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 derivative(x)(2))partial derivative(t)u partial derivative(x)(u + 1/2 u(2)) = 0, is asymptotically stable. Th e large-time dynamics of a solution near a solitary wave are studied b y decomposing the solution into a modulating solitary wave, with speed and phase shift that are functions of t, plus a perturbation. The str ategy of proof follows that used by Pego and Weinstein [24], who consi dered the asymptotic stability of solitary waves of Korteweg-deVries- (KdV-) type equations. For RLW it is necessary to modify the basic ans atz to incorporate a new time scale, which must be determined by the s cheme. Different techniques are also required to analyze the spectral theory of the differential operator that arises in the linearized equa tion for a solitary-wave perturbation. In particular, we use a result of Pruss [27] to show that the linearized operator generates a semigro up with exponentially decaying norm on a certain weighted function spa ce, and we exploit the formal convergence of RLW to KdV under a certai n scaling (KdV scaling) in order to rule out the existence of nonzero eigenvalues of the linearized operator. (C) 1996 John Wiley & Sons, In c.