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Modulational Instability and Variational Structure
Author(s) -
Bronski J. C.,
Hur V. M.
Publication year - 2014
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12029
Subject(s) - modulational instability , instability , floquet theory , mathematics , hamiltonian (control theory) , operator (biology) , mathematical analysis , physics , classical mechanics , mathematical physics , quantum mechanics , nonlinear system , mathematical optimization , biochemistry , chemistry , repressor , transcription factor , gene
We study the modulational instability of periodic traveling waves for a class of Hamiltonian systems in one spatial dimension. We examine how the Jordan block structure of the associated linearized operator bifurcates for small values of the Floquet exponent to derive a criterion governing instability to long wavelengths perturbations in terms of the kinetic and potential energies, the momentum, the mass of the underlying wave, and their derivatives. The dispersion operator of the equation is allowed to be nonlocal, for which Evans function techniques may not be applicable. We illustrate the results by discussing analytically and numerically equations of Korteweg‐de Vries type.