Conservation laws and symmetries of Hunter-Saxton equation: revisited

TL;DR

This paper revisits the Hunter-Saxton equation, revealing its rich symmetry and conservation law structures through transformations, and connects these to well-known integrable systems like the mKdV and Fordy-Gibbons equations.

Contribution

It uncovers new conserved densities, studies symmetry hierarchies under transformations, and links the HS equation to other integrable equations, enhancing understanding of its integrability properties.

Findings

Conserved densities involving arbitrary functions are derived.

Symmetry hierarchies are linearized or related to known equations.

A sixth order hereditary recursion operator is constructed for a fifth order symmetry.

Abstract

Through a reciprocal transformation $\mathcal{T}_0$ induced by the conservation law $\partial_t(u_x^2) = \partial_x(2uu_x^2)$, the Hunter-Saxton (HS) equation $u_{xt} = 2uu_{2x} + u_x^2$ is shown to possess conserved densities involving arbitrary smooth functions, which have their roots in infinitesimal symmetries of $w_t = w^2$, the counterpart of the HS equation under $\mathcal{T}_0$. Hierarchies of commuting symmetries of the HS equation are studied under appropriate changes of variables initiated by $\mathcal{T}_0$, and two of these are linearized while the other is identical to the hierarchy of commuting symmetries admitted by the potential modified Korteweg-de Vries equation. A fifth order symmetry of the HS equation is endowed with a sixth order hereditary recursion operator by its connection with the Fordy-Gibbons equation. These results reveal the origin for the rich and remarkable structures of the HS equation and partially answer the questions raised by Wang [{\it Nonlinearity} {\bf 23}(2010) 2009].