Asymptotic stability of solitary waves for the regularized long-wave equation
Asymptotic stability of solitary waves for the regularized long-wave equation
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DOI：
10.1002/(sici)1097-0312(199604)49:4
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发表时间：
1996-04
影响因子：
3
通讯作者：
Judith R. Miller;M. Weinstein
中科院分区：
文献类型：
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作者：
Judith R. Miller;M. Weinstein
我们证明了正则长波方程的一族孤立波是渐近稳定的。孤立波附近的解决方案的大时间动力学研究分解成调制孤立波的解决方案，与速度和相移的函数的t，加上一个扰动。证明策略遵循Pego和Weinstein [24]所使用的策略，他们考虑了Korteweg-deVries-（KdV-）型方程孤立波的渐近稳定性。对于RLW，有必要修改基本的时间尺度，以纳入一个新的时间尺度，这必须由该计划确定。还需要不同的技术来分析孤立波微扰的线性化方程中出现的微分算子的谱理论。特别地，我们利用Pruss[27]的一个结果证明了线性化算子在某个加权函数空间上生成一个具有指数衰减范数的半群，并利用RLW在某个尺度（KdV尺度）下到KdV的形式收敛性来排除线性化算子的非零特征值的存在性. John Wiley & Sons，Inc.
We show that a family of solitary waves for the regularized long-wave (RLW) equation, is asymptotically stable. The large-time dynamics of a solution near a solitary wave are studied by decomposing the solution into a modulating solitary wave, with speed and phase shift that are functions of t, plus a perturbation. The strategy of proof follows that used by Pego and Weinstein [24], who considered the asymptotic stability of solitary waves of Korteweg-deVries- (KdV-) type equations. For RLW it is necessary to modify the basic ansatz to incorporate a new time scale, which must be determined by the scheme. Different techniques are also required to analyze the spectral theory of the differential operator that arises in the linearized equation for a solitary-wave perturbation. In particular, we use a result of Pruss[27] to show that the linearized operator generates a semigroup with exponentially decaying norm on a certain weighted function space, and we exploit the formal convergence of RLW to KdV under a certain scaling (KdV scaling) in order to rule out the existence of nonzero eigenvalues of the linearized operator. © 1996 John Wiley & Sons, Inc.