2021/05/31 by Jared C. Bronski, Bronski, Jared C., Vera Mikyoung Hur +3 · 1 citation
Mathematics · Physics and Astronomy · #35B10 #35B35 #35C07 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Waves and Solitons #Pattern Formation and Solitons (nlin.PS) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2105.15099
openalex publication_date 2021/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We examine the spectral stability and instability of periodic traveling waves for regularized long-wave models. Examples include the regularized Boussinesq, Benney--Luke, and Benjamin--Bona--Mahony equations. Of particular interest is a striking new instability phenomenon -- spectrum off the imaginary axis extending into infinity. The spectrum of the linearized operator of the generalized Korteweg--de Vries equation, for instance, lies along the imaginary axis outside a bounded set. The spectrum for a regularized long-wave model, by contrast, can vary markedly with the parameters of the periodic traveling waves. We carry out asymptotic spectral analysis to short wavelength perturbations, distinguishing whether the spectrum tends to infinity along the imaginary axis or some curve whose real part is nonzero. We conduct numerical experiments to corroborate our analytical findings.