Asymptotic stability of solitary waves for the regularized long wave equation.

Abstract

We show that a family of solitary waves for the Regularized Long Wave (RLW) equation,$$(I-\partial\sbsp{x}{2})\partial\sb{t}u + \partial\sb{x}(u + {1\over2}u\sp2) = 0\ ,$$is asymptotically stable, by decomposing the solution into a modulating solitary wave with speed and phase shift which are functions of t. The basic outline of the proof follows that used by Pego and Weinstein (1994) to show asymptotic stability of solitary waves for the Korteweg-deVries (KdV) equation. For RLW it is necessary to modify the basic Ansatz to incorporate a new time scale which must be determined by the scheme. New techniques are also required to analyze the spectral theory of the differential operator which arises in the linearized equation for a solitary-wave perturbation. In particular, we use a result of Pruss (1984) to show that the linearized operator generates a semigroup with exponentially decaying norm on a certain weighted function space, and we exploit the formal convergence of RLW to KdV under a certain scaling ("KdV-scaling") in order to rule out nonzero eigenvalues of the linearized operator. The semigroup decay estimates depend, in part, on norm bounds for a certain operator related to the resolvent of the linearized operator. These bounds may be obtained by exploiting special structure of the operator which arises for RLW. However, we also present a more general strategy for bounding the resolvent norm $\Vert(I-C)\sp{-1}\Vert$ when C is a trace-class operator.