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The Modulational Instability for a Generalized Korteweg–de Vries Equation

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Abstract

We study the spectral stability of a family of periodic standing wave solutions to the generalized Korteweg–de Vries in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to perturbations of long wavelength. A study of the normal form of the characteristic polynomial of the monodromy map (the periodic Evans function) in a neighborhood of the origin in the spectral plane leads to two different instability indices. The first, an orientation index, counts modulo 2 the total number of periodic eigenvalues on the real axis. This index is a generalization of the one which governs the stability of the solitary wave. The second, a modulational instability index, provides a necessary and sufficient condition for the existence of a long-wavelength instability. This index is essentially the quantity calculated by Hǎrǎguş and Kapitula in the small amplitude limit. Both of these quantities can be expressed in terms of the map between the constants of integration for the ordinary differential equation defining the traveling waves and the conserved quantities of the partial differential equation. These two indices together provide a good deal of information about the spectrum of the linearized operator. We sketch the connection of this calculation to a study of the linearized operator—in particular we perform a perturbation calculation in terms of the Floquet parameter. This suggests a geometric interpretation attached to the vanishing of the orientation index previously mentioned.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 W Green St, Urbana, IL, 61801, USA

    Jared C. Bronski & Mathew A. Johnson

  2. Department of Mathematics, Indiana University, Rawles Hall, Bloomington, IN, 47405, USA

    Mathew A. Johnson

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  1. Jared C. Bronski
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Correspondence to Jared C. Bronski.

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Communicated by T.-P. Liu

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Bronski, J.C., Johnson, M.A. The Modulational Instability for a Generalized Korteweg–de Vries Equation. Arch Rational Mech Anal 197, 357–400 (2010). https://doi.org/10.1007/s00205-009-0270-5

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