Modulational instabilities of periodic traveling waves in deep water

High-frequency instabilities of Stokes waves

“…Building on previous work by Akers (2015) and Creedon et al (2021a,b), we have developed a formal perturbation method to compute high-frequency instabilities of small-amplitude Stokes wave solutions of Euler's equations in arbitrary depth. This method allows one to approximate an entire high-frequency isola.…”

“…This same approach was first used in Creedon, Deconinck & Trichtchenko (2021a) on the Kawahara equation and in Creedon, Deconinck & Trichtchenko (2021b) on a Boussinesq-Whitham system. An outline of the leading-order calculations of the method in infinite depth is also used by Akers (2015), where the emphasis is on understanding the analyticity properties of the stability spectrum as a function of the boundary conditions imposed on the perturbations (i.e. as a function of the Floquet exponent), and on the connections with resonant interaction theory.…”

High-Frequency Instabilities of a Boussinesq–Whitham System: A Perturbative Approach

“…Remark 3. The WWP shares the same collided eigenvalues with HPT-BW, since (22) is also the dispersion relation of the WWP.…”

“…For a given high-frequency isola of an HPT-BW Stokes wave, we obtain (i) an asymptotic range of Floquet exponents that parameterizes the isola, (ii) an asymptotic estimate for the most unstable spectral element of the isola, (iii) expressions of curves that are asymptotic to the isola, and (iv) wavenumbers for which the given isola is not present. Our approach is inspired by a perturbation method outlined in [22], but modified appropriately for higher-order calculations. We compare all asymptotic results with numerical results computed by the FFH method.…”

Stability of Periodic Traveling Wave Solutions to the Kawahara Equation

“…Away from the resonant regime, it has been noted by McLean [16] that an instability that can be described by an N th order interaction that grows at order N . Akers [1] further discussed the relationship between the N th order interaction and the collision of unstable eigenvalues in the spectral stability problem. He devised a perturbation expansion for these eigenvalues and discussed their radius of analyticity illustrating the mechanism for instabilities.…”