| Abstract: |
| We discuss periodic traveling wave solutions of the reduced Ostrovsky equation, which arises in the context of long surface and internal gravity waves in a rotating fluid. Our aim is to reformulate this local equation into a nonlocal dispersive equation, and study the properties of their traveling waves by analyzing the corresponding convolution kernel. Of particular interest is the existence of a highest, peaked, traveling wave solution, which we obtain as a limiting case at the end of a global bifurcation branch. We show that the regularity at the crest of a highest wave is precisely Lipschitz. While the reduced Ostrovsky equation is of order -2, similar investigations have been done for equations of different orders. This work is part of a longer study with the aim of understanding the interaction of nonlinearities and dispersion, which leads to families of ever-higher waves, ending in a typically singular and highest one. |
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