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Asymptotic Linear Stability of Solitary Water Waves

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Abstract

We prove an asymptotic stability result for the water wave equations linearized around small solitary waves. The equations we consider govern irrotational flow of a fluid with constant density bounded below by a rigid horizontal bottom and above by a free surface under the influence of gravity neglecting surface tension. For sufficiently small amplitude waves, with waveform well-approximated by the well-known sech-squared shape of the KdV soliton, solutions of the linearized equations decay at an exponential rate in an energy norm with exponential weight translated with the wave profile. This holds for all solutions with no component in (that is, symplectically orthogonal to) the two-dimensional neutral-mode space arising from infinitesimal translational and wave-speed variation of solitary waves. We also obtain spectral stability in an unweighted energy norm.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Robert L. Pego

  2. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    Shu-Ming Sun

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  1. Robert L. Pego
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  2. Shu-Ming Sun
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Correspondence to Robert L. Pego.

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Communicated by A. Mielke

This material is based upon work supported by the National Science Foundation under Grant Nos. DMS 06-04420, 09-05723, and 08-07597, and by the Center for Nonlinear Analysis under NSF Grant DMS 06-35983.

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Pego, R.L., Sun, SM. Asymptotic Linear Stability of Solitary Water Waves. Arch Rational Mech Anal 222, 1161–1216 (2016). https://doi.org/10.1007/s00205-016-1021-z

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