色散Korteweg-de Vries流体动力学中的自相似波破碎
Self-similar wave breaking in dispersive Korteweg-de Vries hydrodynamics
摘要：
讨论了非线性波在静止介质中传播时的破缺问题。假定波的轮廓在破碎时刻由坐标$x$的函数$(-x)^{1/n}$（$x<0$，正脉冲）或$-x^{1/n}$（$x>0$，负脉冲）描述。波的演化由Korteweg-de Vries方程控制，导致形成色散冲击波。在正脉冲情况下，色散激波形成在波结构的前缘，而在负脉冲情况下，在其后缘。色散激波的动力学由Whitham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$n>1$，以封闭形式获得了Whitham方程的解。
英文摘要：
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 $(-x)^{1/n}$ ($x<0$, positive pulse) or $-x^{1/n}$ ($x>0$, negative pulse) of the coordinate $x$. Evolution of the wave is governed by the Korteweg-de Vries equation resulting in 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 $n>1$.
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