Asymptotic linear stability of solitary water waves
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 wellapproximated by the wellknown sechsquared 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 twodimensional neutralmode space arising from infinitesimal translational and wavespeed variation of solitary waves. We also obtain spectral stability in an unweighted energy norm.
 Publication:

arXiv eprints
 Pub Date:
 September 2010
 DOI:
 10.48550/arXiv.1009.0494
 arXiv:
 arXiv:1009.0494
 Bibcode:
 2010arXiv1009.0494P
 Keywords:

 Mathematics  Analysis of PDEs;
 Nonlinear Sciences  Pattern Formation and Solitons;
 76B25;
 35C07;
 37K40
 EPrint:
 50 pages, LaTeX with ulem package