Kamchatnov A M
Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow 108840, Russia and Moscow Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny, Moscow 141700, Russia.
Chaos. 2019 Feb;29(2):023106. doi: 10.1063/1.5066038.
We discuss the problem of breaking of a nonlinear wave in the process of its propagation into a medium at rest. It is supposed that the profile of the wave is described at the breaking moment by the function (-x) (x<0, positive pulse) or -x (x>0, negative pulse) of the coordinate x. Evolution of the wave is governed by the Korteweg-de Vries equation resulting in the formation of a dispersive shock wave. In the positive pulse case, the dispersive shock wave forms at the leading edge of the wave structure and in the negative pulse case, at its rear edge. The dynamics of dispersive shock waves is described by the Whitham modulation equations. For power law initial profiles, this dynamics is self-similar and the solution of the Whitham equations is obtained in a closed form for arbitrary n>1.
我们讨论了非线性波在传播到静止介质过程中破裂的问题。假设在破裂时刻，波的轮廓由坐标(x)的函数((-x))（(x < 0)，正脉冲）或(-x)（(x > 0)，负脉冲）描述。波的演化由科特韦格 - 德弗里斯方程控制，导致形成色散冲击波。在正脉冲情况下，色散冲击波在波结构的前沿形成；在负脉冲情况下，在其后沿形成。色散冲击波的动力学由惠特姆调制方程描述。对于幂律初始轮廓，这种动力学是自相似的，并且对于任意(n > 1)，惠特姆方程的解以封闭形式得到。
