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

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., 0⩽ t⩽ O(1/ϵ ^τ) ). Meanwhile, the approximation difference in a suitably weighted space H^1_a(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 0⩽ t<+∞ , the solutions are approximated by a long curve on the family of solitary waves with the exponential decay speed c(t)= c_0e^-β t ( 0<β⩽ 1 ), when the initial values are near a c_0 -speed solitary wave. However, here, the approximation difference merely defined in H^1(ℝ) depends on the damping coefficient ϵ and the exponential decay coefficient β .