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

TL;DR

This paper introduces the r-Hunter-Saxton equation, explores its solutions, blow-up behavior, symmetry reductions, and constructs weak solutions, advancing understanding of this generalized nonlinear PDE.

Contribution

It presents the first formulation and analysis of the r-Hunter-Saxton equation, including solution characterization and weak solution construction.

Findings

Solutions characterized and blow-up times quantified

Piecewise linear functions shown to be weak solutions

Symmetry reductions explored

Abstract

In this work we introduce the r-Hunter-Saxton equation, a generalisation of the Hunter-Saxton equation arising as extremals of an action principle posed in L_r. 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 r-Hunter-Saxton equation.