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Modulated equations of Hamiltonian PDEs and dispersive shocks

Abstract : Motivated by the ongoing study of dispersive shock waves in non integrable systems , we propose and analyze a set of wave parameters for periodic waves of a large class of Hamiltonian partial differential systems-including the generalized Korteweg-de Vries equations and the Euler-Korteweg systems-that are well-behaved in both the small amplitude and small wavelength limits. We use this parametrization to determine fine asymptotic properties of the associated modulation systems, including detailed descriptions of eigenmodes. As a consequence, in the solitary wave limit we prove that modulational instability is decided by the sign of the second derivative-with respect to speed, fixing the endstate-of the Boussinesq moment of instability; and, in the harmonic limit, we identify an explicit modulational instability index, of Benjamin-Feir type.
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https://hal.archives-ouvertes.fr/hal-02365963
Contributor : Sylvie Benzoni-Gavage Connect in order to contact the contributor
Submitted on : Wednesday, February 10, 2021 - 10:15:54 AM
Last modification on : Saturday, September 24, 2022 - 3:36:05 PM

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Sylvie Benzoni-Gavage, Colin Mietka, L. Miguel Rodrigues. Modulated equations of Hamiltonian PDEs and dispersive shocks. Nonlinearity, 2021, 34 (1), pp.578-641. ⟨10.1088/1361-6544/abcb0a⟩. ⟨hal-02365963v2⟩

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