In the setting of a distributional product, we consider a Riemann problem for the Hunter-Saxton equation ${[{u}_{t}+{((1/\mathrm{2}){u}^{\mathrm{2}})}_{x}]}_{x}=(1/\mathrm{2}){u}_{x}^{2}$ in a convenient space of discontinuous functions. With the help of a consistent extension of the classical solution concept, two classes of discontinuous solutions are obtained: one class of conservative solutions and another of dispersive solutions. A necessary and sufficient condition for the propagation of a distributional profile as a travelling wave is also presented, which allows identifying an interesting set of explicit distributional travelling waves. In the paper, we will show some results we have obtained by applying this framework to other equations and systems.

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