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Transverse Dynamics of Two-Dimensional Gravity–Capillary Periodic Water Waves

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Abstract

We study the transverse dynamics of two-dimensional gravity–capillary periodic water waves in the case of large surface tension. In this parameter regime, predictions based on model equations suggest that periodic traveling waves are stable with respect to two-dimensional perturbations, and unstable with respect to three-dimensional perturbations which are periodic in the direction transverse to the direction of propagation. In this paper, we confirm the second prediction. We show that, as solutions of the full water-wave equations, the periodic traveling waves are linearly unstable under such three-dimensional perturbations. In addition, we study the nonlinear bifurcation problem near these transversely unstable two-dimensional periodic waves. We show that a one-parameter family of three-dimensional doubly periodic waves is generated in a dimension-breaking bifurcation. The key step of this approach is the analysis of the purely imaginary spectrum of the linear operator obtained by linearizing the water-wave equations at a periodic traveling wave. Transverse linear instability is then obtained by a perturbation argument for linear operators, and the nonlinear bifurcation problem is studied with the help of a center-manifold reduction and the classical Lyapunov center theorem.

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Notes

  1. Here and in the remainder of the paper \(c\) denotes a positive constant which does not depend upon \(\varepsilon \).

  2. This choice of the function spaces is coming from the recent work [17] and is simpler than the one used in [4].

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Acknowledgments

This paper would have never been written without Klaus Kirchgässner. Thanks to him, I discovered this topic of transverse dynamics and the beauty of nonlinear waves. His advice, support, and encouragements were invaluable over the years. I will always remember his enthusiasm and his kind and caring personality. I would also like to thank Mark Groves and Erik Wahlén for many helpful discussions involving this work. The choice of the function spaces in Sect. 4.2 is due to Erik, and his comments helped to improve and simplify several arguments in the proofs. This work was partially supported by the grant BOND of the French National Research Agency and a BQR-PRES grant of the Federal University of Bourgogne Franche-Comté.

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  1. Laboratoire de Mathématiques, Université de Franche-Comté, 25030 , Besançon cedex, France

    Mariana Haragus

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  1. Mariana Haragus
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Correspondence to Mariana Haragus.

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This paper is dedicated to the memory of Klaus Kirchgässner.

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Haragus, M. Transverse Dynamics of Two-Dimensional Gravity–Capillary Periodic Water Waves. J Dyn Diff Equat 27, 683–703 (2015). https://doi.org/10.1007/s10884-013-9336-z

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