Local conserved-charge hierarchy of sine-Gordon theory (${\partial }_{\eta }{P}_j={\partial }_{\xi }{Q}_j$)

The Bäcklund expansion produces local differential-polynomial currents. In the coordinates $\xi =m(x+t)/2$, $\eta =m(x-t)/2$, their charges are $\int (P_j+Q_j)dx$ with vanishing boundary flux. Derivative improvements contribute no new charge. The first nontrivial higher current can be written $P=u_{\xi \xi }^2-u_{\xi }^4/4$, $Q=\cos u{u}_{\xi }^2$. Continuing, and exchanging light-cone directions, produces the infinite higher-spin hierarchy characterizing classical integrability.