Poisson structure of a modified Hunter–Saxton equation

Abstract

Both the well-known Korteweg–de Vries equation and the Hunter–Saxton equation with a linear wux-term, −utxx = −2ωux + 2uxuxx + uuxxx, ω > 0, are bi-Hamiltonian in the sense that each equation is Hamiltonian with respect to two compatible, but distinct, Poisson brackets. We present a nonlinear change of variables which maps between the two bi-Poisson structures, giving a correspondence between the infinite hierarchies of conservation laws and commuting flows associated with the equations. In particular, under this change of variables this (modified) Hunter–Saxton equation can be viewed as a member of the KdV hierarchy.