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

We show that a family of solitary waves for the regularized long‐wave (RLW) equation,$$(I-\partial^{2}_{x})\partial_t u + \partial_x (u+{1\over 2}u^2) = 0,$$ is asymptotically stable. The large‐time dynamics of a solution near a solitary wave are studied by decomposing the solution into a modulating solitary wave, with speed and phase shift that are functions of t , plus a perturbation. The strategy of proof follows that used by Pego and Weinstein [24], who considered the asymptotic stability of solitary waves of Korteweg‐deVries‐ (KdV‐) type equations. For RLW it is necessary to modify the basic ansatz to incorporate a new time scale, which must be determined by the scheme. Different techniques are also required to analyze the spectral theory of the differential operator that arises in the linearized equation for a solitary‐wave perturbation. In particular, we use a result of Prüss[27] 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 the existence of nonzero eigenvalues of the linearized operator. © 1996 John Wiley & Sons, Inc.