Groves M D, Haragus M, Sun S M
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK.
Philos Trans A Math Phys Eng Sci. 2002 Oct 15;360(1799):2189-243. doi: 10.1098/rsta.2002.1066.
The existence of a line solitary-wave solution to the water-wave problem with strong surface-tension effects was predicted on the basis of a model equation in the celebrated 1895 paper by D. J. Korteweg and G. de Vries and rigorously confirmed a century later by C. J. Amick and K. Kirchgässner in 1989. A model equation derived by B. B. Kadomtsev and V. I. Petviashvili in 1970 suggests that the Korteweg-de Vries line solitary wave belongs to a family of periodically modulated solitary waves which have a solitary-wave profile in the direction of motion and are periodic in the transverse direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. It is shown that the Korteweg-de Vries solitary wave undergoes a dimension-breaking bifurcation that generates a family of periodically modulated solitary waves. The term dimension-breaking phenomenon describes the spontaneous emergence of a spatially inhomogeneous solution of a partial differential equation from a solution which is homogeneous in one or more spatial dimensions.
基于D. J. 科特韦格和G. 德弗里斯1895年著名论文中的一个模型方程，人们预测了具有强表面张力效应的水波问题存在行孤立波解，一个世纪后的1989年，C. J. 阿米克和K. 基尔希加斯纳对此进行了严格证实。B. B. 卡多姆采夫和V. I. 彼得维夏什维利于1970年推导的一个模型方程表明，科特韦格 - 德弗里斯行孤立波属于一类周期调制孤立波，这类波在运动方向上具有孤立波轮廓，在横向方向上是周期性的。本文针对完整的水波问题严格证实了这一预测。结果表明，科特韦格 - 德弗里斯孤立波经历了一个破维分岔，产生了一族周期调制孤立波。术语“破维现象”描述了一个偏微分方程的空间非均匀解从一个在一个或多个空间维度上均匀的解自发出现的情况。
