Conservation laws and symmetries of Hunter-Saxton equation: revisited

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].