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

Abstract

AbstractIn 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 $c_{0}$ c 0 -speed solitary wave, are approximated by a long curve on the family of solitary waves with the time-varying speed $|c(t)-c_{0}|$ | c ( t ) − c 0 | being small, in the long-time period (i.e., $0\leqslant t\leqslant O(\frac{1}{\epsilon ^{\tau}})$ 0 ⩽ t ⩽ O ( 1 ϵ τ ) ). Meanwhile, the approximation difference in a suitably weighted space $H^{1}_{a}(\mathbb{R})$ H a 1 ( 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\leqslant t<+\infty $ 0 ⩽ t < + ∞ , the solutions are approximated by a long curve on the family of solitary waves with the exponential decay speed $c(t)= c_{0}e^{-\beta t}$ c ( t ) = c 0 e − β t ($0<\beta \leqslant 1$ 0 < β ⩽ 1 ), when the initial values are near a $c_{0}$ c 0 -speed solitary wave. However, here, the approximation difference merely defined in $H^{1}(\mathbb{R})$ H 1 ( R ) depends on the damping coefficient ϵ and the exponential decay coefficient β.