We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation partial derivative(t)u + u partial derivative(x)u + partial derivative(x)3u = 0, is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, partial derivative(t)u + partial derivative(x) f(u) + partial derivative(x)3u = 0. In particular, we study the case where f(u) = u(p + 1)/(p + 1), p = 1, 2, 3 (and 3 < p < 4, for u > 0, with f is-an-element-of C4). The same asymptotic stability result for KdV is also proved for the case p = 2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values of p between 3 and 4. (The solitary waves are know to undergo a transition from stability to instability as the parameter p increases beyond the critical value p = 4.) The solution is decomposed into a modulating solitary wave, with time-varying speed c(t) and phase gamma(t) (bound state part), and an infinite dimensional perturbation (radiating part). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. As p --> 4-, the local decay or radiation rate decreases due to the presence of a resonance pole associated with the linearized evolution equation for solitary wave perturbations.
