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Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators

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Abstract

We present a general counting result for the unstable eigenvalues of linear operators of the form J L in which J and L are skew- and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operator K such that the operators J L and J K commute, we prove that the number of unstable eigenvalues of J L is bounded by the number of nonpositive eigenvalues of K. As an application, we discuss the transverse stability of one-dimensional periodic traveling waves in the classical KP-II (Kadomtsev–Petviashvili) equation. We show that these one-dimensional periodic waves are transversely spectrally stable with respect to general two-dimensional bounded perturbations, including periodic and localized perturbations in either the longitudinal or the transverse direction, and that they are transversely linearly stable with respect to doubly periodic perturbations.

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Authors and Affiliations

  1. Institut FEMTO-ST & LMB, Université Bourgogne Franche–Comté, 25030, Besançon, France

    Mariana Haragus

  2. Department of Mathematics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Jin Li & Dmitry E. Pelinovsky

Authors
  1. Mariana Haragus
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  2. Jin Li
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  3. Dmitry E. Pelinovsky
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Correspondence to Dmitry E. Pelinovsky.

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Communicated by W. Schlag

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Haragus, M., Li, J. & Pelinovsky, D.E. Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators. Commun. Math. Phys. 354, 247–268 (2017). https://doi.org/10.1007/s00220-017-2898-6

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  • DOI: https://doi.org/10.1007/s00220-017-2898-6

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