The r-Hunter–Saxton equation, smooth and singular solutions and their approximation

Abstract

<jats:title>Abstract</jats:title> <jats:p>In this work we introduce the <jats:italic>r</jats:italic>-Hunter–Saxton equation, a generalisation of the Hunter–Saxton equation arising as extremals of an action principle posed in <jats:italic>L</jats:italic> <jats:sup> <jats:italic>r</jats:italic> </jats:sup>. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the <jats:italic>r</jats:italic>-Hunter–Saxton equation.</jats:p>