Publication 10-CNA-15
** ****Asymptotic Linear Stability of Solitary Water Waves **

Robert L. Pego

Department of Mathematical Sciences

and Center for Nonlinear Analysis

Carnegie Mellon University

Pittsburgh, PA 15213, USA

rpego@cmu.edu

Shu-Ming Sun

Department of Mathematics

Virginia Polytechnic Institute and State University

Blacksburg, VA 24061, USA

sun@math.vt.edu

**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.

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