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Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit


Authors: Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues and Kevin Zumbrun
Journal: Trans. Amer. Math. Soc. 367 (2015), 2159-2212
MSC (2010): Primary 35B35, 35B10, 35Q53
DOI: https://doi.org/10.1090/S0002-9947-2014-06274-0
Published electronically: July 17, 2014
MathSciNet review: 3286511
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Abstract: We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation $ \partial _t v+v\partial _x v+\partial _x^3 v+\delta \left (\partial _x^2 v +\partial _x^4 v\right )=0, $ $ \delta >0$, in the Korteweg-de Vries limit $ \delta \to 0$, a canonical limit describing small-amplitude weakly unstable thin film flow. More precisely, we carry out a rigorous singular perturbation analysis reducing the problem of spectral stability in this limit to the validation of a set of three conditions, each of which have been numerically analyzed in previous studies and shown to hold simultaneously on a non-empty set of parameter space. The main technical difficulty in our analysis, and one that has not been previously addressed by any authors, is that of obtaining a useful description for $ 0<\delta \ll 1$ of the spectrum of the associated linearized operators in a sufficiently small neighborhood of the origin in the spectral plane. This modulational stability analysis is particularly interesting, relying on direct calculations of a reduced periodic Evans function and using in an essential way an analogy with hyperbolic relaxation theory at the level of the associated Whitham modulation equations. A second technical difficulty is the exclusion of high-frequency instabilities lying between the $ \mathcal {O}(1)$ regime treatable by classical perturbation methods and the $ \gtrsim \delta ^{-1}$ regime excluded by parabolic energy estimates.


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Additional Information

Mathew A. Johnson
Affiliation: Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Boulevard, Lawrence, Kansas 66046
Email: matjohn@ku.edu

Pascal Noble
Affiliation: Institut Camille Jordan, UMR CNRS 5208, Université de Lyon, Université Lyon I, 43 bd du 11 novembre 1918, F - 69622 Villeurbanne Cedex, France
Address at time of publication: Institut de Mathématiques de Toulouse, UMR CNRS 5219, INSA de Toulouse 135, avenue de Rangueil, 31077 Toulouse Cedex 4, France
Email: noble@math.univ-lyon1.fr, Pascal.Noble@math.univ-toulouse.fr

L. Miguel Rodrigues
Affiliation: Institut Camille Jordan, UMR CNRS 5208, Université de Lyon, Université Lyon 1, 43 bd du 11 novembre 1918, F - 69622 Villeurbanne Cedex, France
Email: rodrigues@math.univ-lyon1.fr

Kevin Zumbrun
Affiliation: Department of Mathematics, Indiana University, 831 E. 3rd Street, Bloomington, Indiana 47405
Email: kzumbrun@indiana.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06274-0
Received by editor(s): February 28, 2012
Received by editor(s) in revised form: June 20, 2013
Published electronically: July 17, 2014
Additional Notes: The research of the first author was partially supported under NSF grant no. 1211183.
The research of the second author was partially supported by the French ANR Project no. ANR-09-JCJC-0103-01.
The stay of the third author in Bloomington was supported by French ANR project no. ANR-09-JCJC-0103-01.
The research of the fourth author was partially supported under NSF grant no. DMS-0300487
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.