Integrability structures of the generalized Hunter--Saxton equation

TL;DR

This paper explores the integrability of the generalized Hunter--Saxton equation by deriving Lax pairs, recursion operators, and symmetries, and demonstrates how these structures can be used to construct exact solutions.

Contribution

It provides the first detailed analysis of the integrability structures of the generalized Hunter--Saxton equation, including Lax representation and symmetry algebra.

Findings

Lax representation with nonremovable spectral parameter obtained

Infinite-dimensional Lie algebra of higher symmetries identified

Explicit construction of exact solutions using higher order symmetries

Abstract

We consider integrability structures of the generalized Hunter--Saxton equation. In particular, we obtain the Lax representation with nonremovable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate an infinite-dimensional Lie algebra of higher symmetries, and prove existence of infinite number of cosymmetries of higher order. Further, we give an example of employing the higher order symmetry to constructing exact globally defined solutions for the generalized Hunter--Saxton equation.