STABILITY OF SOLITARY WAVES
STABILITY OF SOLITARY WAVES
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DOI：
10.1098/rspa.1972.0074
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发表时间：
1972-01-01
影响因子：
--
通讯作者：
BENJAMIN, TB
中科院分区：
文献类型：
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作者：
BENJAMIN, TB
Korteweg-de Vries方程描述了长波在广泛的非线性色散系统中的单向传播，众所周知，它具有表示孤立波的解。目前的分析表明，这些溶液是稳定的，从而证实了长期以来人们所假定的一个性质。稳定性的证明取决于Korteweg-de Vries方程的解随时间不变的两个非线性泛函：这两个泛函在§2中介绍过，在§2中我们记得，Boussinesq认识到它们与孤立波稳定性的关系。稳定性理论所依据的原理在§3中得到了解释，并得到了泛函分析的一些基本思想的支持。§4已证明孤波解是稳定的，其中最严格的步骤是用谱理论来完成的。在附录A中提出了一种由变分法推导出来的方法，根据这种方法，证明稳定性所需的结果可以不依赖于§4中所使用的谱理论而得到。在附录B中，显示了稳定性分析如何容易地适用于“正则化长波方程”的孤波解，该方程最近被Benjamin， Bona和Mahony提倡作为Korteweg-de Vries方程的替代方案。在附录C中，证明了与水中孤立波的精确边值问题有关的变分原理：这与本工作中使用的原理（在§2中介绍）相对应，并为证明精确孤立波的稳定性提供了一些前景。
The Korteweg-de Vries equation, which describes the unidirectional propagation of long waves in a wide class of nonlinear dispersive systems, is well known to have solutions representing solitary waves. The present analysis establishes that these solutions are stable, confirming a property that has for a long time been presumed. The demonstration of stability hinges on two nonlinear functionals which for solutions of the Korteweg-de Vries equation are invariant with time: these are introduced in § 2, where it is recalled that Boussinesq recognized their significance in relation to the stability of solitary waves. The principles upon which the stability theory is based are explained in § 3, being supported by a few elementary ideas from functional analysis. A proof that solitary wave solutions are stable is completed in § 4, the most exacting steps of which are accomplished by means of spectral theory. In appendix A a method deriving from the calculus of variations is presented, whereby results needed for the proof of stability may be obtained independently of spectral theory as used in § 4. In appendix B it is shown how the stability analysis may readily be adapted to solitary-wave solutions of the ‘regularized long-wave equation’ that has recently been advocated by Benjamin, Bona & Mahony as an alternative to the Korteweg-de Vries equation. In appendix C a variational principle is demonstrated relating to the exact boundaryvalue problem for solitary waves in water: this is a counterpart to a principle used in the present work (introduced in §2) and offers some prospect of proving the stability of exact solitary waves.