2021/09/17 by Mathew A. Johnson, Johnson, Mathew A., Wesley R. Perkins +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS) #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2109.08459
openalex publication_date 2021/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the stability and nonlinear local dynamics of spectrally stable\nperiodic wave trains of the Korteweg-de Vries / Kuramoto-Sivashinsky equation\nwhen subjected to classes of periodic perturbations. It is known that for each\nN\∈\ℕ, such a T-periodic wave train is asymptotically stable to\nNT-periodic, i.e., subharmonic, perturbations, in the sense that initially\nnearby data will converge asymptotically to a small Galilean boost of the\nunderlying wave, with exponential rates of decay. However, both the allowable\nsize of initial perturbations and the exponential rates of decay depend on N\nand, in fact, tend to zero as N\→\∞, leading to a lack of uniformity in\nsuch subharmonic stability results. Our goal here is to build upon a recent\nmethodology introduced by the authors in the reaction-diffusion setting and\nachieve a subharmonic stability result which is uniform in N. This work is\nmotivated by the dynamics of such wave trains when subjected to perturbations\nwhich are localized (i.e., integrable on the line).\n