Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics

Abstract

The paper provides a direct proof the uniquenessof solutions to the Camassa-Holm equation, based on characteristics.Given a conservative solution $u=u(t,x)$,an equation is introduced which singles out a unique characteristiccurve through each initial point.By studying the evolution of the quantities $u$ and $v= 2\arctan u_x$along each characteristic, it is proved that the Cauchy problem with generalinitial data$u_0\in H^1(\mathbb{R})$ has a unique solution, globally in time.