158 research outputs found

    Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves

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    This paper is concerned with a priori C∞C^\infty regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a diophantine condition are automatically C∞C^\infty. In particular, we prove that the solutions defined by Iooss and Plotnikov are C∞C^\infty. Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.Comment: Corrected versio

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    Letter From the Editors

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    Freak Edge Waves

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    The nonlinear and unsteady dynamics of the edge waves is discussed. Two physical processes: dispersive focusing and nonlinear modulational instability, are studied. Both mechanisms can induce the appearance of the short-living large-amplitude isolated waves and intense wave packets (“freak edge waves”)

    Letter from the Editors

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