| Abstract: |
| In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave is unstable to long wavelength perturbations, provided that (the carrier wave number)$\times$(the undisturbed water depth)$>1.363\dots$. In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous fashion. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are very complicated. One may resort to simple approximate models to gain insight.
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I will begin with wave breaking and ill-posedness in Whitham`s shallow water model and Korteweg-de Vries equations with fractional dispersion. I will then discuss modulational instability of a small amplitude wave of the Whitham equation and an extension to large amplitude waves for a class of Hamiltonian systems, permitting nonlocal dispersion. |
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