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Asymptotic linear stability of solitary water waves Robert L Pego1 and Shu Ming Sun2 August 31 2010 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 i e 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 1 Department of Mathematical Sciences and Center for Nonlinear Analysis Carnegie Mellon University Pittsburgh PA 15213 Email rpego cmu edu 2 Department of Mathematics Virginia Polytechnic Institute and State University Blacksburg VA 24061 Email sun math vt edu 1 2 1 Asymptotic linear stability of solitary water waves Introduction The discovery of solitary water waves by J Scott Russell in 1834 was a seminal event in nonlinear science Russell s observations gave him immediate confidence in the significance of these waves and led him to carry out an extensive program of experiments investigating solitary waves and their interactions 40 But mathematical understanding was slow to develop The first significant steps forward were made by Boussinesq 7 8 9 10 and Rayleigh 38 by carefully balancing long wave and small amplitude approximations The simplest useful model derived by Boussinesq already in 1872 see 10 p 360 and 30 is the famous Korteweg de Vries equation 25 Its sech2 soliton solution approximates the shape of small

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