Subharmonic dynamics of wave trains in the Korteweg‐de Vries/Kuramoto‐Sivashinsky equation. Issue 3 (16th December 2021)

• Record Type: Journal Article
• Authors: Johnson, Mathew A.; Perkins, Wesley R.
• Is Part Of: Studies in applied mathematics. Volume 148:Issue 3(2022)
• Page Start: 1274
• Page End: 1302
• Publication Date: 2021-12-16
• Subjects: Kuramoto‐Sivashinsky -- nonlinear stability -- periodic traveling waves; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12475
• Languages: English
• ISSNs: 0022-2526

Abstract

Abstract: We study the stability and nonlinear local dynamics of spectrally stable periodic wave trains of the Korteweg‐de Vries/Kuramoto‐Sivashinsky equation when subjected to classes of periodic perturbations. It is known that for each N ∈ N $N\in {\mathbb {N}}$, such a T $T$ ‐periodic wave train is asymptotically stable to N T $NT$ ‐periodic, i.e., subharmonic, perturbations, in the sense that initially nearby data will converge asymptotically to a small Galilean boost of the underlying wave, with exponential rates of decay. However, both the allowable size of initial perturbations and the exponential rates of decay depend on N $N$ and, in fact, tend to zero as N → ∞ $N\rightarrow \infty$, leading to a lack of uniformity in such subharmonic stability results. Our goal here is to build upon a recent methodology introduced by the authors in the reaction–diffusion setting and achieve a subharmonic stability result, which is uniform in N $N$ . This work is motivated by the dynamics of such wave trains when subjected to perturbations that are localized (i.e., integrable on the line).