Asymptotic linear stability of solitary water wavesRobert L. Pego1and Shu-Ming Sun2August 31, 2010AbstractWe prove an asymptotic stability result for the water wave equations linearized aroundsmall solitary waves. The equations we consider govern irrotational flow of a fluid with constantdensity bounded below by a rigid horizontal bottom and above by a free surface under theinfluence of gravity neglecting surface tension. For sufficiently small amplitude waves, withwaveform well-approximated by the well-known sech-squared shape of the KdV solito n, s olutionsof the linearized equations decay at an exponential rate in an energy nor m with exponentialweight translated with the wave profile. This holds for all so lutions with no component in (i.e.,symplectically orthogonal to) the two-dimensional neutral-mode space arising from infinitesimaltranslational and wave-speed variation of solitary waves. We also obtain spectral stability in anunweighted energy norm.1Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh,PA 15213. Email: [email protected] of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061. Email:[email protected] Asymptotic linear stability of solitary water waves1 IntroductionThe discovery of solitary water waves by J. Scott Russell in 1834 was a seminal event in nonlinearscience. Russell’s ob s ervations gave him immediate confidence in the significance of these waves,and led him to carry out an extensive program of experiments investigating solitary waves an d theirinteractions [40]. But mathematical understanding was slow to develop. The first significant stepsforward were made by Boussinesq [7, 8, 9, 10] and Rayleigh [38] by carefully balancing long-waveand small-amplitude approximations. The simplest usefu l model (derived by Boussinesq alreadyin 1872, see [10, p. 360] and [30]) is the famous K orteweg-de Vries equation [25]. Its sech2solitonsolution approximates the shape of small-amplitude solitary water waves.Given the status of the KdV equation as an approximate model, it is important to understandwhether the soliton solutions of the KdV equation are approximations of some solutions of a moreexact water wave model with similar properties. In this paper, we focus on questions of stabilityfor exact solitary wave solutions of the Euler equations that govern incompressible and irrotationalmotions of an inviscid, constant-density fluid of finite depth. The fluid occupies a two-dimensionaldomain whose lower boundary is a flat rigid bottom and whose upper boundary is a fr ee surfacethat forms an interface with air of n egligible density and viscosity. Surface tension on the freesurface is neglected.For these water wave equations, the existence of solitary wave solutions with shape well-approximated by the KdV soliton was proved by Lavrent’ev [27], Friedrichs and Hyers [13] andBeale [1]. If the surface tension is positive and small, finite-energy, single-hump solitary wavesare not known to exist, and indeed, exact traveling waves approximated by the KdV soliton maynot exist without ‘ripples at infinity’ [2, 42]. For large surface tension, solitary water waves ofdepression exist [42], but the relevant physical regime corresponds to water dep th less than 0.5 cm.Explaining the stability of solitary water waves mathematically remains a very challenging prob-lem, despite considerable physical and numerical evidence. Remarkably, a valuable step forwardwas made already by Boussinesq [9, 10], who argued for their stability based on a quantity he calledthe ‘moment of instability,’ which he showed was invariant in time based on the KdV approxima-tion. Over a century later, Benjamin [3] made use of the same quantity as a Hamiltonian energy,constrained by a time-invariant momentum functional, to develop a rigorous variational methodto prove orbital stability for the set of solitary-wave solutions of the KdV equation. Benjamin’sarguments were improved and perfected by Bona [5].Variational methods for orbital stability and instability in Hamiltonian wave equations, basedon the use of energy-momentum functionals, were subsequently greatly advanced by many authors.Notably, the general theory of Grillakis et al. [20, 21] has been applied extensively to m any physicalsystems. Using variational methods of this type for the case of solitary water waves of depressionfor the Euler equations with large surface tension, orbital stability conditional on global existencewas obtained by Mielke [29] and Buffoni [11]. For small surface tension, such variational stabilityresults have also been obtained recently by Groves and Wahlen [22] for oscillatory traveling wavepackets of finite energy (also called solitary waves by several authors).For solitary waves with zero surface tension, however, it appears hopeless to study stabilityusing variational methods based on constrained minimization. As remarked by Bona and Sachs [6],R. L. Pego and S.-M. Sun 3the usual energy-momentum functional is highly indefinite in this case—The second variation lacksthe finite-dimensional indefiniteness property key to the success of current variational methods.Regarding the stability of solitary waves with zero surface tension, the only existing rigorous workappears to be the recent paper of Lin [28], which addresses the linear instability of large waves closeto the wave of maximum height.The present study involves a direct analysis of the Euler equations linearized about a small-amplitude solitary wave solution. The linearized equations have a natural two-dimensional spaceof neutral modes arising from infinitesimal shifts and changes in wave speed of solitary waves. Wededuce asymptotic stability for solutions in a space of perturbations naturally constrained to omitthese neutral-mo de components, being symplectically orthogonal to them. Asymptotic stability isobtained in a norm that is weighted spatially to decay exponentially behind the wave profile. Thetime decay of such a norm corresponds to unidirectional scattering behavior for wave perturbations.The weighted-norm linear stability analysis is also used to obtain a spectral stability r esult in anunweighted energy norm. Our main results are stated precisely in section 3.The use of exponential weights to obtain nonlinear asymptotic orbital stability for solitary waveswas developed for KdV solitons by Pego and Weinstein [36], for regularized long-wave