The propagation and collision behavior of \(\varvec{\delta }'\) waves in a model of three partial differential equations

Both propagation and collision behavior of \(\delta '\) ${\delta }^{\prime }$ waves in a model of three partial differential equations are investigated. These are Cauchy problems for this model with initial values involving the first-order derivative of Dirac measure. Based on an \(\alpha \) $\alpha$ -solution concept defined in the framework of multiplication of distributions, we obtain a unique \(\alpha \) $\alpha$ -solution for the propagation process of an \(\delta '\) ${\delta }^{\prime }$ wave. This \(\alpha \) $\alpha$ -solution shows rigorously the location, velocity and strengths of the \(\delta '\) ${\delta }^{\prime }$ wave. Besides, with the help of this result, we also derive an \(\alpha \) $\alpha$ -solution for the collision behavior of two \(\delta '\) ${\delta }^{\prime }$ waves. Such \(\alpha \) $\alpha$ -solution not only reveals, respectively, the location, speed and strengths of each one of the two \(\delta '\) ${\delta }^{\prime }$ waves before their interaction, but also describes the location, velocity and strengths of a new emerged \(\delta '\) ${\delta }^{\prime }$ wave after their collision. Moreover, this work can be applied to investigate the Cauchy problems for a system of partial differential equations with general initial values involving the derivative of the Dirac measure.