The long-time behavior of solitary waves for the weakly damped KdV equation

Abstract

In this paper, we first introduce the long-time behavior stability of solitary waves for the weakly damped Korteweg–de Vries equation. More concretely, solutions of the dissipative system with the initial values near a c_{0}-speed solitary wave, are approximated by a long curve on the family of solitary waves with the time-varying speed |c(t)-c_{0}| being small, in the long-time period (i.e., 0leqslant tleqslant O(frac{1}{epsilon ^{tau}})). Meanwhile, the approximation difference in a suitably weighted space H^{1}_{a}(mathbb{R}) is of the order of the damping coefficient and of some kind of exponential weight form. As a comparison, we also study the long-time behavior stability, i.e., for 0leqslant t<+infty , the solutions are approximated by a long curve on the family of solitary waves with the exponential decay speed c(t)= c_{0}e^{-beta t} (0<beta leqslant 1), when the initial values are near a c_{0}-speed solitary wave. However, here, the approximation difference merely defined in H^{1}(mathbb{R}) depends on the damping coefficient ϵ and the exponential decay coefficient β.