2018/10/04 by Mathew A. Johnson, Johnson, Mathew A., Gregory D. Lyng +3
Computer Science · Engineering · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics and Thin Films #Nonlinear Dynamics and Pattern Formation #Nonlinear Photonic Systems #Pattern Formation and Solitons (nlin.PS)
paper · pdf · doi:10.48550/arxiv.1810.02233
openalex publication_date 2018/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the stability and dynamics of traveling-front solutions of a\nmodified Kuramoto--Sivashinsky equation arising in the modeling of nanoscale\nripple patterns that form when a nominally flat solid surface is bombarded with\na broad ion beam at an oblique angle of incidence. Structurally, the linearized\noperators associated with these fronts have unstable essential\nspectrum---corresponding to instability of the spatially asymptotic\nstates---and stable point spectrum---corresponding to stability of the\ntransition profile of the front. We show that these waves are linearly\norbitally asymptotically stable in appropriate exponentially weighted spaces.\nWhile the technical device of exponential weights allows us to accommodate the\nunstable essential spectrum of individual waves in our linear analysis, it does\nnot shed light on the long-time pattern formation that is observed\nexperimentally and in numerical simulations. To begin to address this issue, we\nconsider a periodic array of unstable front and back solutions. While not an\nexact solution of the governing equation, this periodic pattern mimics\nexperimentally observed phenomena. Our numerical experiments suggest that the\nconvecting instabilities associated with each individual wave are damped as\nthey pass through transition layers and that this stabilization mechanism\nunderlies the pattern formation seen in experiments.\n