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Journal of Physical Studies 10(1), 24–28 (2006)
DOI: https://doi.org/10.30970/jps.10.24 ON MODULATION INSTABILITY OF STOKES WAVES WITHIN THE EXTENDED SET OF NLSE WITH THE MIDDLE FLOW EQUATIONYu. Sedletsky
Institute of Physics, National Academy of Sciences of Ukraine, |
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For Stokes waves on the surface of ideal fluid layer, the set of non-linear Schrödinger equation (NLSE) for the by-pass first harmonics and equation for the zeroth harmonics is extended with full linear dispersion taken into account in both equations. Without traditional assumption about the motion of middle flow with the group velocity on the frequency of rapid filling, the 4-order equation for the perturbation frequency is derived and modulation instability (MI) of the waves under consideration is studied. The interaction is demonstrated for disperse branches corresponding to four roots of this equation. The appearence of MI bands remaining described within NLSE is also shown. It follows from the analysis of the obtained expressions that the limit $kh=1.363$ ($h$ is the fluid depth, $k$ is the wave number) found by Benjamin, Feir, Whitham, Hasimoto and Ono for the transition between stable and instable fluid is such only in a particular case of small amplitude of unperturbed wave and wave numbers of the perturbing wave.
PACS number(s): 05.45.-a, 05.45.Yv, 47.35.+i