Linear instability and uniqueness of the peaked periodic wave in the\n reduced Ostrovsky equation

2018/04/10 by Anna Geyer, Geyer, Anna, Dmitry E. Pelinovsky +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Mathematical Physics (math-ph) #Pattern Formation and Solitons (nlin.PS) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1804.03788

openalex publication_date 2018/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Stability of the peaked periodic waves in the reduced Ostrovsky equation has\nremained an open problem for a long time. In order to solve this problem we\nobtain sharp bounds on the exponential growth of the L² norm of co-periodic\nperturbations to the peaked periodic wave, from which it follows that the\npeaked periodic wave is linearly unstable. We also prove that the peaked\nperiodic wave with parabolic profile is the unique peaked wave in the space of\nperiodic L² functions with zero mean and a single minimum per period.\n

Citations

• One-parameter semigroups for linear evolution equations
• An index theorem for the stability of periodic traveling waves of KdV type

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• Orbital stability of periodic waves in the class of reduced Ostrovsky\n equations
• Growth of perturbations to the peaked periodic waves in the Camassa-Holm equation,
• High-frequency instabilities of the Ostrovsky equation
• Spectral stability for classical periodic waves of the Ostrovsky and\n short pulse models