The Hunter-Saxton equation: remarkable structures of symmetries and conserved densities

TL;DR

This paper uncovers complex algebraic and geometric structures in the Hunter-Saxton equation, including infinite symmetries, conserved densities, and recursion operators, revealing new mathematical insights into its integrability.

Contribution

It introduces new recursion operators, classifies compatible operators, and explicitly constructs infinite hierarchies of symmetries and conserved densities for the Hunter-Saxton equation.

Findings

Discovery of three Nijenhuis recursion operators, two of which are new.

Explicit generation of infinitely many conserved densities.

Classification of compatible anti-symmetric operators.

Abstract

In this paper, we present extraordinary algebraic and geometrical structures for the Hunter-Saxton equation: infinitely many commuting and non-commuting $x,t$-independent higher order symmetries and conserved densities. Using a recursive relation, we explicitly generate infinitely many higher order conserved densities dependent on arbitrary parameters. We find three Nijenhuis recursion operators resulting from Hamiltonian pairs, of which two are new. They generate three hierarchies of commuting local symmetries. Finally, we give a local recursion operator depending on an arbitrary parameter. As a by-product, we classify all anti-symmetric operators of a definite form that are compatible with the Hamiltonian operator $D_x^{-1}$.