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Wilton Ripples in Weakly Nonlinear Models of Water Waves: Existence and Computation

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Abstract

In this contribution, we prove that small amplitude, resonant harmonic, spatially periodic traveling waves (Wilton ripples) exist in a family of weakly nonlinear PDEs which model water waves. The proof is inspired by that of Reeder and Shinbrot (Arch. Rat. Mech. Anal. 77:321–347, 1981) and complements the authors’ recent, independent result proven by a perturbative technique (Akers and Nicholls 2021). The method is based on a Banach Fixed Point Iteration and, in addition to proving that this iteration has Wilton ripples as a fixed point, we use it as a numerical method for simulating these solutions. The output of this numerical scheme and its performance are evaluated against a quasi-Newton iteration.

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Authors and Affiliations

  1. Department of Mathematics, Air Force Institute of Technology, Wright Patterson Air Force Base, Dayton, OH, 45433, USA

    Benjamin Akers

  2. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    David P. Nicholls

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  1. Benjamin Akers
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B.A. was supported during the preparation of this manuscript by the Air Force Office of Sponsored Research and the Office of Naval Research. D.P.N. gratefully acknowledges support from the National Science Foundation through Grant No. DMS–1813033.

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Akers, B., Nicholls, D.P. Wilton Ripples in Weakly Nonlinear Models of Water Waves: Existence and Computation. Water Waves 3, 491–511 (2021). https://doi.org/10.1007/s42286-021-00052-2

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