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Modulational Instability of Small Amplitude Wave Trains in the Novikov Equation
Ehrman, Brett
Ehrman, Brett
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Abstract
We study the spectral stability of smooth, small-amplitude periodic traveling wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. Specifically, we investigate the L^2(R)-spectrum of the associated linearized operator, which in this case is an integro-differential operator with periodic coefficients, in a neighborhood of the origin in the spectral plane. Our analysis shows that such small-amplitude periodic solutions are spectrally unstable to long-wavelength perturbations if the wave number if greater than a critical value, bearing out the famous Benmajin-Feir instability for the Novikov equation. On the other hand, such waves with wave number less than the critical value are shown to be spectrally stable. Our methods are based on applying spectral perturbation theory to the associated linearization.
Description
These are the slides from a presentation given at The Thirteenth International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory on 04/15/2025.
Date
2025-04-15
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University of Kansas
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Keywords
Nonlinear waves, Stability, Novikov