of periodic states. New equations describing the nonlinear evolution of the pattern wavenumber on time-dependent domains are derived, and the results compared with those on fixed domains. Pattern coarsening on time-dependent domains is contrasted with that on fixed domains with the help of the Cahn–Hilliard equation extended here to time-dependent domains. Parallel results for the evolution of the Benjamin–Feir instability on time-dependent domains are also given. Communicated by Paul K Newton. In memory of Professor Jerrold E. Marsden, teacher, colleague, and friend. E. Knobloch Department of Physics, University of California, Berkeley, CA 94720, USA R. Krechetnikov (B) Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA e-mail: [email protected] Present Address: R. Krechetnikov University of Alberta, Edmonton, AB T6G 2G1, Canada