Asymptotic stability of solitary waves
Asymptotic stability of solitary waves
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DOI：
10.1007/bf02101705
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发表时间：
1994-08
影响因子：
2.4
通讯作者：
R. Pego;M. Weinstein
中科院分区：
文献类型：
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作者：
R. Pego;M. Weinstein
我们证明了Korteweg-de弗里斯方程的孤立波族（1-孤子）是渐近稳定的。我们的方法也适用于一类广义Korteweg-de弗里斯方程的孤波解。特别地，我们研究了f（u）=up+1/（p+1），p=1，2，3（且3<p<4，对于u>0，f ∈C4）的情形。对于KdV的渐近稳定性结果也被证明为对情形ep =2（修正的Korteweg-de弗里斯方程）。我们还证明了渐近稳定的孤立波的家庭，但所有的p值在3和4之间的有限数量。(The已知当参数p增加到超过临界值p =4时，孤立波经历从稳定到不稳定的转变。该解被分解为具有时变速度c（t）和相位γ（t）的调制孤立波（束缚态部分）和无限维扰动（辐射部分）。的扰动示出指数衰减的时间，在局部意义上相对于一个帧移动的孤立波。Asp→4−时，由于存在与孤立波扰动线性化演化方程相关的共振极，局域衰变或辐射率降低。
We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equationis asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations,In particular, we study the case wheref(u)=up+1/(p+1),p=1, 2, 3 (and 3<p<4, foru>0, withf∈C4). The same asymptotic stability result for KdV is also proved for the casep=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 ofpbetween 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameterpincreases beyond the critical valuep=4.) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t)and phase γ(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. Asp→4−, the local decay or radiation rate decreases due to the presence of aresonance poleassociated with the linearized evolution equation for solitary wave perturbations.